TSTP Solution File: DAT095_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT095_1 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n004.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 14:36:43 EDT 2022
% Result : Theorem 0.64s 0.64s
% Output : Proof 0.67s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 68
% Syntax : Number of formulae : 168 ( 62 unt; 7 typ; 0 def)
% Number of atoms : 534 ( 375 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 578 ( 238 ~; 202 |; 30 &)
% ( 97 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of FOOLs : 33 ( 33 fml; 0 var)
% Number arithmetic : 1509 ( 34 atm; 344 fun; 923 num; 208 var)
% Number of types : 2 ( 0 usr; 1 ari)
% Number of type conns : 10 ( 6 >; 4 *; 0 +; 0 <<)
% Number of predicates : 12 ( 6 usr; 3 prp; 0-3 aty)
% Number of functors : 11 ( 6 usr; 4 con; 0-2 aty)
% Number of variables : 392 ( 351 !; 0 ?; 392 :)
% Comments :
%------------------------------------------------------------------------------
tff(count_type,type,
count: ( $int * list ) > $int ).
tff(cons_type,type,
cons: ( $int * list ) > list ).
tff(append_type,type,
append: ( list * list ) > list ).
tff(nil_type,type,
nil: list ).
tff(tail_type,type,
tail: list > list ).
tff(head_type,type,
head: list > $int ).
tff(in_type,type,
in: ( $int * list ) > $o ).
tff(1,plain,
^ [K: $int,L: list] :
refl(
( ( cons(K,L) != nil )
<=> ( cons(K,L) != nil ) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [K: $int,L: list] : ( cons(K,L) != nil )
<=> ! [K: $int,L: list] : ( cons(K,L) != nil ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [K: $int,L: list] : ( cons(K,L) != nil )
<=> ! [K: $int,L: list] : ( cons(K,L) != nil ) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [K: $int,L: list] : ( cons(K,L) != nil ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l4) ).
tff(5,plain,
! [K: $int,L: list] : ( cons(K,L) != nil ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [K: $int,L: list] : ( cons(K,L) != nil ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [K: $int,L: list] : ( cons(K,L) != nil ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [K: $int,L: list] : ( cons(K,L) != nil )
| ( cons(0,nil) != nil ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
cons(0,nil) != nil,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,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(11,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)],[10]) ).
tff(12,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(13,axiom,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l3) ).
tff(14,plain,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,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(18,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(19,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)],[18,17]) ).
tff(20,plain,
cons(0,nil) = cons(head(cons(0,nil)),tail(cons(0,nil))),
inference(unit_resolution,[status(thm)],[19,16,9]) ).
tff(21,plain,
cons(head(cons(0,nil)),tail(cons(0,nil))) = cons(0,nil),
inference(symmetry,[status(thm)],[20]) ).
tff(22,plain,
append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil) = append(cons(0,nil),nil),
inference(monotonicity,[status(thm)],[21]) ).
tff(23,plain,
append(cons(0,nil),nil) = append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil),
inference(symmetry,[status(thm)],[22]) ).
tff(24,plain,
^ [I: $int,K: list,L: list] :
refl(
( ( append(cons(I,K),L) = cons(I,append(K,L)) )
<=> ( append(cons(I,K),L) = cons(I,append(K,L)) ) )),
inference(bind,[status(th)],]) ).
tff(25,plain,
( ! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) )
<=> ! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,plain,
( ! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) )
<=> ! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(27,axiom,
! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l_7) ).
tff(28,plain,
! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) ),
inference(skolemize,[status(sab)],[28]) ).
tff(30,plain,
! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) ),
inference(modus_ponens,[status(thm)],[29,25]) ).
tff(31,plain,
( ~ ! [I: $int,K: list,L: list] : ( append(cons(I,K),L) = cons(I,append(K,L)) )
| ( append(cons(0,nil),nil) = cons(0,append(nil,nil)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(32,plain,
append(cons(0,nil),nil) = cons(0,append(nil,nil)),
inference(unit_resolution,[status(thm)],[31,30]) ).
tff(33,plain,
cons(0,append(nil,nil)) = append(cons(0,nil),nil),
inference(symmetry,[status(thm)],[32]) ).
tff(34,plain,
^ [L: list] :
refl(
( ( append(nil,L) = L )
<=> ( append(nil,L) = L ) )),
inference(bind,[status(th)],]) ).
tff(35,plain,
( ! [L: list] : ( append(nil,L) = L )
<=> ! [L: list] : ( append(nil,L) = L ) ),
inference(quant_intro,[status(thm)],[34]) ).
tff(36,plain,
( ! [L: list] : ( append(nil,L) = L )
<=> ! [L: list] : ( append(nil,L) = L ) ),
inference(rewrite,[status(thm)],]) ).
tff(37,axiom,
! [L: list] : ( append(nil,L) = L ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l_6) ).
tff(38,plain,
! [L: list] : ( append(nil,L) = L ),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,plain,
! [L: list] : ( append(nil,L) = L ),
inference(skolemize,[status(sab)],[38]) ).
tff(40,plain,
! [L: list] : ( append(nil,L) = L ),
inference(modus_ponens,[status(thm)],[39,35]) ).
tff(41,plain,
( ~ ! [L: list] : ( append(nil,L) = L )
| ( append(nil,nil) = nil ) ),
inference(quant_inst,[status(thm)],]) ).
tff(42,plain,
append(nil,nil) = nil,
inference(unit_resolution,[status(thm)],[41,40]) ).
tff(43,plain,
cons(0,append(nil,nil)) = cons(0,nil),
inference(monotonicity,[status(thm)],[42]) ).
tff(44,plain,
cons(0,nil) = cons(0,append(nil,nil)),
inference(symmetry,[status(thm)],[43]) ).
tff(45,plain,
cons(head(cons(0,nil)),tail(cons(0,nil))) = append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil),
inference(transitivity,[status(thm)],[21,44,33,23]) ).
tff(46,plain,
cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) = cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)),
inference(monotonicity,[status(thm)],[45]) ).
tff(47,plain,
count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))) = count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))),
inference(monotonicity,[status(thm)],[46]) ).
tff(48,plain,
( ( count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))) != count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))) )
| $lesseq($sum(count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(49,plain,
$lesseq($sum(count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))),0),
inference(unit_resolution,[status(thm)],[48,47]) ).
tff(50,plain,
cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) = cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),
inference(symmetry,[status(thm)],[46]) ).
tff(51,plain,
( in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
<=> in(0,cons(0,nil)) ),
inference(monotonicity,[status(thm)],[21]) ).
tff(52,plain,
( in(0,cons(0,nil))
<=> in(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ),
inference(symmetry,[status(thm)],[51]) ).
tff(53,plain,
^ [N: $int,L: list] :
refl(
( ( in(N,L)
<=> ~ $lesseq(count(N,L),0) )
<=> ( in(N,L)
<=> ~ $lesseq(count(N,L),0) ) )),
inference(bind,[status(th)],]) ).
tff(54,plain,
( ! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) )
<=> ! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ) ),
inference(quant_intro,[status(thm)],[53]) ).
tff(55,plain,
( ! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) )
<=> ! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ) ),
inference(rewrite,[status(thm)],]) ).
tff(56,plain,
^ [N: $int,L: list] :
rewrite(
( ( in(N,L)
<=> $greater(count(N,L),0) )
<=> ( in(N,L)
<=> ~ $lesseq(count(N,L),0) ) )),
inference(bind,[status(th)],]) ).
tff(57,plain,
( ! [N: $int,L: list] :
( in(N,L)
<=> $greater(count(N,L),0) )
<=> ! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ) ),
inference(quant_intro,[status(thm)],[56]) ).
tff(58,axiom,
! [N: $int,L: list] :
( in(N,L)
<=> $greater(count(N,L),0) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_8) ).
tff(59,plain,
! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ),
inference(skolemize,[status(sab)],[60]) ).
tff(62,plain,
! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) ),
inference(modus_ponens,[status(thm)],[61,54]) ).
tff(63,plain,
( ~ ! [N: $int,L: list] :
( in(N,L)
<=> ~ $lesseq(count(N,L),0) )
| ( in(0,cons(0,nil))
<=> ~ $lesseq(count(0,cons(0,nil)),0) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(64,plain,
( in(0,cons(0,nil))
<=> ~ $lesseq(count(0,cons(0,nil)),0) ),
inference(unit_resolution,[status(thm)],[63,62]) ).
tff(65,plain,
^ [K: $int,H: $int,T: list] :
refl(
( ( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
<=> ( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ) )),
inference(bind,[status(th)],]) ).
tff(66,plain,
( ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,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 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ) )),
inference(bind,[status(th)],]) ).
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 ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,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(70,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)],[69]) ).
tff(71,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(72,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(73,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(74,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)],[73]) ).
tff(75,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)],[74,72]) ).
tff(76,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(77,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[76,75]) ).
tff(78,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[77,71]) ).
tff(79,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)],[78,70]) ).
tff(80,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[79,68]) ).
tff(81,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ),
inference(skolemize,[status(sab)],[80]) ).
tff(82,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[81,66]) ).
tff(83,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,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 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(84,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(85,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(86,plain,
( ( 0 = 0 )
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(87,plain,
$sum(0,0) = 0,
inference(rewrite,[status(thm)],]) ).
tff(88,plain,
$product(-1,0) = 0,
inference(rewrite,[status(thm)],]) ).
tff(89,plain,
$sum(0,$product(-1,0)) = $sum(0,0),
inference(monotonicity,[status(thm)],[88]) ).
tff(90,plain,
$sum(0,$product(-1,0)) = 0,
inference(transitivity,[status(thm)],[89,87]) ).
tff(91,plain,
( ( $sum(0,$product(-1,0)) = 0 )
<=> ( 0 = 0 ) ),
inference(monotonicity,[status(thm)],[90]) ).
tff(92,plain,
( ( $sum(0,$product(-1,0)) = 0 )
<=> $true ),
inference(transitivity,[status(thm)],[91,86]) ).
tff(93,plain,
( ( $sum(0,$product(-1,0)) != 0 )
<=> ~ $true ),
inference(monotonicity,[status(thm)],[92]) ).
tff(94,plain,
( ( $sum(0,$product(-1,0)) != 0 )
<=> $false ),
inference(transitivity,[status(thm)],[93,85]) ).
tff(95,plain,
( ( ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) )
<=> ( $false
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[94]) ).
tff(96,plain,
( ( ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[95,84]) ).
tff(97,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[96]) ).
tff(98,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(transitivity,[status(thm)],[97,83]) ).
tff(99,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(100,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[99,98]) ).
tff(101,plain,
$sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1,
inference(unit_resolution,[status(thm)],[100,82]) ).
tff(102,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(103,plain,
$lesseq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1),
inference(unit_resolution,[status(thm)],[102,101]) ).
tff(104,plain,
^ [K: $int] :
refl(
( ( count(K,nil) = 0 )
<=> ( count(K,nil) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(105,plain,
( ! [K: $int] : ( count(K,nil) = 0 )
<=> ! [K: $int] : ( count(K,nil) = 0 ) ),
inference(quant_intro,[status(thm)],[104]) ).
tff(106,plain,
( ! [K: $int] : ( count(K,nil) = 0 )
<=> ! [K: $int] : ( count(K,nil) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(107,axiom,
! [K: $int] : ( count(K,nil) = 0 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a) ).
tff(108,plain,
! [K: $int] : ( count(K,nil) = 0 ),
inference(modus_ponens,[status(thm)],[107,106]) ).
tff(109,plain,
! [K: $int] : ( count(K,nil) = 0 ),
inference(skolemize,[status(sab)],[108]) ).
tff(110,plain,
! [K: $int] : ( count(K,nil) = 0 ),
inference(modus_ponens,[status(thm)],[109,105]) ).
tff(111,plain,
( ~ ! [K: $int] : ( count(K,nil) = 0 )
| ( count(0,nil) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(112,plain,
count(0,nil) = 0,
inference(unit_resolution,[status(thm)],[111,110]) ).
tff(113,plain,
( ( count(0,nil) != 0 )
| $greatereq(count(0,nil),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(114,plain,
$greatereq(count(0,nil),0),
inference(unit_resolution,[status(thm)],[113,112]) ).
tff(115,plain,
( ~ $lesseq(count(0,cons(0,nil)),0)
| ~ $greatereq(count(0,nil),0)
| ~ $lesseq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(116,plain,
~ $lesseq(count(0,cons(0,nil)),0),
inference(unit_resolution,[status(thm)],[115,114,103]) ).
tff(117,plain,
( ~ ( in(0,cons(0,nil))
<=> ~ $lesseq(count(0,cons(0,nil)),0) )
| in(0,cons(0,nil))
| $lesseq(count(0,cons(0,nil)),0) ),
inference(tautology,[status(thm)],]) ).
tff(118,plain,
in(0,cons(0,nil)),
inference(unit_resolution,[status(thm)],[117,116,64]) ).
tff(119,plain,
in(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),
inference(modus_ponens,[status(thm)],[118,52]) ).
tff(120,plain,
^ [M: $int,N: $int,L: list,L1: list] :
refl(
( ( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
<=> ( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
inference(bind,[status(th)],]) ).
tff(121,plain,
( ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ) ),
inference(quant_intro,[status(thm)],[120]) ).
tff(122,plain,
^ [M: $int,N: $int,L: list,L1: list] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( in(N,L)
& ( L1 = cons(M,L) ) )
<=> ~ ( ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
<=> ~ ~ ( ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
rewrite(
( ~ ~ ( ~ in(N,L)
| ( L1 != cons(M,L) ) )
<=> ( ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
<=> ( ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
( ( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) )
<=> ( ~ in(N,L)
| ( L1 != cons(M,L) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) ) )),
rewrite(
( ( ~ in(N,L)
| ( L1 != cons(M,L) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) )
<=> ( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
( ( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) )
<=> ( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ) )),
inference(bind,[status(th)],]) ).
tff(123,plain,
( ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ) ),
inference(quant_intro,[status(thm)],[122]) ).
tff(124,plain,
^ [M: $int,N: $int,L: list,L1: list] :
rewrite(
( ( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L1),$product(-1,count(N,L))) = 0 ) )
<=> ( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(125,plain,
( ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L1),$product(-1,count(N,L))) = 0 ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[124]) ).
tff(126,plain,
^ [M: $int,N: $int,L: list,L1: list] :
rewrite(
( ( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) )
<=> ( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L1),$product(-1,count(N,L))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(127,plain,
( ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L1),$product(-1,count(N,L))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[126]) ).
tff(128,plain,
( ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(129,plain,
( ~ ~ ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(130,plain,
( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( in(N,L)
& ( L1 = cons(M,L) ) )
=> ( count(N,L1) = count(N,L) ) )
<=> ~ ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(131,plain,
( ~ ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( in(N,L)
& ( L1 = cons(M,L) ) )
=> ( count(N,L1) = count(N,L) ) )
<=> ~ ~ ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ) ),
inference(monotonicity,[status(thm)],[130]) ).
tff(132,plain,
( ~ ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( in(N,L)
& ( L1 = cons(M,L) ) )
=> ( count(N,L1) = count(N,L) ) )
<=> ! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ) ),
inference(transitivity,[status(thm)],[131,129]) ).
tff(133,axiom,
~ ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( in(N,L)
& ( L1 = cons(M,L) ) )
=> ( count(N,L1) = count(N,L) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',c) ).
tff(134,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ),
inference(modus_ponens,[status(thm)],[133,132]) ).
tff(135,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( count(N,L1) = count(N,L) ) ),
inference(modus_ponens,[status(thm)],[134,128]) ).
tff(136,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L1),$product(-1,count(N,L))) = 0 ) ),
inference(modus_ponens,[status(thm)],[135,127]) ).
tff(137,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) ),
inference(modus_ponens,[status(thm)],[136,125]) ).
tff(138,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ~ ( in(N,L)
& ( L1 = cons(M,L) ) )
| ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 ) ),
inference(skolemize,[status(sab)],[137]) ).
tff(139,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ),
inference(modus_ponens,[status(thm)],[138,123]) ).
tff(140,plain,
! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) ),
inference(modus_ponens,[status(thm)],[139,121]) ).
tff(141,plain,
( ( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) )
<=> ( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(142,plain,
( ( ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) )
<=> ( ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(143,plain,
( ( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) )
<=> ( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) ) ),
inference(monotonicity,[status(thm)],[142]) ).
tff(144,plain,
( ( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) )
<=> ( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) ) ),
inference(transitivity,[status(thm)],[143,141]) ).
tff(145,plain,
( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(146,plain,
( ~ ! [M: $int,N: $int,L: list,L1: list] :
( ( $sum(count(N,L),$product(-1,count(N,L1))) = 0 )
| ~ in(N,L)
| ( L1 != cons(M,L) ) )
| ~ in(0,cons(head(cons(0,nil)),tail(cons(0,nil))))
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0 )
| ( cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil)) != cons(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) ) ),
inference(modus_ponens,[status(thm)],[145,144]) ).
tff(147,plain,
$sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) = 0,
inference(unit_resolution,[status(thm)],[146,140,119,50]) ).
tff(148,plain,
( ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))) != 0 )
| $greatereq($sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(149,plain,
$greatereq($sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,append(cons(head(cons(0,nil)),tail(cons(0,nil))),nil))))),0),
inference(unit_resolution,[status(thm)],[148,147]) ).
tff(150,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(151,plain,
( ( $false
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) )
<=> ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(152,plain,
( ( ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) )
<=> ( $false
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[94]) ).
tff(153,plain,
( ( ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) )
<=> ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ),
inference(transitivity,[status(thm)],[152,151]) ).
tff(154,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[153]) ).
tff(155,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ) ),
inference(transitivity,[status(thm)],[154,150]) ).
tff(156,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(157,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,T),$product(-1,count(K,cons(H,T)))) = -1 ) )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1 ) ),
inference(modus_ponens,[status(thm)],[156,155]) ).
tff(158,plain,
$sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) = -1,
inference(unit_resolution,[status(thm)],[157,82]) ).
tff(159,plain,
( ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))) != -1 )
| $lesseq($sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(160,plain,
$lesseq($sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,cons(0,cons(head(cons(0,nil)),tail(cons(0,nil))))))),-1),
inference(unit_resolution,[status(thm)],[159,158]) ).
tff(161,plain,
$false,
inference(theory_lemma,[status(thm)],[160,149,49]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : DAT095_1 : TPTP v8.1.0. Released v6.1.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n004.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 31 01:47:19 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.64/0.64 % SZS status Theorem
% 0.64/0.64 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------