YES Problem: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X Proof: DP Processor: DPs: 2nd#(cons(X,XS)) -> activate#(XS) 2nd#(cons(X,XS)) -> head#(activate(XS)) take#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) 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__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X TDG Processor: DPs: 2nd#(cons(X,XS)) -> activate#(XS) 2nd#(cons(X,XS)) -> head#(activate(XS)) take#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) 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__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X graph: sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) -> sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) -> sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> activate#(X1) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> activate#(X2) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__s(X)) -> s#(activate(X)) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__s(X)) -> activate#(X) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__from(X)) -> from#(activate(X)) sel#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__from(X)) -> activate#(X) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> activate#(X1) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> activate#(X2) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__s(X)) -> s#(activate(X)) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__s(X)) -> activate#(X) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__from(X)) -> from#(activate(X)) take#(s(N),cons(X,XS)) -> activate#(XS) -> activate#(n__from(X)) -> activate#(X) activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) -> take#(s(N),cons(X,XS)) -> activate#(XS) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__take(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__take(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) -> activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> activate#(X) -> activate#(n__take(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__take(X1,X2)) -> take#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__take(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) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> activate#(X1) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__take(X1,X2)) -> activate#(X2) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__s(X)) -> s#(activate(X)) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__s(X)) -> activate#(X) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__from(X)) -> from#(activate(X)) 2nd#(cons(X,XS)) -> activate#(XS) -> activate#(n__from(X)) -> activate#(X) SCC Processor: #sccs: 2 #rules: 7 #arcs: 52/144 DPs: sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(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))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X Qed DPs: activate#(n__from(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> activate#(X1) activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) -> activate#(XS) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1 + 0, [activate#](x0) = x0, [sel](x0, x1) = x0 + x1 + 7, [n__take](x0, x1) = 7x0 + x1 + 0, [s](x0) = x0 + 0, [nil] = 0, [take](x0, x1) = 7x0 + x1 + 0, [0] = 0, [activate](x0) = x0, [2nd](x0) = x0 + 0, [head](x0) = x0 + 7, [cons](x0, x1) = x0 + x1 + 7, [n__from](x0) = x0 + 7, [n__s](x0) = x0 + 0, [from](x0) = x0 + 7 orientation: activate#(n__from(X)) = X + 7 >= X = activate#(X) activate#(n__s(X)) = X + 0 >= X = activate#(X) activate#(n__take(X1,X2)) = 7X1 + X2 + 0 >= X2 = activate#(X2) activate#(n__take(X1,X2)) = 7X1 + X2 + 0 >= X1 = activate#(X1) activate#(n__take(X1,X2)) = 7X1 + X2 + 0 >= X2 + 0 = take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) = X + XS + 7 >= XS = activate#(XS) from(X) = X + 7 >= X + 7 = cons(X,n__from(n__s(X))) head(cons(X,XS)) = X + XS + 7 >= X = X 2nd(cons(X,XS)) = X + XS + 7 >= XS + 7 = head(activate(XS)) take(0(),XS) = XS + 7 >= 0 = nil() take(s(N),cons(X,XS)) = 7N + X + XS + 7 >= 7N + X + XS + 7 = cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) = X + XS + 7 >= X = X sel(s(N),cons(X,XS)) = N + X + XS + 7 >= N + XS + 7 = sel(N,activate(XS)) from(X) = X + 7 >= X + 7 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) take(X1,X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = n__take(X1,X2) activate(n__from(X)) = X + 7 >= X + 7 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__take(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(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__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) -> activate#(XS) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0 + x1 + 0, [activate#](x0) = x0 + 0, [sel](x0, x1) = x0 + 5x1, [n__take](x0, x1) = x0 + x1 + 0, [s](x0) = x0, [nil] = 3, [take](x0, x1) = x0 + x1 + 0, [0] = 3, [activate](x0) = x0 + 0, [2nd](x0) = 7x0 + 0, [head](x0) = 2x0 + 4, [cons](x0, x1) = x0 + x1 + 0, [n__from](x0) = 1x0 + 2, [n__s](x0) = x0, [from](x0) = 1x0 + 2 orientation: activate#(n__from(X)) = 1X + 2 >= X + 0 = activate#(X) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) activate#(n__take(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) = N + X + XS + 0 >= XS + 0 = activate#(XS) from(X) = 1X + 2 >= 1X + 2 = cons(X,n__from(n__s(X))) head(cons(X,XS)) = 2X + 2XS + 4 >= X = X 2nd(cons(X,XS)) = 7X + 7XS + 7 >= 2XS + 4 = head(activate(XS)) take(0(),XS) = XS + 3 >= 3 = nil() take(s(N),cons(X,XS)) = N + X + XS + 0 >= N + X + XS + 0 = cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) = 5X + 5XS + 5 >= X = X sel(s(N),cons(X,XS)) = N + 5X + 5XS + 5 >= N + 5XS + 5 = sel(N,activate(XS)) from(X) = 1X + 2 >= 1X + 2 = n__from(X) s(X) = X >= X = n__s(X) take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__take(X1,X2) activate(n__from(X)) = 1X + 2 >= 1X + 2 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = take(activate(X1),activate(X2)) activate(X) = X + 0 >= X = X problem: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X2) activate#(n__take(X1,X2)) -> take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) -> activate#(XS) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0 + x1 + 0, [activate#](x0) = x0 + 2, [sel](x0, x1) = 6x0 + x1, [n__take](x0, x1) = 2x0 + 2x1, [s](x0) = x0 + 0, [nil] = 1, [take](x0, x1) = 2x0 + 2x1, [0] = 0, [activate](x0) = x0, [2nd](x0) = 4x0 + 1, [head](x0) = 4x0, [cons](x0, x1) = x0 + x1 + 2, [n__from](x0) = x0 + 2, [n__s](x0) = x0 + 0, [from](x0) = x0 + 2 orientation: activate#(n__s(X)) = X + 2 >= X + 2 = activate#(X) activate#(n__take(X1,X2)) = 2X1 + 2X2 + 2 >= X2 + 2 = activate#(X2) activate#(n__take(X1,X2)) = 2X1 + 2X2 + 2 >= X1 + X2 + 0 = take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) = N + X + XS + 2 >= XS + 2 = activate#(XS) from(X) = X + 2 >= X + 2 = cons(X,n__from(n__s(X))) head(cons(X,XS)) = 4X + 4XS + 6 >= X = X 2nd(cons(X,XS)) = 4X + 4XS + 6 >= 4XS = head(activate(XS)) take(0(),XS) = 2XS + 2 >= 1 = nil() take(s(N),cons(X,XS)) = 2N + 2X + 2XS + 4 >= 2N + X + 2XS + 2 = cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) = X + XS + 6 >= X = X sel(s(N),cons(X,XS)) = 6N + X + XS + 6 >= 6N + XS = sel(N,activate(XS)) from(X) = X + 2 >= X + 2 = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) take(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = n__take(X1,X2) activate(n__from(X)) = X + 2 >= X + 2 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__take(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = take(activate(X1),activate(X2)) activate(X) = X >= X = X problem: DPs: activate#(n__s(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X2) take#(s(N),cons(X,XS)) -> activate#(XS) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 2 #arcs: 26/9 DPs: activate#(n__s(X)) -> activate#(X) activate#(n__take(X1,X2)) -> activate#(X2) TRS: from(X) -> cons(X,n__from(n__s(X))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(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))) head(cons(X,XS)) -> X 2nd(cons(X,XS)) -> head(activate(XS)) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1,X2) -> n__take(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) activate(X) -> X Qed