MAYBE Problem: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Proof: DP Processor: DPs: a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> mark#(L) a__length#(cons(N,L)) -> a__length#(mark(L)) a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(zeros()) -> a__zeros#() mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) TDG Processor: DPs: a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> mark#(L) a__length#(cons(N,L)) -> a__length#(mark(L)) a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(zeros()) -> a__zeros#() mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) graph: a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X1) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X2) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(length(X)) -> a__length#(mark(X)) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(length(X)) -> mark#(X) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__take#(s(M),cons(N,IL)) -> mark#(N) -> mark#(zeros()) -> a__zeros#() a__length#(cons(N,L)) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> mark#(L) a__length#(cons(N,L)) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__length#(cons(N,L)) -> mark#(L) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__length#(cons(N,L)) -> mark#(L) -> mark#(take(X1,X2)) -> mark#(X1) a__length#(cons(N,L)) -> mark#(L) -> mark#(take(X1,X2)) -> mark#(X2) a__length#(cons(N,L)) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__length#(cons(N,L)) -> mark#(L) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__length#(cons(N,L)) -> mark#(L) -> mark#(and(X1,X2)) -> mark#(X1) a__length#(cons(N,L)) -> mark#(L) -> mark#(zeros()) -> a__zeros#() mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__length#(mark(L)) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> mark#(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#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) 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#(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#(zeros()) -> a__zeros#() 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#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) 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#(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#(zeros()) -> a__zeros#() mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> a__length#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> a__zeros#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> 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#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) 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#(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#(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#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) 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#(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#(zeros()) -> a__zeros#() 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#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) 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#(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#(zeros()) -> a__zeros#() SCC Processor: #sccs: 1 #rules: 13 #arcs: 96/196 DPs: a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> mark#(L) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = x1, [a__length#](x0) = x0, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [length](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + x1, [s](x0) = x0, [a__length](x0) = x0 + 1, [nil] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = IL + N >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) a__and#(tt(),X) = X >= X = mark#(X) mark#(length(X)) = X + 1 >= X = mark#(X) mark#(length(X)) = X + 1 >= X = a__length#(mark(X)) a__length#(cons(N,L)) = L + N >= L = mark#(L) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = L + N >= L = a__length#(mark(L)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__and(tt(),X) = X >= X = mark(X) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = s(a__length(mark(L))) a__take(0(),IL) = IL >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + M + N >= IL + M + N = cons(mark(N),take(M,IL)) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = X >= X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) problem: DPs: a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) a__length#(cons(N,L)) -> mark#(L) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = 0, [a__length#](x0) = 1, [mark#](x0) = 0, [a__and#](x0, x1) = 0, [length](x0) = 0, [and](x0, x1) = 0, [take](x0, x1) = 0, [a__take](x0, x1) = x1, [s](x0) = 0, [a__length](x0) = 0, [nil] = 0, [mark](x0) = 0, [a__and](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = 0 >= 0 = mark#(N) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) a__and#(tt(),X) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 1 >= 0 = mark#(L) mark#(take(X1,X2)) = 0 >= 0 = mark#(X2) mark#(take(X1,X2)) = 0 >= 0 = mark#(X1) mark#(take(X1,X2)) = 0 >= 0 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) mark#(s(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__length#(mark(L)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__and(tt(),X) = 0 >= 0 = mark(X) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = s(a__length(mark(L))) a__take(0(),IL) = IL >= 0 = nil() a__take(s(M),cons(N,IL)) = 0 >= 0 = cons(mark(N),take(M,IL)) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__length(X) = 0 >= 0 = length(X) a__take(X1,X2) = X2 >= 0 = take(X1,X2) problem: DPs: a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = x1, [a__length#](x0) = 1, [mark#](x0) = x0, [a__and#](x0, x1) = x1 + 1, [length](x0) = 0, [and](x0, x1) = x0 + x1 + 1, [take](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + x1, [s](x0) = x0, [a__length](x0) = 0, [nil] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1 + 1, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__take#(s(M),cons(N,IL)) = N >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__and#(mark(X1),X2) a__and#(tt(),X) = X + 1 >= X = mark#(X) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__length#(mark(L)) a__zeros() = 1 >= 0 = cons(0(),zeros()) a__and(tt(),X) = X + 1 >= X = mark(X) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = s(a__length(mark(L))) a__take(0(),IL) = IL >= 0 = nil() a__take(s(M),cons(N,IL)) = M + N >= N = cons(mark(N),take(M,IL)) mark(zeros()) = 1 >= 1 = a__zeros() mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = X >= X = s(mark(X)) a__zeros() = 1 >= 1 = zeros() a__and(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) a__length(X) = 0 >= 0 = length(X) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) problem: DPs: a__take#(s(M),cons(N,IL)) -> mark#(N) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = x1 + 1, [a__length#](x0) = 1, [mark#](x0) = x0 + 1, [a__and#](x0, x1) = x0 + 1, [length](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + x1, [s](x0) = x0, [a__length](x0) = 0, [nil] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 1, [cons](x0, x1) = x0 + 1, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__take#(s(M),cons(N,IL)) = N + 2 >= N + 1 = mark#(N) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = a__and#(mark(X1),X2) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = mark#(X1) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__length#(mark(L)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__and(tt(),X) = X + 1 >= X = mark(X) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = s(a__length(mark(L))) a__take(0(),IL) = IL >= 0 = nil() a__take(s(M),cons(N,IL)) = M + N + 1 >= N + 1 = cons(mark(N),take(M,IL)) mark(zeros()) = 1 >= 1 = a__zeros() mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 1 >= 1 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = X >= X = s(mark(X)) a__zeros() = 1 >= 1 = zeros() a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__length(X) = 0 >= 0 = length(X) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) problem: DPs: mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 1, [a__and#](x0, x1) = 1, [length](x0) = 0, [and](x0, x1) = 0, [take](x0, x1) = 0, [a__take](x0, x1) = 0, [s](x0) = 0, [a__length](x0) = 0, [nil] = 0, [mark](x0) = 0, [a__and](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(take(X1,X2)) = 1 >= 1 = mark#(X2) mark#(take(X1,X2)) = 1 >= 1 = mark#(X1) mark#(take(X1,X2)) = 1 >= 0 = a__take#(mark(X1),mark(X2)) mark#(s(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__length#(mark(L)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__and(tt(),X) = 0 >= 0 = mark(X) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = s(a__length(mark(L))) a__take(0(),IL) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 0 >= 0 = cons(mark(N),take(M,IL)) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__length(X) = 0 >= 0 = length(X) a__take(X1,X2) = 0 >= 0 = take(X1,X2) problem: DPs: mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = 0, [mark#](x0) = 1, [a__and#](x0, x1) = 0, [length](x0) = 0, [and](x0, x1) = 0, [take](x0, x1) = 0, [a__take](x0, x1) = 0, [s](x0) = 0, [a__length](x0) = 0, [nil] = 0, [mark](x0) = 0, [a__and](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: mark#(and(X1,X2)) = 1 >= 0 = a__and#(mark(X1),X2) mark#(take(X1,X2)) = 1 >= 1 = mark#(X2) mark#(take(X1,X2)) = 1 >= 1 = mark#(X1) mark#(s(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__length#(mark(L)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__and(tt(),X) = 0 >= 0 = mark(X) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = s(a__length(mark(L))) a__take(0(),IL) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 0 >= 0 = cons(mark(N),take(M,IL)) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__length(X) = 0 >= 0 = length(X) a__take(X1,X2) = 0 >= 0 = take(X1,X2) problem: DPs: mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = 0, [mark#](x0) = x0 + 1, [length](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = x0 + x1 + 1, [a__take](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [a__length](x0) = 0, [nil] = 0, [mark](x0) = x0, [a__and](x0, x1) = x1, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: mark#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__length#(mark(L)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__and(tt(),X) = X >= X = mark(X) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = s(a__length(mark(L))) a__take(0(),IL) = IL + 1 >= 0 = nil() a__take(s(M),cons(N,IL)) = M + 1 >= 0 = cons(mark(N),take(M,IL)) mark(zeros()) = 0 >= 0 = a__zeros() mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(nil()) = 0 >= 0 = nil() mark(s(X)) = X >= X = s(mark(X)) a__zeros() = 0 >= 0 = zeros() a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__length(X) = 0 >= 0 = length(X) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) problem: DPs: mark#(s(X)) -> mark#(X) a__length#(cons(N,L)) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__and(tt(),X) -> mark(X) a__length(nil()) -> 0() a__length(cons(N,L)) -> s(a__length(mark(L))) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> cons(mark(N),take(M,IL)) mark(zeros()) -> a__zeros() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) a__zeros() -> zeros() a__and(X1,X2) -> and(X1,X2) a__length(X) -> length(X) a__take(X1,X2) -> take(X1,X2) Open