YES Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Proof: DP Processor: DPs: a__U11#(tt(),L) -> mark#(L) a__U11#(tt(),L) -> a__length#(mark(L)) a__and#(tt(),X) -> mark#(X) a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatIList#(V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2)) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N)) a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) TDG Processor: DPs: a__U11#(tt(),L) -> mark#(L) a__U11#(tt(),L) -> a__length#(mark(L)) a__and#(tt(),X) -> mark#(X) a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatIList#(V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2)) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N)) a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) graph: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNatIList#(V) -> a__isNatList#(V) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) a__isNatIList#(V) -> a__isNatList#(V) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) -> a__and#(tt(),X) -> mark#(X) a__isNat#(length(V1)) -> a__isNatList#(V1) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) a__isNat#(length(V1)) -> a__isNatList#(V1) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) a__and#(tt(),X) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(length(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(zeros()) -> a__zeros#() a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N)) -> a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) -> a__U11#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) -> a__U11#(tt(),L) -> mark#(L) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) -> a__U11#(tt(),L) -> a__length#(mark(L)) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) -> a__U11#(tt(),L) -> mark#(L) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNat#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(length(V1)) -> a__isNatList#(V1) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2)) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(V) -> a__isNatList#(V) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N)) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__isNatList#(L) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(length(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() a__U11#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) a__U11#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N)) a__U11#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__isNatList#(L) a__U11#(tt(),L) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__U11#(tt(),L) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__U11#(tt(),L) -> mark#(L) -> mark#(isNatIList(X)) -> a__isNatIList#(X) a__U11#(tt(),L) -> mark#(L) -> mark#(isNatList(X)) -> a__isNatList#(X) a__U11#(tt(),L) -> mark#(L) -> mark#(isNat(X)) -> a__isNat#(X) a__U11#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__U11#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> mark#(X1) a__U11#(tt(),L) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__U11#(tt(),L) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__U11#(tt(),L) -> mark#(L) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),L) -> mark#(L) -> mark#(U11(X1,X2)) -> mark#(X1) a__U11#(tt(),L) -> mark#(L) -> mark#(zeros()) -> a__zeros#() SCC Processor: #sccs: 1 #rules: 24 #arcs: 117/625 DPs: a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatList(V2)) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNat(N)) a__length#(cons(N,L)) -> a__U11#(a__and(a__isNatList(L),isNat(N)),L) a__U11#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNat(V1),isNatIList(V2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Matrix Interpretation Processor: dim=1 interpretation: [a__isNatIList#](x0) = 2x0 + 7, [a__isNatList#](x0) = x0 + 4, [a__isNat#](x0) = x0 + 5, [a__and#](x0, x1) = x0 + 2x1 + 3, [a__length#](x0) = 2x0, [mark#](x0) = 2x0 + 6, [a__U11#](x0, x1) = x0 + 2x1 + 5, [and](x0, x1) = x0 + x1 + 1, [U11](x0, x1) = x0 + 3x1 + 4, [isNat](x0) = x0, [isNatList](x0) = x0, [nil] = 4, [isNatIList](x0) = x0 + 3, [a__isNatIList](x0) = x0 + 4, [a__isNatList](x0) = x0 + 1, [length](x0) = x0 + 3, [a__isNat](x0) = x0 + 4, [a__and](x0, x1) = x0 + x1 + 1, [s](x0) = x0 + 1, [a__length](x0) = x0 + 3, [mark](x0) = x0 + 4, [a__U11](x0, x1) = x0 + 3x1 + 4, [tt] = 4, [cons](x0, x1) = x0 + 4x1 + 4, [zeros] = 0, [0] = 0, [a__zeros] = 4 orientation: a__isNatIList#(cons(V1,V2)) = 2V1 + 8V2 + 15 >= V1 + 5 = a__isNat#(V1) a__isNat#(length(V1)) = V1 + 8 >= V1 + 4 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = V1 + 4V2 + 8 >= V1 + 5 = a__isNat#(V1) a__isNat#(s(V1)) = V1 + 6 >= V1 + 5 = a__isNat#(V1) a__isNatList#(cons(V1,V2)) = V1 + 4V2 + 8 >= V1 + 2V2 + 7 = a__and#(a__isNat(V1),isNatList(V2)) a__and#(tt(),X) = 2X + 7 >= 2X + 6 = mark#(X) mark#(U11(X1,X2)) = 2X1 + 6X2 + 14 >= 2X1 + 6 = mark#(X1) mark#(U11(X1,X2)) = 2X1 + 6X2 + 14 >= X1 + 2X2 + 9 = a__U11#(mark(X1),X2) a__U11#(tt(),L) = 2L + 9 >= 2L + 6 = mark#(L) mark#(length(X)) = 2X + 12 >= 2X + 6 = mark#(X) mark#(length(X)) = 2X + 12 >= 2X + 8 = a__length#(mark(X)) a__length#(cons(N,L)) = 8L + 2N + 8 >= L + 4 = a__isNatList#(L) a__length#(cons(N,L)) = 8L + 2N + 8 >= L + 2N + 4 = a__and#(a__isNatList(L),isNat(N)) a__length#(cons(N,L)) = 8L + 2N + 8 >= 3L + N + 7 = a__U11#(a__and(a__isNatList(L),isNat(N)),L) a__U11#(tt(),L) = 2L + 9 >= 2L + 8 = a__length#(mark(L)) mark#(and(X1,X2)) = 2X1 + 2X2 + 8 >= 2X1 + 6 = mark#(X1) mark#(and(X1,X2)) = 2X1 + 2X2 + 8 >= X1 + 2X2 + 7 = a__and#(mark(X1),X2) mark#(isNat(X)) = 2X + 6 >= X + 5 = a__isNat#(X) mark#(isNatList(X)) = 2X + 6 >= X + 4 = a__isNatList#(X) mark#(isNatIList(X)) = 2X + 12 >= 2X + 7 = a__isNatIList#(X) a__isNatIList#(V) = 2V + 7 >= V + 4 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 2V1 + 8V2 + 15 >= V1 + 2V2 + 13 = a__and#(a__isNat(V1),isNatIList(V2)) mark#(cons(X1,X2)) = 2X1 + 8X2 + 14 >= 2X1 + 6 = mark#(X1) mark#(s(X)) = 2X + 8 >= 2X + 6 = mark#(X) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__U11(tt(),L) = 3L + 8 >= L + 8 = s(a__length(mark(L))) a__and(tt(),X) = X + 5 >= X + 4 = mark(X) a__isNat(0()) = 4 >= 4 = tt() a__isNat(length(V1)) = V1 + 7 >= V1 + 1 = a__isNatList(V1) a__isNat(s(V1)) = V1 + 5 >= V1 + 4 = a__isNat(V1) a__isNatIList(V) = V + 4 >= V + 1 = a__isNatList(V) a__isNatIList(zeros()) = 4 >= 4 = tt() a__isNatIList(cons(V1,V2)) = V1 + 4V2 + 8 >= V1 + V2 + 8 = a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) = 5 >= 4 = tt() a__isNatList(cons(V1,V2)) = V1 + 4V2 + 5 >= V1 + V2 + 5 = a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) = 7 >= 0 = 0() a__length(cons(N,L)) = 4L + N + 7 >= 4L + N + 6 = a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) = 4 >= 4 = a__zeros() mark(U11(X1,X2)) = X1 + 3X2 + 8 >= X1 + 3X2 + 8 = a__U11(mark(X1),X2) mark(length(X)) = X + 7 >= X + 7 = a__length(mark(X)) mark(and(X1,X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = a__and(mark(X1),X2) mark(isNat(X)) = X + 4 >= X + 4 = a__isNat(X) mark(isNatList(X)) = X + 4 >= X + 1 = a__isNatList(X) mark(isNatIList(X)) = X + 7 >= X + 4 = a__isNatIList(X) mark(cons(X1,X2)) = X1 + 4X2 + 8 >= X1 + 4X2 + 8 = cons(mark(X1),X2) mark(0()) = 4 >= 0 = 0() mark(tt()) = 8 >= 4 = tt() mark(s(X)) = X + 5 >= X + 5 = s(mark(X)) mark(nil()) = 8 >= 4 = nil() a__zeros() = 4 >= 0 = zeros() a__U11(X1,X2) = X1 + 3X2 + 4 >= X1 + 3X2 + 4 = U11(X1,X2) a__length(X) = X + 3 >= X + 3 = length(X) a__and(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) a__isNat(X) = X + 4 >= X = isNat(X) a__isNatList(X) = X + 1 >= X = isNatList(X) a__isNatIList(X) = X + 4 >= X + 3 = isNatIList(X) problem: DPs: TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> s(a__length(mark(L))) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatIList(V2)) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__and(a__isNat(V1),isNatList(V2)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(a__and(a__isNatList(L),isNat(N)),L) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__length(X) -> length(X) a__and(X1,X2) -> and(X1,X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) Qed