YES Problem: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 3x0, [pairNs] = 0, [repItems](x0) = 4x0 + 5, [a__repItems](x0) = 4x0 + 5, [a__tail](x0) = 3x0, [zip](x0, x1) = 4x0 + 2x1 + 4, [pair](x0, x1) = 2x0 + x1, [a__zip](x0, x1) = 4x0 + 2x1 + 4, [take](x0, x1) = x0 + 2x1 + 2, [nil] = 1, [a__take](x0, x1) = x0 + 2x1 + 4, [s](x0) = x0, [mark](x0) = 2x0, [a__incr](x0) = 4x0, [a__oddNs] = 0, [cons](x0, x1) = 4x0 + x1, [incr](x0) = 4x0, [oddNs] = 0, [0] = 0, [a__pairNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 16X + 4XS >= 8X + 4XS = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = 2XS + 4 >= 1 = nil() a__take(s(N),cons(X,XS)) = N + 8X + 2XS + 4 >= N + 8X + 2XS + 2 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = 2XS + 8 >= 1 = nil() a__zip(X,nil()) = 4X + 6 >= 1 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 16X + 4XS + 8Y + 2YS + 4 >= 16X + 4XS + 8Y + 2YS + 4 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = 12X + 3XS >= 2XS = mark(XS) a__repItems(nil()) = 9 >= 1 = nil() a__repItems(cons(X,XS)) = 16X + 4XS + 5 >= 12X + 4XS + 5 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = 8X >= 8X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 8X1 + 4X2 + 8 >= 8X1 + 4X2 + 4 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = 6X >= 6X = a__tail(mark(X)) mark(repItems(X)) = 8X + 10 >= 8X + 5 = a__repItems(mark(X)) mark(cons(X1,X2)) = 8X1 + 2X2 >= 8X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(nil()) = 2 >= 1 = nil() mark(pair(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = 4X >= 4X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = X1 + 2X2 + 4 >= X1 + 2X2 + 2 = take(X1,X2) a__zip(X1,X2) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = zip(X1,X2) a__tail(X) = 3X >= 3X = tail(X) a__repItems(X) = 4X + 5 >= 4X + 5 = repItems(X) problem: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 2x0 + 1, [pairNs] = 0, [repItems](x0) = 2x0, [a__repItems](x0) = 4x0, [a__tail](x0) = 2x0 + 2, [zip](x0, x1) = 4x0 + x1, [pair](x0, x1) = x0 + x1, [a__zip](x0, x1) = 4x0 + 2x1, [take](x0, x1) = x0 + 6x1, [a__take](x0, x1) = x0 + 4x1, [s](x0) = 2x0, [mark](x0) = 2x0, [a__incr](x0) = 4x0, [a__oddNs] = 0, [cons](x0, x1) = 2x0 + 2x1, [incr](x0) = 4x0, [oddNs] = 0, [0] = 0, [a__pairNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 8X + 8XS >= 8X + 8XS = cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) = 8X + 8XS + 4Y + 4YS >= 4X + 8XS + 4Y + 2YS = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = 4X + 4XS + 2 >= 2XS = mark(XS) a__repItems(cons(X,XS)) = 8X + 8XS >= 8X + 8XS = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = 8X >= 8X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = 2X1 + 12X2 >= 2X1 + 8X2 = a__take(mark(X1),mark(X2)) mark(tail(X)) = 4X + 2 >= 4X + 2 = a__tail(mark(X)) mark(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 2X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 4X >= 4X = s(mark(X)) mark(pair(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = 4X >= 4X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__zip(X1,X2) = 4X1 + 2X2 >= 4X1 + X2 = zip(X1,X2) a__tail(X) = 2X + 2 >= 2X + 1 = tail(X) a__repItems(X) = 4X >= 2X = repItems(X) problem: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__repItems(X) -> repItems(X) DP Processor: DPs: a__oddNs#() -> a__pairNs#() a__oddNs#() -> a__incr#(a__pairNs()) a__incr#(cons(X,XS)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) mark#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__repItems(X) -> repItems(X) TDG Processor: DPs: a__oddNs#() -> a__pairNs#() a__oddNs#() -> a__incr#(a__pairNs()) a__incr#(cons(X,XS)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) mark#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__repItems(X) -> repItems(X) graph: a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(pair(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(pair(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(take(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(take(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(oddNs()) -> a__oddNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(incr(X)) -> a__incr#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(incr(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(pairNs()) -> a__pairNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() mark#(tail(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() 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#(pairNs()) -> a__pairNs#() mark#(pair(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(oddNs()) -> a__oddNs#() mark#(pair(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(pairNs()) -> a__pairNs#() mark#(pair(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() mark#(pair(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(pairNs()) -> a__pairNs#() mark#(take(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(pairNs()) -> a__pairNs#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(pairNs()) -> a__pairNs#() mark#(s(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() 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#(pairNs()) -> a__pairNs#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() 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#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(incr(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() 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#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> a__incr#(mark(X)) -> a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() -> a__oddNs#() -> a__incr#(a__pairNs()) mark#(oddNs()) -> a__oddNs#() -> a__oddNs#() -> a__pairNs#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() 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#(pairNs()) -> a__pairNs#() a__oddNs#() -> a__incr#(a__pairNs()) -> a__incr#(cons(X,XS)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 12 #arcs: 136/289 DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__repItems(X) -> repItems(X) Usable Rule Processor: DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> cons(0(),incr(oddNs())) a__pairNs() -> pairNs() a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__incr(X) -> incr(X) a__oddNs() -> a__incr(a__pairNs()) a__oddNs() -> oddNs() Semantic Labeling Processor: dimension: 1 usable rules: interpretation: [tail](x0) = x0, [pairNs] = 1, [a__tail](x0) = x0, [pair](x0, x1) = x0 + x1 + 1, [take](x0, x1) = x0 + x1 + 1, [a__take](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [mark](x0) = x0, [a__incr](x0) = x0, [a__oddNs] = 1, [cons](x0, x1) = x0 + 1, [incr](x0) = x0, [oddNs] = 1, [0] = 0, [a__pairNs] = 1 labeled: mark# a__incr# mark a__oddNs# usable (for model): mark# incr a__incr# mark cons oddNs a__oddNs# a__pairNs take tail s pair pairNs a__incr a__oddNs a__take a__tail 0 argument filtering: pi(a__pairNs) = [] pi(0) = [] pi(oddNs) = [] pi(incr) = 0 pi(cons) = 1 pi(a__oddNs) = [] pi(a__incr) = 0 pi(mark) = 0 pi(s) = 0 pi(a__take) = 0 pi(take) = [] pi(pair) = 1 pi(a__tail) = [] pi(pairNs) = [] pi(tail) = 0 pi(a__oddNs#) = [] pi(a__incr#) = [] pi(mark#) = [] precedence: a__oddNs# > a__incr# > mark# ~ tail ~ pairNs ~ a__tail ~ pair ~ take ~ a__take ~ s ~ mark ~ a__incr ~ a__oddNs ~ cons ~ incr ~ oddNs ~ 0 ~ a__pairNs problem: DPs: mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> cons(0(),incr(oddNs())) a__pairNs() -> pairNs() a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__incr(X) -> incr(X) a__oddNs() -> a__incr(a__pairNs()) a__oddNs() -> oddNs() Restore Modifier: DPs: mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__repItems(X) -> repItems(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__zip(X1,X2) -> zip(X1,X2) a__repItems(X) -> repItems(X) Qed