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 Usable Rule Processor: 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: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) interpretation: [take#](x0, x1) = x0 + x1 + 0, [activate#](x0) = x0 + 1, [n__take](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [nil] = 1, [take](x0, x1) = x0 + x1 + 0, [0] = 4, [activate](x0) = x0 + 1, [cons](x0, x1) = x0 + x1 + 1, [n__from](x0) = 5x0 + 7, [n__s](x0) = x0 + 0, [from](x0) = 5x0 + 7 orientation: activate#(n__from(X)) = 5X + 7 >= X + 1 = activate#(X) activate#(n__s(X)) = X + 1 >= X + 1 = activate#(X) activate#(n__take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = activate#(X2) activate#(n__take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = activate#(X1) activate#(n__take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) = N + X + XS + 1 >= XS + 1 = activate#(XS) activate(n__from(X)) = 5X + 7 >= 5X + 7 = from(activate(X)) activate(n__s(X)) = X + 1 >= X + 1 = s(activate(X)) activate(n__take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(activate(X1),activate(X2)) activate(X) = X + 1 >= X = X from(X) = 5X + 7 >= 5X + 7 = cons(X,n__from(n__s(X))) from(X) = 5X + 7 >= 5X + 7 = n__from(X) s(X) = X + 1 >= X + 0 = n__s(X) take(0(),XS) = XS + 4 >= 1 = nil() take(s(N),cons(X,XS)) = N + X + XS + 1 >= N + X + XS + 1 = cons(X,n__take(N,activate(XS))) take(X1,X2) = X1 + X2 + 0 >= X1 + X2 = n__take(X1,X2) problem: DPs: 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: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) Restore Modifier: DPs: 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 Usable Rule Processor: DPs: 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: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) interpretation: [take#](x0, x1) = 7x0 + x1 + 0, [activate#](x0) = x0, [n__take](x0, x1) = 7x0 + x1 + 7, [s](x0) = x0 + 0, [nil] = 0, [take](x0, x1) = 7x0 + x1 + 7, [0] = 0, [activate](x0) = x0 + 0, [cons](x0, x1) = x1 + 0, [n__from](x0) = x0, [n__s](x0) = x0 + 0, [from](x0) = x0 + 0 orientation: activate#(n__s(X)) = X + 0 >= X = activate#(X) activate#(n__take(X1,X2)) = 7X1 + X2 + 7 >= X2 = activate#(X2) activate#(n__take(X1,X2)) = 7X1 + X2 + 7 >= X1 = activate#(X1) activate#(n__take(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) = 7N + XS + 7 >= XS = activate#(XS) activate(n__from(X)) = X + 0 >= X + 0 = from(activate(X)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__take(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = take(activate(X1),activate(X2)) activate(X) = X + 0 >= X = X from(X) = X + 0 >= X + 0 = cons(X,n__from(n__s(X))) from(X) = X + 0 >= X = n__from(X) s(X) = X + 0 >= X + 0 = n__s(X) take(0(),XS) = XS + 7 >= 0 = nil() take(s(N),cons(X,XS)) = 7N + XS + 7 >= 7N + XS + 7 = cons(X,n__take(N,activate(XS))) take(X1,X2) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = n__take(X1,X2) 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: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) Restore Modifier: 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 Usable Rule Processor: 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: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) interpretation: [take#](x0, x1) = 4x1 + 0, [activate#](x0) = 1x0, [n__take](x0, x1) = 4x1 + 0, [s](x0) = x0, [nil] = 0, [take](x0, x1) = 4x1 + 0, [0] = 0, [activate](x0) = x0, [cons](x0, x1) = x1, [n__from](x0) = 2, [n__s](x0) = x0, [from](x0) = 2 orientation: activate#(n__s(X)) = 1X >= 1X = activate#(X) activate#(n__take(X1,X2)) = 5X2 + 1 >= 1X2 = activate#(X2) activate#(n__take(X1,X2)) = 5X2 + 1 >= 4X2 + 0 = take#(activate(X1),activate(X2)) take#(s(N),cons(X,XS)) = 4XS + 0 >= 1XS = activate#(XS) activate(n__from(X)) = 2 >= 2 = from(activate(X)) activate(n__s(X)) = X >= X = s(activate(X)) activate(n__take(X1,X2)) = 4X2 + 0 >= 4X2 + 0 = take(activate(X1),activate(X2)) activate(X) = X >= X = X from(X) = 2 >= 2 = cons(X,n__from(n__s(X))) from(X) = 2 >= 2 = n__from(X) s(X) = X >= X = n__s(X) take(0(),XS) = 4XS + 0 >= 0 = nil() take(s(N),cons(X,XS)) = 4XS + 0 >= 4XS + 0 = cons(X,n__take(N,activate(XS))) take(X1,X2) = 4X2 + 0 >= 4X2 + 0 = n__take(X1,X2) problem: DPs: activate#(n__s(X)) -> activate#(X) TRS: 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 from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) take(X1,X2) -> n__take(X1,X2) Restore Modifier: DPs: activate#(n__s(X)) -> activate#(X) 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