YES Problem: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Proof: DP Processor: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) TDG Processor: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) graph: a__after#(0(),XS) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__after#(0(),XS) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__after#(0(),XS) -> mark#(XS) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__after#(0(),XS) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X1) a__after#(0(),XS) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X2) a__after#(0(),XS) -> mark#(XS) -> mark#(from(X)) -> a__from#(mark(X)) a__after#(0(),XS) -> mark#(XS) -> mark#(from(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X2) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> a__from#(mark(X)) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(after(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(after(X1,X2)) -> mark#(X2) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(from(X)) -> a__from#(mark(X)) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(from(X)) -> mark#(X) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(s(N),cons(X,XS)) -> mark#(N) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(s(N),cons(X,XS)) -> mark#(XS) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(0(),XS) -> mark#(XS) mark#(after(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(after(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X2) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(after(X1,X2)) -> mark#(X2) -> mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X2) -> mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(after(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(after(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(after(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) 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#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) 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#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__after#](x0, x1) = x0 + x1 + 4, [mark#](x0) = x0, [a__from#](x0) = x0 + 3, [after](x0, x1) = 3x0 + x1 + 5, [a__after](x0, x1) = 3x0 + x1 + 5, [0] = 0, [cons](x0, x1) = x0 + x1, [from](x0) = x0 + 3, [s](x0) = x0 + 3, [mark](x0) = x0, [a__from](x0) = x0 + 3 orientation: a__from#(X) = X + 3 >= X = mark#(X) a__after#(0(),XS) = XS + 4 >= XS = mark#(XS) a__after#(s(N),cons(X,XS)) = N + X + XS + 4 >= XS = mark#(XS) a__after#(s(N),cons(X,XS)) = N + X + XS + 4 >= N = mark#(N) a__after#(s(N),cons(X,XS)) = N + X + XS + 4 >= N + XS + 4 = a__after#(mark(N),mark(XS)) mark#(from(X)) = X + 3 >= X = mark#(X) mark#(from(X)) = X + 3 >= X + 3 = a__from#(mark(X)) mark#(after(X1,X2)) = 3X1 + X2 + 5 >= X2 = mark#(X2) mark#(after(X1,X2)) = 3X1 + X2 + 5 >= X1 = mark#(X1) mark#(after(X1,X2)) = 3X1 + X2 + 5 >= X1 + X2 + 4 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X + 3 >= X = mark#(X) a__from(X) = X + 3 >= X + 3 = cons(mark(X),from(s(X))) a__after(0(),XS) = XS + 5 >= XS = mark(XS) a__after(s(N),cons(X,XS)) = 3N + X + XS + 6 >= 3N + XS + 5 = a__after(mark(N),mark(XS)) mark(from(X)) = X + 3 >= X + 3 = a__from(mark(X)) mark(after(X1,X2)) = 3X1 + X2 + 5 >= 3X1 + X2 + 5 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X + 3 >= X + 3 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = X + 3 >= X + 3 = from(X) a__after(X1,X2) = 3X1 + X2 + 5 >= 3X1 + X2 + 5 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__after#](x0, x1) = 3x0 + 4x1 + 0, [mark#](x0) = 3x0, [a__from#](x0) = 3x0 + 0, [after](x0, x1) = x0 + 1x1 + 1, [a__after](x0, x1) = x0 + 1x1 + 1, [0] = 0, [cons](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 1, [s](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__from](x0) = x0 + 1 orientation: a__from#(X) = 3X + 0 >= 3X = mark#(X) a__after#(0(),XS) = 4XS + 3 >= 3XS = mark#(XS) a__after#(s(N),cons(X,XS)) = 3N + 4X + 4XS + 4 >= 3XS = mark#(XS) a__after#(s(N),cons(X,XS)) = 3N + 4X + 4XS + 4 >= 3N = mark#(N) a__after#(s(N),cons(X,XS)) = 3N + 4X + 4XS + 4 >= 3N + 4XS + 4 = a__after#(mark(N),mark(XS)) mark#(from(X)) = 3X + 4 >= 3X = mark#(X) mark#(from(X)) = 3X + 4 >= 3X + 3 = a__from#(mark(X)) mark#(after(X1,X2)) = 3X1 + 4X2 + 4 >= 3X2 = mark#(X2) mark#(after(X1,X2)) = 3X1 + 4X2 + 4 >= 3X1 + 4X2 + 4 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 3X1 + 3X2 + 3 >= 3X1 = mark#(X1) mark#(s(X)) = 3X + 3 >= 3X = mark#(X) a__from(X) = X + 1 >= X + 1 = cons(mark(X),from(s(X))) a__after(0(),XS) = 1XS + 1 >= XS + 0 = mark(XS) a__after(s(N),cons(X,XS)) = N + 1X + 1XS + 1 >= N + 1XS + 1 = a__after(mark(N),mark(XS)) mark(from(X)) = X + 1 >= X + 1 = a__from(mark(X)) mark(after(X1,X2)) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = X + 1 >= X + 1 = from(X) a__after(X1,X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__after#](x0, x1) = 4x0 + 4x1 + 0, [mark#](x0) = x0 + 0, [a__from#](x0) = x0 + 0, [after](x0, x1) = 4x0 + 4x1 + 4, [a__after](x0, x1) = 4x0 + 4x1 + 4, [0] = 1, [cons](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 0, [s](x0) = x0, [mark](x0) = x0 + 0, [a__from](x0) = x0 + 0 orientation: a__from#(X) = X + 0 >= X + 0 = mark#(X) a__after#(s(N),cons(X,XS)) = 4N + 4X + 4XS + 4 >= N + 0 = mark#(N) a__after#(s(N),cons(X,XS)) = 4N + 4X + 4XS + 4 >= 4N + 4XS + 4 = a__after#(mark(N),mark(XS)) mark#(from(X)) = X + 0 >= X + 0 = mark#(X) mark#(from(X)) = X + 0 >= X + 0 = a__from#(mark(X)) mark#(after(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__after(0(),XS) = 4XS + 5 >= XS + 0 = mark(XS) a__after(s(N),cons(X,XS)) = 4N + 4X + 4XS + 4 >= 4N + 4XS + 4 = a__after(mark(N),mark(XS)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(after(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__from(X) = X + 0 >= X + 0 = from(X) a__after(X1,X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) SCC Processor: #sccs: 2 #rules: 6 #arcs: 72/49 DPs: mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [a__from#](x0) = 2x0 + 0, [after](x0, x1) = x0 + x1, [a__after](x0, x1) = x0 + x1 + 0, [0] = 0, [cons](x0, x1) = x0 + x1 + 0, [from](x0) = 2x0 + 3, [s](x0) = x0, [mark](x0) = x0 + 0, [a__from](x0) = 2x0 + 3 orientation: mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(from(X)) = 2X + 3 >= X = mark#(X) mark#(from(X)) = 2X + 3 >= 2X + 2 = a__from#(mark(X)) a__from#(X) = 2X + 0 >= X = mark#(X) mark#(s(X)) = X >= X = mark#(X) a__from(X) = 2X + 3 >= 2X + 3 = cons(mark(X),from(s(X))) a__after(0(),XS) = XS + 0 >= XS + 0 = mark(XS) a__after(s(N),cons(X,XS)) = N + X + XS + 0 >= N + XS + 0 = a__after(mark(N),mark(XS)) mark(from(X)) = 2X + 3 >= 2X + 3 = a__from(mark(X)) mark(after(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = 2X + 3 >= 2X + 3 = from(X) a__after(X1,X2) = X1 + X2 + 0 >= X1 + X2 = after(X1,X2) problem: DPs: mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) SCC Processor: #sccs: 1 #rules: 2 #arcs: 17/9 DPs: mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Qed DPs: a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__after#](x0, x1) = 4x0, [after](x0, x1) = 2x0 + x1, [a__after](x0, x1) = 2x0 + x1, [0] = 0, [cons](x0, x1) = x1, [from](x0) = 0, [s](x0) = 1x0, [mark](x0) = x0, [a__from](x0) = 0 orientation: a__after#(s(N),cons(X,XS)) = 5N >= 4N = a__after#(mark(N),mark(XS)) a__from(X) = 0 >= 0 = cons(mark(X),from(s(X))) a__after(0(),XS) = XS + 2 >= XS = mark(XS) a__after(s(N),cons(X,XS)) = 3N + XS >= 2N + XS = a__after(mark(N),mark(XS)) mark(from(X)) = 0 >= 0 = a__from(mark(X)) mark(after(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X2 >= X2 = cons(mark(X1),X2) mark(s(X)) = 1X >= 1X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = 0 >= 0 = from(X) a__after(X1,X2) = 2X1 + X2 >= 2X1 + X2 = after(X1,X2) problem: DPs: TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Qed