YES Problem: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Proof: DP Processor: DPs: first#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) 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)) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X TDG Processor: DPs: first#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) 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)) activate#(n__first(X1,X2)) -> activate#(X2) activate#(n__first(X1,X2)) -> activate#(X1) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X graph: sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) -> sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) -> sel#(s(X),cons(Y,Z)) -> activate#(Z) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X1) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__first(X1,X2)) -> activate#(X2) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(activate(X)) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> activate#(X) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(activate(X)) sel#(s(X),cons(Y,Z)) -> activate#(Z) -> 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#(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#(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)) -> 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)) -> first#(activate(X1),activate(X2)) -> first#(s(X),cons(Y,Z)) -> activate#(Z) 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__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__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__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) 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) 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) SCC Processor: #sccs: 2 #rules: 7 #arcs: 45/100 DPs: sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(sel#) = 0 problem: DPs: TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed DPs: activate#(n__from(X)) -> activate#(X) activate#(n__s(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) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0 + 0, [first#](x0, x1) = x1 + 2, [sel](x0, x1) = x0 + 1x1 + 14, [n__first](x0, x1) = 1x0 + x1 + 4, [activate](x0) = x0, [s](x0) = x0, [nil] = 4, [first](x0, x1) = 1x0 + x1 + 4, [0] = 6, [cons](x0, x1) = x0 + x1, [n__from](x0) = 1x0, [n__s](x0) = x0, [from](x0) = 1x0 orientation: activate#(n__from(X)) = 1X + 0 >= X + 0 = activate#(X) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = 1X1 + X2 + 4 >= X2 + 0 = activate#(X2) activate#(n__first(X1,X2)) = 1X1 + X2 + 4 >= X1 + 0 = activate#(X1) activate#(n__first(X1,X2)) = 1X1 + X2 + 4 >= X2 + 2 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = Y + Z + 2 >= Z + 0 = activate#(Z) from(X) = 1X >= 1X = cons(X,n__from(n__s(X))) first(0(),Z) = Z + 7 >= 4 = nil() first(s(X),cons(Y,Z)) = 1X + Y + Z + 4 >= 1X + Y + Z + 4 = cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) = 1X + 1Z + 14 >= X = X sel(s(X),cons(Y,Z)) = X + 1Y + 1Z + 14 >= X + 1Z + 14 = sel(X,activate(Z)) from(X) = 1X >= 1X = n__from(X) s(X) = X >= X = n__s(X) first(X1,X2) = 1X1 + X2 + 4 >= 1X1 + X2 + 4 = n__first(X1,X2) activate(n__from(X)) = 1X >= 1X = from(activate(X)) activate(n__s(X)) = X >= X = s(activate(X)) activate(n__first(X1,X2)) = 1X1 + X2 + 4 >= 1X1 + X2 + 4 = first(activate(X1),activate(X2)) activate(X) = X >= X = X problem: DPs: activate#(n__from(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) 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) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = x0 + 0, [first#](x0, x1) = x1 + 0, [sel](x0, x1) = 3x0 + 7x1 + 0, [n__first](x0, x1) = x0 + x1, [activate](x0) = x0, [s](x0) = x0 + 0, [nil] = 0, [first](x0, x1) = x0 + x1, [0] = 3, [cons](x0, x1) = x0 + x1 + 0, [n__from](x0) = 1x0 + 1, [n__s](x0) = x0 + 0, [from](x0) = 1x0 + 1 orientation: activate#(n__from(X)) = 1X + 1 >= X + 0 = activate#(X) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = Y + Z + 0 >= Z + 0 = activate#(Z) from(X) = 1X + 1 >= 1X + 1 = cons(X,n__from(n__s(X))) first(0(),Z) = Z + 3 >= 0 = nil() first(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) = 7X + 7Z + 7 >= X = X sel(s(X),cons(Y,Z)) = 3X + 7Y + 7Z + 7 >= 3X + 7Z + 0 = sel(X,activate(Z)) from(X) = 1X + 1 >= 1X + 1 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) first(X1,X2) = X1 + X2 >= X1 + X2 = n__first(X1,X2) activate(n__from(X)) = 1X + 1 >= 1X + 1 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__first(X1,X2)) = X1 + X2 >= X1 + X2 = first(activate(X1),activate(X2)) activate(X) = X >= X = X problem: DPs: activate#(n__s(X)) -> activate#(X) 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) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [activate#](x0) = 3x0, [first#](x0, x1) = 5x0 + 4x1 + 0, [sel](x0, x1) = 4x0 + x1 + 0, [n__first](x0, x1) = 2x0 + 2x1 + 2, [activate](x0) = x0 + 0, [s](x0) = x0, [nil] = 0, [first](x0, x1) = 2x0 + 2x1 + 2, [0] = 2, [cons](x0, x1) = x0 + x1, [n__from](x0) = 2x0 + 2, [n__s](x0) = x0, [from](x0) = 2x0 + 2 orientation: activate#(n__s(X)) = 3X >= 3X = activate#(X) activate#(n__first(X1,X2)) = 5X1 + 5X2 + 5 >= 3X2 = activate#(X2) activate#(n__first(X1,X2)) = 5X1 + 5X2 + 5 >= 5X1 + 4X2 + 5 = first#(activate(X1),activate(X2)) first#(s(X),cons(Y,Z)) = 5X + 4Y + 4Z + 0 >= 3Z = activate#(Z) from(X) = 2X + 2 >= 2X + 2 = cons(X,n__from(n__s(X))) first(0(),Z) = 2Z + 4 >= 0 = nil() first(s(X),cons(Y,Z)) = 2X + 2Y + 2Z + 2 >= 2X + Y + 2Z + 2 = cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) = X + Z + 6 >= X = X sel(s(X),cons(Y,Z)) = 4X + Y + Z + 0 >= 4X + Z + 0 = sel(X,activate(Z)) from(X) = 2X + 2 >= 2X + 2 = n__from(X) s(X) = X >= X = n__s(X) first(X1,X2) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = n__first(X1,X2) activate(n__from(X)) = 2X + 2 >= 2X + 2 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__first(X1,X2)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = first(activate(X1),activate(X2)) activate(X) = X + 0 >= X = X problem: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 1 #arcs: 26/4 DPs: activate#(n__s(X)) -> activate#(X) TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Subterm Criterion Processor: simple projection: pi(activate#) = 0 problem: DPs: TRS: from(X) -> cons(X,n__from(n__s(X))) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) from(X) -> n__from(X) s(X) -> n__s(X) first(X1,X2) -> n__first(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(X) -> X Qed