YES Problem: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) Proof: DP Processor: DPs: a__odds#() -> a__pairs#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) mark#(nats()) -> a__nats#() mark#(pairs()) -> a__pairs#() mark#(odds()) -> a__odds#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) TDG Processor: DPs: a__odds#() -> a__pairs#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) mark#(nats()) -> a__nats#() mark#(pairs()) -> a__pairs#() mark#(odds()) -> a__odds#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) graph: a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(tail(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(head(X)) -> a__head#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(head(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(incr(X)) -> a__incr#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(incr(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(odds()) -> a__odds#() a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(pairs()) -> a__pairs#() a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(nats()) -> a__nats#() a__head#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__head#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(odds()) -> a__odds#() a__head#(cons(X,XS)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() a__head#(cons(X,XS)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(tail(X)) -> a__tail#(mark(X)) -> a__tail#(cons(X,XS)) -> mark#(XS) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(tail(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(tail(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(head(X)) -> a__head#(mark(X)) -> a__head#(cons(X,XS)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(head(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(head(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(s(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(s(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(odds()) -> a__odds#() -> a__odds#() -> a__incr#(a__pairs()) mark#(odds()) -> a__odds#() -> a__odds#() -> a__pairs#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(odds()) -> a__odds#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairs()) -> a__pairs#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats()) -> a__nats#() mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(odds()) -> a__odds#() mark#(incr(X)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() mark#(incr(X)) -> mark#(X) -> mark#(nats()) -> a__nats#() mark#(incr(X)) -> a__incr#(mark(X)) -> a__incr#(cons(X,XS)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(odds()) -> a__odds#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pairs()) -> a__pairs#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(nats()) -> a__nats#() a__odds#() -> a__incr#(a__pairs()) -> a__incr#(cons(X,XS)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 13 #arcs: 94/256 DPs: a__tail#(cons(X,XS)) -> mark#(XS) mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__tail#](x0) = x0 + 0, [a__head#](x0) = x0 + 0, [mark#](x0) = x0, [a__incr#](x0) = x0 + 0, [a__odds#] = 0, [nil] = 0, [tail](x0) = 7x0 + 7, [head](x0) = x0 + 0, [pairs] = 0, [a__tail](x0) = 7x0 + 7, [a__head](x0) = x0 + 0, [s](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__incr](x0) = x0 + 0, [a__odds] = 0, [odds] = 0, [a__pairs] = 0, [cons](x0, x1) = x0 + x1 + 0, [incr](x0) = x0 + 0, [nats] = 0, [0] = 0, [a__nats] = 0 orientation: a__tail#(cons(X,XS)) = X + XS + 0 >= XS = mark#(XS) mark#(odds()) = 0 >= 0 = a__odds#() a__odds#() = 0 >= 0 = a__incr#(a__pairs()) a__incr#(cons(X,XS)) = X + XS + 0 >= X = mark#(X) mark#(incr(X)) = X + 0 >= X = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) mark#(head(X)) = X + 0 >= X = mark#(X) mark#(head(X)) = X + 0 >= X + 0 = a__head#(mark(X)) a__head#(cons(X,XS)) = X + XS + 0 >= X = mark#(X) mark#(tail(X)) = 7X + 7 >= X = mark#(X) mark#(tail(X)) = 7X + 7 >= X + 0 = a__tail#(mark(X)) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) a__nats() = 0 >= 0 = cons(0(),incr(nats())) a__pairs() = 0 >= 0 = cons(0(),incr(odds())) a__odds() = 0 >= 0 = a__incr(a__pairs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) = X + XS + 0 >= X + 0 = mark(X) a__tail(cons(X,XS)) = 7X + 7XS + 7 >= XS + 0 = mark(XS) mark(nats()) = 0 >= 0 = a__nats() mark(pairs()) = 0 >= 0 = a__pairs() mark(odds()) = 0 >= 0 = a__odds() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = 7X + 7 >= 7X + 7 = a__tail(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) a__nats() = 0 >= 0 = nats() a__pairs() = 0 >= 0 = pairs() a__odds() = 0 >= 0 = odds() a__incr(X) = X + 0 >= X + 0 = incr(X) a__head(X) = X + 0 >= X + 0 = head(X) a__tail(X) = 7X + 7 >= 7X + 7 = tail(X) problem: DPs: a__tail#(cons(X,XS)) -> mark#(XS) mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 10 #arcs: 77/121 DPs: mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__head#](x0) = x0 + 2, [mark#](x0) = x0, [a__incr#](x0) = x0 + 0, [a__odds#] = 0, [nil] = 7, [tail](x0) = 1x0 + 1, [head](x0) = 1x0 + 7, [pairs] = 0, [a__tail](x0) = 1x0 + 1, [a__head](x0) = 1x0 + 7, [s](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__incr](x0) = x0 + 0, [a__odds] = 0, [odds] = 0, [a__pairs] = 0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0 + 0, [nats] = 3, [0] = 0, [a__nats] = 3 orientation: mark#(odds()) = 0 >= 0 = a__odds#() a__odds#() = 0 >= 0 = a__incr#(a__pairs()) a__incr#(cons(X,XS)) = X + XS + 0 >= X = mark#(X) mark#(incr(X)) = X + 0 >= X = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) mark#(head(X)) = 1X + 7 >= X = mark#(X) mark#(head(X)) = 1X + 7 >= X + 2 = a__head#(mark(X)) a__head#(cons(X,XS)) = X + XS + 2 >= X = mark#(X) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) a__nats() = 3 >= 3 = cons(0(),incr(nats())) a__pairs() = 0 >= 0 = cons(0(),incr(odds())) a__odds() = 0 >= 0 = a__incr(a__pairs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) = 1X + 1XS + 7 >= X + 0 = mark(X) a__tail(cons(X,XS)) = 1X + 1XS + 1 >= XS + 0 = mark(XS) mark(nats()) = 3 >= 3 = a__nats() mark(pairs()) = 0 >= 0 = a__pairs() mark(odds()) = 0 >= 0 = a__odds() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(head(X)) = 1X + 7 >= 1X + 7 = a__head(mark(X)) mark(tail(X)) = 1X + 1 >= 1X + 1 = a__tail(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 7 >= 7 = nil() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) a__nats() = 3 >= 3 = nats() a__pairs() = 0 >= 0 = pairs() a__odds() = 0 >= 0 = odds() a__incr(X) = X + 0 >= X + 0 = incr(X) a__head(X) = 1X + 7 >= 1X + 7 = head(X) a__tail(X) = 1X + 1 >= 1X + 1 = tail(X) problem: DPs: mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 7 #arcs: 46/64 DPs: mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) a__incr#(cons(X,XS)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [a__odds#] = 2, [nil] = 7, [tail](x0) = x0, [head](x0) = x0, [pairs] = 2, [a__tail](x0) = x0, [a__head](x0) = x0, [s](x0) = x0 + 0, [mark](x0) = x0, [a__incr](x0) = x0 + 0, [a__odds] = 2, [odds] = 2, [a__pairs] = 2, [cons](x0, x1) = 1x0 + x1 + 1, [incr](x0) = x0 + 0, [nats] = 5, [0] = 0, [a__nats] = 5 orientation: mark#(odds()) = 2 >= 2 = a__odds#() a__odds#() = 2 >= 2 = a__incr#(a__pairs()) a__incr#(cons(X,XS)) = 1X + XS + 1 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = 1X1 + X2 + 1 >= X1 + 0 = mark#(X1) a__nats() = 5 >= 5 = cons(0(),incr(nats())) a__pairs() = 2 >= 2 = cons(0(),incr(odds())) a__odds() = 2 >= 2 = a__incr(a__pairs()) a__incr(cons(X,XS)) = 1X + XS + 1 >= 1X + XS + 1 = cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) = 1X + XS + 1 >= X = mark(X) a__tail(cons(X,XS)) = 1X + XS + 1 >= XS = mark(XS) mark(nats()) = 5 >= 5 = a__nats() mark(pairs()) = 2 >= 2 = a__pairs() mark(odds()) = 2 >= 2 = a__odds() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(head(X)) = X >= X = a__head(mark(X)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 7 >= 7 = nil() mark(cons(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = cons(mark(X1),X2) a__nats() = 5 >= 5 = nats() a__pairs() = 2 >= 2 = pairs() a__odds() = 2 >= 2 = odds() a__incr(X) = X + 0 >= X + 0 = incr(X) a__head(X) = X >= X = head(X) a__tail(X) = X >= X = tail(X) problem: DPs: mark#(odds()) -> a__odds#() a__odds#() -> a__incr#(a__pairs()) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(s(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) SCC Processor: #sccs: 1 #rules: 2 #arcs: 23/25 DPs: mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__nats() -> cons(0(),incr(nats())) a__pairs() -> cons(0(),incr(odds())) a__odds() -> a__incr(a__pairs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__head(cons(X,XS)) -> mark(X) a__tail(cons(X,XS)) -> mark(XS) mark(nats()) -> a__nats() mark(pairs()) -> a__pairs() mark(odds()) -> a__odds() mark(incr(X)) -> a__incr(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) a__nats() -> nats() a__pairs() -> pairs() a__odds() -> odds() a__incr(X) -> incr(X) a__head(X) -> head(X) a__tail(X) -> tail(X) Qed