YES Problem: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Proof: DP Processor: DPs: first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X TDG Processor: DPs: first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X graph: activate#(n__from(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X2) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> activate#(X1) -> activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(Z) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(activate(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(activate(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> activate#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X1) first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X2) SCC Processor: #sccs: 1 #rules: 6 #arcs: 36/64 DPs: activate#(n__from(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__s(X)) -> activate#(X) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [activate#](x0) = x0, [first#](x0, x1) = x0 + x1, [n__from](x0) = x0 + 0, [n__s](x0) = x0 + 0, [from](x0) = x0 + 0, [n__first](x0, x1) = x0 + 4x1, [activate](x0) = x0, [cons](x0, x1) = x1 + 0, [s](x0) = x0 + 0, [nil] = 1, [first](x0, x1) = x0 + 4x1, [0] = 6 orientation: activate#(n__from(X)) = X + 0 >= X = activate#(X) activate#(n__first(X1,X2)) = X1 + 4X2 >= X2 = activate#(X2) activate#(n__first(X1,X2)) = X1 + 4X2 >= X1 = activate#(X1) activate#(n__first(X1,X2)) = X1 + 4X2 >= X1 + X2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + Z + 0 >= Z = activate#(Z) activate#(n__s(X)) = X + 0 >= X = activate#(X) first(0(),X) = 4X + 6 >= 1 = nil() first(s(X),cons(Y,Z)) = X + 4Z + 4 >= X + 4Z + 0 = cons(Y,n__first(X,activate(Z))) from(X) = X + 0 >= X + 0 = cons(X,n__from(n__s(X))) first(X1,X2) = X1 + 4X2 >= X1 + 4X2 = n__first(X1,X2) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) activate(n__first(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = first(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__from(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__s(X)) -> activate#(X) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: activate#(n__from(X)) -> activate#(X) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__s(X)) -> activate#(X) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [activate#](x0) = x0 + 0, [first#](x0, x1) = x0 + x1 + 0, [n__from](x0) = 1x0 + 1, [n__s](x0) = x0 + 0, [from](x0) = 1x0 + 1, [n__first](x0, x1) = x0 + 4x1 + 1, [activate](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [nil] = 2, [first](x0, x1) = x0 + 4x1 + 1, [0] = 5 orientation: activate#(n__from(X)) = 1X + 1 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = X1 + 4X2 + 1 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = X1 + 4X2 + 1 >= X1 + X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = X + Y + Z + 0 >= Z + 0 = activate#(Z) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) first(0(),X) = 4X + 5 >= 2 = nil() first(s(X),cons(Y,Z)) = X + 4Y + 4Z + 4 >= X + Y + 4Z + 1 = cons(Y,n__first(X,activate(Z))) from(X) = 1X + 1 >= 1X + 1 = cons(X,n__from(n__s(X))) first(X1,X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = n__first(X1,X2) from(X) = 1X + 1 >= 1X + 1 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) activate(n__first(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = first(activate(X1),activate(X2)) activate(n__from(X)) = 1X + 1 >= 1X + 1 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__s(X)) -> activate#(X) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) activate#(n__s(X)) -> activate#(X) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [activate#](x0) = x0 + -16, [first#](x0, x1) = 5x1 + -16, [n__from](x0) = x0 + 0, [n__s](x0) = 3x0 + -8, [from](x0) = x0 + 0, [n__first](x0, x1) = x0 + 5x1, [activate](x0) = x0, [cons](x0, x1) = x0 + -5x1, [s](x0) = 3x0 + -8, [nil] = 0, [first](x0, x1) = x0 + 5x1, [0] = 0 orientation: activate#(n__first(X1,X2)) = X1 + 5X2 + -16 >= X1 + -16 = activate#(X1) activate#(n__first(X1,X2)) = X1 + 5X2 + -16 >= 5X2 + -16 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 5Y + Z + -16 >= Z + -16 = activate#(Z) activate#(n__s(X)) = 3X + -8 >= X + -16 = activate#(X) first(0(),X) = 5X + 0 >= 0 = nil() first(s(X),cons(Y,Z)) = 3X + 5Y + Z + -8 >= -5X + Y + Z = cons(Y,n__first(X,activate(Z))) from(X) = X + 0 >= X + -5 = cons(X,n__from(n__s(X))) first(X1,X2) = X1 + 5X2 >= X1 + 5X2 = n__first(X1,X2) from(X) = X + 0 >= X + 0 = n__from(X) s(X) = 3X + -8 >= 3X + -8 = n__s(X) activate(n__first(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = first(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = 3X + -8 >= 3X + -8 = s(activate(X)) activate(X) = X >= X = X problem: DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) -> activate#(Z) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [activate#](x0) = 2x0, [first#](x0, x1) = 2x0 + 3x1 + 0, [n__from](x0) = x0, [n__s](x0) = x0, [from](x0) = x0 + 0, [n__first](x0, x1) = x0 + 2x1 + 2, [activate](x0) = x0 + 0, [cons](x0, x1) = x1, [s](x0) = x0 + 0, [nil] = 0, [first](x0, x1) = x0 + 2x1 + 2, [0] = 0 orientation: activate#(n__first(X1,X2)) = 2X1 + 4X2 + 4 >= 2X1 = activate#(X1) activate#(n__first(X1,X2)) = 2X1 + 4X2 + 4 >= 2X1 + 3X2 + 3 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 2X + 3Z + 2 >= 2Z = activate#(Z) first(0(),X) = 2X + 2 >= 0 = nil() first(s(X),cons(Y,Z)) = X + 2Z + 2 >= X + 2Z + 2 = cons(Y,n__first(X,activate(Z))) from(X) = X + 0 >= X = cons(X,n__from(n__s(X))) first(X1,X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = n__first(X1,X2) from(X) = X + 0 >= X = n__from(X) s(X) = X + 0 >= X = n__s(X) activate(n__first(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = first(activate(X1),activate(X2)) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X + 0 >= X = X problem: DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 1 #arcs: 26/4 DPs: activate#(n__first(X1,X2)) -> activate#(X1) TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(activate#) = 0 problem: DPs: TRS: first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) first(X1,X2) -> n__first(X1,X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Qed