MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) tail#(mark(X)) -> tail#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) tail#(ok(X)) -> tail#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) CDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) tail#(mark(X)) -> tail#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) tail#(ok(X)) -> tail#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X) top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X)) top#(ok(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X) top#(ok(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X) top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(tail(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X)) proper#(tail(X)) -> tail#(proper(X)) -> tail#(ok(X)) -> tail#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X2) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) -> proper#(tail(X)) -> tail#(proper(X)) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) tail#(ok(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X) tail#(mark(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(tail(X)) -> tail#(active(X)) -> tail#(mark(X)) -> tail#(X) active#(tail(X)) -> active#(X) -> active#(zeros()) -> cons#(0(),zeros()) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1) active#(tail(X)) -> active#(X) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X) active#(tail(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X)) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(mark(X1),X2) -> cons#(X1,X2) active#(cons(X1,X2)) -> cons#(active(X1),X2) -> cons#(ok(X1),ok(X2)) -> cons#(X1,X2) active#(cons(X1,X2)) -> active#(X1) -> active#(zeros()) -> cons#(0(),zeros()) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) -> active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) -> active#(tail(X)) -> tail#(active(X)) SCC Processor: #sccs: 5 #rules: 11 #arcs: 54/324 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(tail(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0 + 1, [mark](x0) = x0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: proper#(tail(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = X + XS + 3 >= XS = mark(XS) active(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(active(X1),X2) active(tail(X)) = X + 2 >= X + 2 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(tail(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = 0, [proper](x0) = x0, [tail](x0) = x0 + 1, [mark](x0) = 0, [cons](x0, x1) = x0 + 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: active#(tail(X)) = X + 2 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = X + 3 >= 0 = mark(XS) active(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(active(X1),X2) active(tail(X)) = X + 2 >= X + 2 = tail(active(X)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) tail(mark(X)) = 1 >= 0 = mark(tail(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) tail(ok(X)) = 1 >= 0 = ok(tail(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [tail](x0) = x0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = 0 >= 0 = mark(XS) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(tail(X)) = 0 >= 0 = tail(active(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(tail(X)) = 1 >= 1 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [tail](x0) = x0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: cons#(mark(X1),X2) = X1 + 1 >= X1 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(tail(X)) = 0 >= 0 = tail(proper(X)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) tail(ok(X)) = 0 >= 0 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [tail#](x0) = x0, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: tail#(mark(X)) = X + 1 >= X = tail#(X) tail#(ok(X)) = X >= X = tail#(X) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = X + XS + 1 >= XS + 1 = mark(XS) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(tail(X)) = X >= X = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) tail(ok(X)) = X >= X = ok(tail(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: tail#(ok(X)) -> tail#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [tail#](x0) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [tail](x0) = x0, [mark](x0) = 1, [cons](x0, x1) = x0, [0] = 1, [active](x0) = 1, [zeros] = 1 orientation: tail#(ok(X)) = X + 1 >= X = tail#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = 1 >= 1 = mark(XS) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) active(tail(X)) = 1 >= 1 = tail(active(X)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) tail(mark(X)) = 1 >= 1 = mark(tail(X)) proper(zeros()) = 2 >= 2 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed