TSTP Solution File: DAT087_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT087_1 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n022.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 0.18s 0.50s
% Output : Proof 0.44s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 29
% Syntax : Number of formulae : 67 ( 22 unt; 3 typ; 0 def)
% Number of atoms : 174 ( 150 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 168 ( 70 ~; 53 |; 0 &)
% ( 38 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of FOOLs : 12 ( 12 fml; 0 var)
% Number arithmetic : 693 ( 6 atm; 220 fun; 372 num; 95 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 4 ( 2 >; 2 *; 0 +; 0 <<)
% Number of predicates : 10 ( 5 usr; 3 prp; 0-3 aty)
% Number of functors : 8 ( 3 usr; 4 con; 0-2 aty)
% Number of variables : 151 ( 134 !; 0 ?; 151 :)
% Comments :
%------------------------------------------------------------------------------
tff(count_type,type,
count: ( $int * list ) > $int ).
tff(cons_type,type,
cons: ( $int * list ) > list ).
tff(nil_type,type,
nil: list ).
tff(1,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(2,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)],[1]) ).
tff(3,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(4,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)],[3]) ).
tff(5,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(6,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)],[5]) ).
tff(7,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(8,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(9,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(10,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)],[9]) ).
tff(11,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)],[10,8]) ).
tff(12,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(13,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[13,7]) ).
tff(15,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)],[14,6]) ).
tff(16,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)],[15,4]) ).
tff(17,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)],[16]) ).
tff(18,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)],[17,2]) ).
tff(19,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 ) )
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,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 ) )
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(20,plain,
( ( $false
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) )
<=> ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(21,plain,
( ( $sum($product(-1,count(1,nil)),count(1,cons(1,nil))) = 1 )
<=> ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(22,plain,
$sum(count(1,cons(1,nil)),$product(-1,count(1,nil))) = $sum($product(-1,count(1,nil)),count(1,cons(1,nil))),
inference(rewrite,[status(thm)],]) ).
tff(23,plain,
( ( $sum(count(1,cons(1,nil)),$product(-1,count(1,nil))) = 1 )
<=> ( $sum($product(-1,count(1,nil)),count(1,cons(1,nil))) = 1 ) ),
inference(monotonicity,[status(thm)],[22]) ).
tff(24,plain,
( ( $sum(count(1,cons(1,nil)),$product(-1,count(1,nil))) = 1 )
<=> ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[23,21]) ).
tff(25,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(26,plain,
( ( 0 = 0 )
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(27,plain,
$sum(1,-1) = 0,
inference(rewrite,[status(thm)],]) ).
tff(28,plain,
$product(-1,1) = -1,
inference(rewrite,[status(thm)],]) ).
tff(29,plain,
$sum(1,$product(-1,1)) = $sum(1,-1),
inference(monotonicity,[status(thm)],[28]) ).
tff(30,plain,
$sum(1,$product(-1,1)) = 0,
inference(transitivity,[status(thm)],[29,27]) ).
tff(31,plain,
( ( $sum(1,$product(-1,1)) = 0 )
<=> ( 0 = 0 ) ),
inference(monotonicity,[status(thm)],[30]) ).
tff(32,plain,
( ( $sum(1,$product(-1,1)) = 0 )
<=> $true ),
inference(transitivity,[status(thm)],[31,26]) ).
tff(33,plain,
( ( $sum(1,$product(-1,1)) != 0 )
<=> ~ $true ),
inference(monotonicity,[status(thm)],[32]) ).
tff(34,plain,
( ( $sum(1,$product(-1,1)) != 0 )
<=> $false ),
inference(transitivity,[status(thm)],[33,25]) ).
tff(35,plain,
( ( ( $sum(1,$product(-1,1)) != 0 )
| ( $sum(count(1,cons(1,nil)),$product(-1,count(1,nil))) = 1 ) )
<=> ( $false
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[34,24]) ).
tff(36,plain,
( ( ( $sum(1,$product(-1,1)) != 0 )
| ( $sum(count(1,cons(1,nil)),$product(-1,count(1,nil))) = 1 ) )
<=> ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[35,20]) ).
tff(37,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 ) )
| ( $sum(1,$product(-1,1)) != 0 )
| ( $sum(count(1,cons(1,nil)),$product(-1,count(1,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 ) )
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[36]) ).
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 ) )
| ( $sum(1,$product(-1,1)) != 0 )
| ( $sum(count(1,cons(1,nil)),$product(-1,count(1,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 ) )
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ) ),
inference(transitivity,[status(thm)],[37,19]) ).
tff(39,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 ) )
| ( $sum(1,$product(-1,1)) != 0 )
| ( $sum(count(1,cons(1,nil)),$product(-1,count(1,nil))) = 1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(40,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 ) )
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[39,38]) ).
tff(41,plain,
$sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = -1,
inference(unit_resolution,[status(thm)],[40,18]) ).
tff(42,plain,
( ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) != -1 )
| $lesseq($sum(count(1,nil),$product(-1,count(1,cons(1,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(43,plain,
$lesseq($sum(count(1,nil),$product(-1,count(1,cons(1,nil)))),-1),
inference(unit_resolution,[status(thm)],[42,41]) ).
tff(44,plain,
( ~ $greatereq($sum(count(1,nil),$product(-1,count(1,cons(1,nil)))),0)
| ~ $lesseq($sum(count(1,nil),$product(-1,count(1,cons(1,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(45,plain,
~ $greatereq($sum(count(1,nil),$product(-1,count(1,cons(1,nil)))),0),
inference(unit_resolution,[status(thm)],[44,43]) ).
tff(46,plain,
( ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) != 0 )
| $greatereq($sum(count(1,nil),$product(-1,count(1,cons(1,nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(47,plain,
$sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) != 0,
inference(unit_resolution,[status(thm)],[46,45]) ).
tff(48,plain,
^ [N: $int,L: list] :
refl(
( ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 )
<=> ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(49,plain,
( ! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 )
<=> ! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ) ),
inference(quant_intro,[status(thm)],[48]) ).
tff(50,plain,
^ [N: $int,L: list] :
rewrite(
( ( count(N,L) = count(N,cons(1,L)) )
<=> ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(51,plain,
( ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) )
<=> ! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ) ),
inference(quant_intro,[status(thm)],[50]) ).
tff(52,plain,
( ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) )
<=> ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(53,plain,
( ~ ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) )
<=> ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(54,plain,
( ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) )
<=> ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(55,plain,
( ~ ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) )
<=> ~ ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ) ),
inference(monotonicity,[status(thm)],[54]) ).
tff(56,plain,
( ~ ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) )
<=> ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ) ),
inference(transitivity,[status(thm)],[55,53]) ).
tff(57,axiom,
~ ~ ! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',c) ).
tff(58,plain,
! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [N: $int,L: list] : ( count(N,L) = count(N,cons(1,L)) ),
inference(modus_ponens,[status(thm)],[58,52]) ).
tff(60,plain,
! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ),
inference(modus_ponens,[status(thm)],[59,51]) ).
tff(61,plain,
! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ),
inference(skolemize,[status(sab)],[60]) ).
tff(62,plain,
! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 ),
inference(modus_ponens,[status(thm)],[61,49]) ).
tff(63,plain,
( ~ ! [N: $int,L: list] : ( $sum(count(N,L),$product(-1,count(N,cons(1,L)))) = 0 )
| ( $sum(count(1,nil),$product(-1,count(1,cons(1,nil)))) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(64,plain,
$false,
inference(unit_resolution,[status(thm)],[63,62,47]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : DAT087_1 : TPTP v8.1.0. Released v6.1.0.
% 0.06/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.32 % Computer : n022.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Wed Aug 31 01:51:26 EDT 2022
% 0.12/0.32 % CPUTime :
% 0.12/0.32 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.32 Usage: tptp [options] [-file:]file
% 0.12/0.32 -h, -? prints this message.
% 0.12/0.32 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.32 -m, -model generate model.
% 0.12/0.32 -p, -proof generate proof.
% 0.12/0.32 -c, -core generate unsat core of named formulas.
% 0.12/0.32 -st, -statistics display statistics.
% 0.12/0.32 -t:timeout set timeout (in second).
% 0.12/0.32 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.32 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.32 -<param>:<value> configuration parameter and value.
% 0.12/0.32 -o:<output-file> file to place output in.
% 0.18/0.50 % SZS status Theorem
% 0.18/0.50 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------