YES Problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(take(0(),XS)) -> mark(nil()) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(nil(),XS)) -> mark(nil()) active(zip(X,nil())) -> mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(tail(cons(X,XS))) -> mark(XS) active(repItems(nil())) -> mark(nil()) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [repItems](x0) = 2x0, [tail](x0) = 2x0, [pair](x0, x1) = x0 + 2x1, [zip](x0, x1) = 2x0 + 4x1, [nil] = 0, [take](x0, x1) = x0 + 2x1 + 2, [s](x0) = 4x0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [incr](x0) = 4x0, [oddNs] = 0, [0] = 0, [active](x0) = x0, [pairNs] = 0 orientation: active(pairNs()) = 0 >= 0 = mark(cons(0(),incr(oddNs()))) active(oddNs()) = 0 >= 0 = mark(incr(pairNs())) active(incr(cons(X,XS))) = 16X + 4XS >= 16X + 4XS = mark(cons(s(X),incr(XS))) active(take(0(),XS)) = 2XS + 2 >= 0 = mark(nil()) active(take(s(N),cons(X,XS))) = 4N + 8X + 2XS + 2 >= N + 4X + 2XS + 2 = mark(cons(X,take(N,XS))) active(zip(nil(),XS)) = 4XS >= 0 = mark(nil()) active(zip(X,nil())) = 2X >= 0 = mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) = 8X + 2XS + 16Y + 4YS >= 4X + 2XS + 8Y + 4YS = mark(cons(pair(X,Y),zip(XS,YS))) active(tail(cons(X,XS))) = 8X + 2XS >= XS = mark(XS) active(repItems(nil())) = 0 >= 0 = mark(nil()) active(repItems(cons(X,XS))) = 8X + 2XS >= 8X + 2XS = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(active(X1),X2) active(incr(X)) = 4X >= 4X = incr(active(X)) active(s(X)) = 4X >= 4X = s(active(X)) active(take(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = take(active(X1),X2) active(take(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = take(X1,active(X2)) active(zip(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = zip(active(X1),X2) active(zip(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = zip(X1,active(X2)) active(pair(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = pair(active(X1),X2) active(pair(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = pair(X1,active(X2)) active(tail(X)) = 2X >= 2X = tail(active(X)) active(repItems(X)) = 2X >= 2X = repItems(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = 4X >= 4X = mark(incr(X)) s(mark(X)) = 4X >= 4X = mark(s(X)) take(mark(X1),X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = mark(take(X1,X2)) zip(mark(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(zip(X1,X2)) zip(X1,mark(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = mark(zip(X1,X2)) pair(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = mark(pair(X1,X2)) tail(mark(X)) = 2X >= 2X = mark(tail(X)) repItems(mark(X)) = 2X >= 2X = mark(repItems(X)) proper(pairNs()) = 0 >= 0 = ok(pairNs()) proper(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 4X >= 4X = incr(proper(X)) proper(oddNs()) = 0 >= 0 = ok(oddNs()) proper(s(X)) = 4X >= 4X = s(proper(X)) proper(take(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(zip(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = zip(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = pair(proper(X1),proper(X2)) proper(tail(X)) = 2X >= 2X = tail(proper(X)) proper(repItems(X)) = 2X >= 2X = repItems(proper(X)) cons(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = 4X >= 4X = ok(incr(X)) s(ok(X)) = 4X >= 4X = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = ok(take(X1,X2)) zip(ok(X1),ok(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = ok(zip(X1,X2)) pair(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(pair(X1,X2)) tail(ok(X)) = 2X >= 2X = ok(tail(X)) repItems(ok(X)) = 2X >= 2X = ok(repItems(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(nil(),XS)) -> mark(nil()) active(zip(X,nil())) -> mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(tail(cons(X,XS))) -> mark(XS) active(repItems(nil())) -> mark(nil()) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [repItems](x0) = 2x0, [tail](x0) = 2x0 + 1, [pair](x0, x1) = 2x0 + x1, [zip](x0, x1) = 2x0 + 2x1, [nil] = 0, [take](x0, x1) = x0 + 4x1, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [oddNs] = 6, [0] = 0, [active](x0) = x0, [pairNs] = 6 orientation: active(pairNs()) = 6 >= 6 = mark(cons(0(),incr(oddNs()))) active(oddNs()) = 6 >= 6 = mark(incr(pairNs())) active(incr(cons(X,XS))) = X + XS >= X + XS = mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) = N + 4X + 4XS >= N + X + 4XS = mark(cons(X,take(N,XS))) active(zip(nil(),XS)) = 2XS >= 0 = mark(nil()) active(zip(X,nil())) = 2X >= 0 = mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) = 2X + 2XS + 2Y + 2YS >= 2X + 2XS + Y + 2YS = mark(cons(pair(X,Y),zip(XS,YS))) active(tail(cons(X,XS))) = 2X + 2XS + 1 >= XS = mark(XS) active(repItems(nil())) = 0 >= 0 = mark(nil()) active(repItems(cons(X,XS))) = 2X + 2XS >= 2X + 2XS = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(incr(X)) = X >= X = incr(active(X)) active(s(X)) = X >= X = s(active(X)) active(take(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = take(active(X1),X2) active(take(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = take(X1,active(X2)) active(zip(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = zip(active(X1),X2) active(zip(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = zip(X1,active(X2)) active(pair(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = pair(active(X1),X2) active(pair(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = pair(X1,active(X2)) active(tail(X)) = 2X + 1 >= 2X + 1 = tail(active(X)) active(repItems(X)) = 2X >= 2X = repItems(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = X >= X = mark(incr(X)) s(mark(X)) = X >= X = mark(s(X)) take(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 4X2 >= X1 + 4X2 = mark(take(X1,X2)) zip(mark(X1),X2) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(zip(X1,X2)) zip(X1,mark(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = mark(zip(X1,X2)) pair(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = mark(pair(X1,X2)) tail(mark(X)) = 2X + 1 >= 2X + 1 = mark(tail(X)) repItems(mark(X)) = 2X >= 2X = mark(repItems(X)) proper(pairNs()) = 6 >= 6 = ok(pairNs()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(oddNs()) = 6 >= 6 = ok(oddNs()) proper(s(X)) = X >= X = s(proper(X)) proper(take(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(zip(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = zip(proper(X1),proper(X2)) proper(pair(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = pair(proper(X1),proper(X2)) proper(tail(X)) = 2X + 1 >= 2X + 1 = tail(proper(X)) proper(repItems(X)) = 2X >= 2X = repItems(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 4X2 >= X1 + 4X2 = ok(take(X1,X2)) zip(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(zip(X1,X2)) pair(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(pair(X1,X2)) tail(ok(X)) = 2X + 1 >= 2X + 1 = ok(tail(X)) repItems(ok(X)) = 2X >= 2X = ok(repItems(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(nil(),XS)) -> mark(nil()) active(zip(X,nil())) -> mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(nil())) -> mark(nil()) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [repItems](x0) = 2x0, [tail](x0) = 4x0, [pair](x0, x1) = x0 + 2x1, [zip](x0, x1) = x0 + 4x1 + 1, [nil] = 0, [take](x0, x1) = 4x0 + 2x1, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [oddNs] = 4, [0] = 0, [active](x0) = x0, [pairNs] = 4 orientation: active(pairNs()) = 4 >= 4 = mark(cons(0(),incr(oddNs()))) active(oddNs()) = 4 >= 4 = mark(incr(pairNs())) active(incr(cons(X,XS))) = X + XS >= X + XS = mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) = 4N + 2X + 2XS >= 4N + X + 2XS = mark(cons(X,take(N,XS))) active(zip(nil(),XS)) = 4XS + 1 >= 0 = mark(nil()) active(zip(X,nil())) = X + 1 >= 0 = mark(nil()) active(zip(cons(X,XS),cons(Y,YS))) = X + XS + 4Y + 4YS + 1 >= X + XS + 2Y + 4YS + 1 = mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(nil())) = 0 >= 0 = mark(nil()) active(repItems(cons(X,XS))) = 2X + 2XS >= 2X + 2XS = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(incr(X)) = X >= X = incr(active(X)) active(s(X)) = X >= X = s(active(X)) active(take(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = take(active(X1),X2) active(take(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = take(X1,active(X2)) active(zip(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = zip(active(X1),X2) active(zip(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = zip(X1,active(X2)) active(pair(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = pair(active(X1),X2) active(pair(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = pair(X1,active(X2)) active(tail(X)) = 4X >= 4X = tail(active(X)) active(repItems(X)) = 2X >= 2X = repItems(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = X >= X = mark(incr(X)) s(mark(X)) = X >= X = mark(s(X)) take(mark(X1),X2) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(take(X1,X2)) take(X1,mark(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(take(X1,X2)) zip(mark(X1),X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = mark(zip(X1,X2)) zip(X1,mark(X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = mark(zip(X1,X2)) pair(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = mark(pair(X1,X2)) tail(mark(X)) = 4X >= 4X = mark(tail(X)) repItems(mark(X)) = 2X >= 2X = mark(repItems(X)) proper(pairNs()) = 4 >= 4 = ok(pairNs()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(oddNs()) = 4 >= 4 = ok(oddNs()) proper(s(X)) = X >= X = s(proper(X)) proper(take(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(zip(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = zip(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = pair(proper(X1),proper(X2)) proper(tail(X)) = 4X >= 4X = tail(proper(X)) proper(repItems(X)) = 2X >= 2X = repItems(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) take(ok(X1),ok(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = ok(take(X1,X2)) zip(ok(X1),ok(X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = ok(zip(X1,X2)) pair(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(pair(X1,X2)) tail(ok(X)) = 4X >= 4X = ok(tail(X)) repItems(ok(X)) = 2X >= 2X = ok(repItems(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(nil())) -> mark(nil()) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0, [ok](x0) = x0, [proper](x0) = x0, [repItems](x0) = 4x0 + 2, [tail](x0) = 2x0 + 4, [pair](x0, x1) = x0 + x1, [zip](x0, x1) = x0 + 2x1, [nil] = 0, [take](x0, x1) = 2x0 + x1, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [incr](x0) = x0, [oddNs] = 4, [0] = 0, [active](x0) = x0, [pairNs] = 4 orientation: active(pairNs()) = 4 >= 4 = mark(cons(0(),incr(oddNs()))) active(oddNs()) = 4 >= 4 = mark(incr(pairNs())) active(incr(cons(X,XS))) = 4X + XS >= 4X + XS = mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) = 2N + 4X + XS >= 2N + 4X + XS = mark(cons(X,take(N,XS))) active(zip(cons(X,XS),cons(Y,YS))) = 4X + XS + 8Y + 2YS >= 4X + XS + 4Y + 2YS = mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(nil())) = 2 >= 0 = mark(nil()) active(repItems(cons(X,XS))) = 16X + 4XS + 2 >= 8X + 4XS + 2 = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(active(X1),X2) active(incr(X)) = X >= X = incr(active(X)) active(s(X)) = X >= X = s(active(X)) active(take(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = take(active(X1),X2) active(take(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = take(X1,active(X2)) active(zip(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = zip(active(X1),X2) active(zip(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = zip(X1,active(X2)) active(pair(X1,X2)) = X1 + X2 >= X1 + X2 = pair(active(X1),X2) active(pair(X1,X2)) = X1 + X2 >= X1 + X2 = pair(X1,active(X2)) active(tail(X)) = 2X + 4 >= 2X + 4 = tail(active(X)) active(repItems(X)) = 4X + 2 >= 4X + 2 = repItems(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = X >= X = mark(incr(X)) s(mark(X)) = X >= X = mark(s(X)) take(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(take(X1,X2)) take(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = mark(take(X1,X2)) zip(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(zip(X1,X2)) zip(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = mark(zip(X1,X2)) pair(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(pair(X1,X2)) tail(mark(X)) = 2X + 4 >= 2X + 4 = mark(tail(X)) repItems(mark(X)) = 4X + 2 >= 4X + 2 = mark(repItems(X)) proper(pairNs()) = 4 >= 4 = ok(pairNs()) proper(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = X >= X = incr(proper(X)) proper(oddNs()) = 4 >= 4 = ok(oddNs()) proper(s(X)) = X >= X = s(proper(X)) proper(take(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = take(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(zip(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = zip(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + X2 >= X1 + X2 = pair(proper(X1),proper(X2)) proper(tail(X)) = 2X + 4 >= 2X + 4 = tail(proper(X)) proper(repItems(X)) = 4X + 2 >= 4X + 2 = repItems(proper(X)) cons(ok(X1),ok(X2)) = 4X1 + X2 >= 4X1 + X2 = ok(cons(X1,X2)) incr(ok(X)) = X >= X = ok(incr(X)) s(ok(X)) = X >= X = ok(s(X)) take(ok(X1),ok(X2)) = 2X1 + X2 >= 2X1 + X2 = ok(take(X1,X2)) zip(ok(X1),ok(X2)) = X1 + 2X2 >= X1 + 2X2 = ok(zip(X1,X2)) pair(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(pair(X1,X2)) tail(ok(X)) = 2X + 4 >= 2X + 4 = ok(tail(X)) repItems(ok(X)) = 4X + 2 >= 4X + 2 = ok(repItems(X)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 1 0] [1], [1 0 0] [ok](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [repItems](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [pair](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [zip](x0, x1) = [1 0 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [0] [nil] = [1] [0], [1 1 0] [1 0 0] [take](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [cons](x0, x1) = [1 0 0]x0 + x1 [0 0 0] , [1 0 0] [incr](x0) = [1 0 0]x0 [0 0 1] , [0] [oddNs] = [0] [1], [0] [0] = [1] [0], [active](x0) = x0 , [0] [pairNs] = [0] [0] orientation: [0] [0] active(pairNs()) = [0] >= [0] = mark(cons(0(),incr(oddNs()))) [0] [0] [0] [0] active(oddNs()) = [0] >= [0] = mark(incr(pairNs())) [1] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(incr(cons(X,XS))) = [1 0 0]X + [1 0 0]XS >= [1 0 0]X + [1 0 0]XS = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1] [1 1 0] [1 0 0] [1 0 0] active(take(s(N),cons(X,XS))) = [0 0 0]N + [1 0 0]X + [0 1 0]XS + [0] >= [0 0 0]N + [1 0 0]X + [0 1 0]XS = mark(cons(X,take(N,XS))) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(zip(cons(X,XS),cons(Y,YS))) = [1 0 0]X + [1 0 0]XS + [1 0 0]Y + [1 0 0]YS >= [1 0 0]X + [1 0 0]XS + [1 0 0]Y + [1 0 0]YS = mark(cons(pair(X,Y),zip(XS,YS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 0] [1 1 0] [2 0 0] [1 1 0] active(repItems(cons(X,XS))) = [2 0 0]X + [1 1 0]XS >= [2 0 0]X + [1 1 0]XS = mark(cons(X,cons(X,repItems(XS)))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [1 0 0]X1 + X2 >= [1 0 0]X1 + X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(incr(X)) = [1 0 0]X >= [1 0 0]X = incr(active(X)) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(take(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = take(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] active(take(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = take(X1,active(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(zip(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = zip(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(zip(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = zip(X1,active(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(pair(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = pair(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(pair(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = pair(X1,active(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] active(repItems(X)) = [1 1 0]X >= [1 1 0]X = repItems(active(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [1 0 0]X1 + X2 >= [1 0 0]X1 + [0 1 0]X2 = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(mark(X)) = [1 0 0]X >= [1 0 0]X = mark(incr(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] take(mark(X1),X2) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = mark(take(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] take(X1,mark(X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = mark(take(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] zip(mark(X1),X2) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = mark(zip(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] zip(X1,mark(X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = mark(zip(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] pair(mark(X1),X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = mark(pair(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] pair(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = mark(pair(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tail(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] repItems(mark(X)) = [1 1 0]X >= [1 1 0]X = mark(repItems(X)) [0 0 0] [0 0 0] [0] [0] proper(pairNs()) = [0] >= [0] = ok(pairNs()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [1 0 0]X1 + [0 1 0]X2 >= [1 0 0]X1 + [0 1 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(incr(X)) = [1 0 0]X >= [1 0 0]X = incr(proper(X)) [0 0 0] [0 0 0] [0] [0] proper(oddNs()) = [0] >= [0] = ok(oddNs()) [0] [0] [1 0 0] [0] [1 0 0] [0] proper(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(proper(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] proper(take(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = take(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(nil()) = [1] >= [1] = ok(nil()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(zip(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = zip(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(pair(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = pair(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(proper(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(repItems(X)) = [1 1 0]X >= [1 1 0]X = repItems(proper(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [1 0 0]X1 + [0 1 0]X2 >= [1 0 0]X1 + [0 1 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(ok(X)) = [1 0 0]X >= [1 0 0]X = ok(incr(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] take(ok(X1),ok(X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = ok(take(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] zip(ok(X1),ok(X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = ok(zip(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] pair(ok(X1),ok(X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = ok(pair(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] repItems(ok(X)) = [1 1 0]X >= [1 1 0]X = ok(repItems(X)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 1 0] [1] [0 1 0] [1] problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=2 interpretation: [1 2] [2] [top](x0) = [0 0]x0 + [0], [2 0] [ok](x0) = [0 1]x0, [proper](x0) = x0, [2 0] [0] [repItems](x0) = [0 1]x0 + [2], [2 0] [tail](x0) = [0 1]x0, [1 0] [pair](x0, x1) = [0 2]x0 + x1, [2 0] [0] [zip](x0, x1) = [0 2]x0 + x1 + [2], [0] [nil] = [0], [2 0] [take](x0, x1) = x0 + [0 1]x1, [s](x0) = x0, [0] [mark](x0) = x0 + [1], [2 0] [1 0] [cons](x0, x1) = [0 1]x0 + [0 0]x1, [2 0] [0] [incr](x0) = [0 1]x0 + [1], [0] [oddNs] = [3], [0] [0] = [0], [active](x0) = x0, [0] [pairNs] = [1] orientation: [0] [0] active(pairNs()) = [1] >= [1] = mark(cons(0(),incr(oddNs()))) [0] [0] active(oddNs()) = [3] >= [3] = mark(incr(pairNs())) [4 0] [2 0] [0] [2 0] [2 0] [0] active(incr(cons(X,XS))) = [0 1]X + [0 0]XS + [1] >= [0 1]X + [0 0]XS + [1] = mark(cons(s(X),incr(XS))) [4 0] [2 0] [2 0] [1 0] [0] [2 0] [2 0] [2 0] [1 0] [0] active(zip(cons(X,XS),cons(Y,YS))) = [0 2]X + [0 0]XS + [0 1]Y + [0 0]YS + [2] >= [0 2]X + [0 0]XS + [0 1]Y + [0 0]YS + [1] = mark(cons(pair(X,Y),zip(XS,YS))) [4 0] [2 0] [0] [4 0] [2 0] [0] active(repItems(cons(X,XS))) = [0 1]X + [0 0]XS + [2] >= [0 1]X + [0 0]XS + [1] = mark(cons(X,cons(X,repItems(XS)))) [2 0] [1 0] [2 0] [1 0] active(cons(X1,X2)) = [0 1]X1 + [0 0]X2 >= [0 1]X1 + [0 0]X2 = cons(active(X1),X2) [2 0] [0] [2 0] [0] active(incr(X)) = [0 1]X + [1] >= [0 1]X + [1] = incr(active(X)) active(s(X)) = X >= X = s(active(X)) [2 0] [2 0] active(take(X1,X2)) = X1 + [0 1]X2 >= X1 + [0 1]X2 = take(active(X1),X2) [2 0] [2 0] active(take(X1,X2)) = X1 + [0 1]X2 >= X1 + [0 1]X2 = take(X1,active(X2)) [2 0] [0] [2 0] [0] active(zip(X1,X2)) = [0 2]X1 + X2 + [2] >= [0 2]X1 + X2 + [2] = zip(active(X1),X2) [2 0] [0] [2 0] [0] active(zip(X1,X2)) = [0 2]X1 + X2 + [2] >= [0 2]X1 + X2 + [2] = zip(X1,active(X2)) [1 0] [1 0] active(pair(X1,X2)) = [0 2]X1 + X2 >= [0 2]X1 + X2 = pair(active(X1),X2) [1 0] [1 0] active(pair(X1,X2)) = [0 2]X1 + X2 >= [0 2]X1 + X2 = pair(X1,active(X2)) [2 0] [2 0] active(tail(X)) = [0 1]X >= [0 1]X = tail(active(X)) [2 0] [0] [2 0] [0] active(repItems(X)) = [0 1]X + [2] >= [0 1]X + [2] = repItems(active(X)) [2 0] [1 0] [0] [2 0] [1 0] [0] cons(mark(X1),X2) = [0 1]X1 + [0 0]X2 + [1] >= [0 1]X1 + [0 0]X2 + [1] = mark(cons(X1,X2)) [2 0] [0] [2 0] [0] incr(mark(X)) = [0 1]X + [2] >= [0 1]X + [2] = mark(incr(X)) [0] [0] s(mark(X)) = X + [1] >= X + [1] = mark(s(X)) [2 0] [0] [2 0] [0] take(mark(X1),X2) = X1 + [0 1]X2 + [1] >= X1 + [0 1]X2 + [1] = mark(take(X1,X2)) [2 0] [0] [2 0] [0] take(X1,mark(X2)) = X1 + [0 1]X2 + [1] >= X1 + [0 1]X2 + [1] = mark(take(X1,X2)) [2 0] [0] [2 0] [0] zip(mark(X1),X2) = [0 2]X1 + X2 + [4] >= [0 2]X1 + X2 + [3] = mark(zip(X1,X2)) [2 0] [0] [2 0] [0] zip(X1,mark(X2)) = [0 2]X1 + X2 + [3] >= [0 2]X1 + X2 + [3] = mark(zip(X1,X2)) [1 0] [0] [1 0] [0] pair(mark(X1),X2) = [0 2]X1 + X2 + [2] >= [0 2]X1 + X2 + [1] = mark(pair(X1,X2)) [1 0] [0] [1 0] [0] pair(X1,mark(X2)) = [0 2]X1 + X2 + [1] >= [0 2]X1 + X2 + [1] = mark(pair(X1,X2)) [2 0] [0] [2 0] [0] tail(mark(X)) = [0 1]X + [1] >= [0 1]X + [1] = mark(tail(X)) [2 0] [0] [2 0] [0] repItems(mark(X)) = [0 1]X + [3] >= [0 1]X + [3] = mark(repItems(X)) [0] [0] proper(pairNs()) = [1] >= [1] = ok(pairNs()) [2 0] [1 0] [2 0] [1 0] proper(cons(X1,X2)) = [0 1]X1 + [0 0]X2 >= [0 1]X1 + [0 0]X2 = cons(proper(X1),proper(X2)) [0] [0] proper(0()) = [0] >= [0] = ok(0()) [2 0] [0] [2 0] [0] proper(incr(X)) = [0 1]X + [1] >= [0 1]X + [1] = incr(proper(X)) [0] [0] proper(oddNs()) = [3] >= [3] = ok(oddNs()) proper(s(X)) = X >= X = s(proper(X)) [2 0] [2 0] proper(take(X1,X2)) = X1 + [0 1]X2 >= X1 + [0 1]X2 = take(proper(X1),proper(X2)) [0] [0] proper(nil()) = [0] >= [0] = ok(nil()) [2 0] [0] [2 0] [0] proper(zip(X1,X2)) = [0 2]X1 + X2 + [2] >= [0 2]X1 + X2 + [2] = zip(proper(X1),proper(X2)) [1 0] [1 0] proper(pair(X1,X2)) = [0 2]X1 + X2 >= [0 2]X1 + X2 = pair(proper(X1),proper(X2)) [2 0] [2 0] proper(tail(X)) = [0 1]X >= [0 1]X = tail(proper(X)) [2 0] [0] [2 0] [0] proper(repItems(X)) = [0 1]X + [2] >= [0 1]X + [2] = repItems(proper(X)) [4 0] [2 0] [4 0] [2 0] cons(ok(X1),ok(X2)) = [0 1]X1 + [0 0]X2 >= [0 1]X1 + [0 0]X2 = ok(cons(X1,X2)) [4 0] [0] [4 0] [0] incr(ok(X)) = [0 1]X + [1] >= [0 1]X + [1] = ok(incr(X)) [2 0] [2 0] s(ok(X)) = [0 1]X >= [0 1]X = ok(s(X)) [2 0] [4 0] [2 0] [4 0] take(ok(X1),ok(X2)) = [0 1]X1 + [0 1]X2 >= [0 1]X1 + [0 1]X2 = ok(take(X1,X2)) [4 0] [2 0] [0] [4 0] [2 0] [0] zip(ok(X1),ok(X2)) = [0 2]X1 + [0 1]X2 + [2] >= [0 2]X1 + [0 1]X2 + [2] = ok(zip(X1,X2)) [2 0] [2 0] [2 0] [2 0] pair(ok(X1),ok(X2)) = [0 2]X1 + [0 1]X2 >= [0 2]X1 + [0 1]X2 = ok(pair(X1,X2)) [4 0] [4 0] tail(ok(X)) = [0 1]X >= [0 1]X = ok(tail(X)) [4 0] [0] [4 0] [0] repItems(ok(X)) = [0 1]X + [2] >= [0 1]X + [2] = ok(repItems(X)) [1 2] [4] [1 2] [2] top(mark(X)) = [0 0]X + [0] >= [0 0]X + [0] = top(proper(X)) [2 2] [2] [1 2] [2] top(ok(X)) = [0 0]X + [0] >= [0 0]X + [0] = top(active(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(oddNs()) -> mark(incr(pairNs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(pairNs()) -> ok(pairNs()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(oddNs()) -> ok(oddNs()) proper(s(X)) -> s(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) tail(ok(X)) -> ok(tail(X)) repItems(ok(X)) -> ok(repItems(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = x0 + 7, [ok](x0) = 4x0 + 1, [proper](x0) = 5x0, [repItems](x0) = x0, [tail](x0) = 4x0, [pair](x0, x1) = 4x0 + x1, [zip](x0, x1) = x0 + x1, [nil] = 4, [take](x0, x1) = 5x0 + 2x1 + 2, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [incr](x0) = x0, [oddNs] = 4, [0] = 1, [active](x0) = 4x0, [pairNs] = 2 orientation: active(pairNs()) = 8 >= 8 = mark(cons(0(),incr(oddNs()))) active(oddNs()) = 16 >= 2 = mark(incr(pairNs())) active(incr(cons(X,XS))) = 16X + 4XS >= 4X + XS = mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) = 16X + 4XS + 16Y + 4YS >= 16X + XS + 4Y + YS = mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) = 16X + 4XS >= 8X + XS = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = 16X1 + 4X2 >= 16X1 + X2 = cons(active(X1),X2) active(incr(X)) = 4X >= 4X = incr(active(X)) active(s(X)) = 4X >= 4X = s(active(X)) active(take(X1,X2)) = 20X1 + 8X2 + 8 >= 20X1 + 2X2 + 2 = take(active(X1),X2) active(take(X1,X2)) = 20X1 + 8X2 + 8 >= 5X1 + 8X2 + 2 = take(X1,active(X2)) active(zip(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = zip(active(X1),X2) active(zip(X1,X2)) = 4X1 + 4X2 >= X1 + 4X2 = zip(X1,active(X2)) active(pair(X1,X2)) = 16X1 + 4X2 >= 16X1 + X2 = pair(active(X1),X2) active(pair(X1,X2)) = 16X1 + 4X2 >= 4X1 + 4X2 = pair(X1,active(X2)) active(tail(X)) = 16X >= 16X = tail(active(X)) active(repItems(X)) = 4X >= 4X = repItems(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = X >= X = mark(incr(X)) s(mark(X)) = X >= X = mark(s(X)) take(mark(X1),X2) = 5X1 + 2X2 + 2 >= 5X1 + 2X2 + 2 = mark(take(X1,X2)) take(X1,mark(X2)) = 5X1 + 2X2 + 2 >= 5X1 + 2X2 + 2 = mark(take(X1,X2)) zip(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(zip(X1,X2)) zip(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(zip(X1,X2)) pair(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 4X1 + X2 >= 4X1 + X2 = mark(pair(X1,X2)) tail(mark(X)) = 4X >= 4X = mark(tail(X)) repItems(mark(X)) = X >= X = mark(repItems(X)) proper(pairNs()) = 10 >= 9 = ok(pairNs()) proper(cons(X1,X2)) = 20X1 + 5X2 >= 20X1 + 5X2 = cons(proper(X1),proper(X2)) proper(0()) = 5 >= 5 = ok(0()) proper(incr(X)) = 5X >= 5X = incr(proper(X)) proper(oddNs()) = 20 >= 17 = ok(oddNs()) proper(s(X)) = 5X >= 5X = s(proper(X)) proper(take(X1,X2)) = 25X1 + 10X2 + 10 >= 25X1 + 10X2 + 2 = take(proper(X1),proper(X2)) proper(nil()) = 20 >= 17 = ok(nil()) proper(zip(X1,X2)) = 5X1 + 5X2 >= 5X1 + 5X2 = zip(proper(X1),proper(X2)) proper(pair(X1,X2)) = 20X1 + 5X2 >= 20X1 + 5X2 = pair(proper(X1),proper(X2)) proper(tail(X)) = 20X >= 20X = tail(proper(X)) proper(repItems(X)) = 5X >= 5X = repItems(proper(X)) cons(ok(X1),ok(X2)) = 16X1 + 4X2 + 5 >= 16X1 + 4X2 + 1 = ok(cons(X1,X2)) incr(ok(X)) = 4X + 1 >= 4X + 1 = ok(incr(X)) s(ok(X)) = 4X + 1 >= 4X + 1 = ok(s(X)) take(ok(X1),ok(X2)) = 20X1 + 8X2 + 9 >= 20X1 + 8X2 + 9 = ok(take(X1,X2)) zip(ok(X1),ok(X2)) = 4X1 + 4X2 + 2 >= 4X1 + 4X2 + 1 = ok(zip(X1,X2)) pair(ok(X1),ok(X2)) = 16X1 + 4X2 + 5 >= 16X1 + 4X2 + 1 = ok(pair(X1,X2)) tail(ok(X)) = 16X + 4 >= 16X + 1 = ok(tail(X)) repItems(ok(X)) = 4X + 1 >= 4X + 1 = ok(repItems(X)) top(ok(X)) = 4X + 8 >= 4X + 7 = top(active(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(s(X)) -> s(proper(X)) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(tail(X)) -> tail(proper(X)) proper(repItems(X)) -> repItems(proper(X)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) repItems(ok(X)) -> ok(repItems(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = x0, [proper](x0) = 2x0, [repItems](x0) = 2x0 + 2, [tail](x0) = 2x0, [pair](x0, x1) = 4x0 + x1 + 4, [zip](x0, x1) = 4x0 + x1 + 4, [take](x0, x1) = x0 + x1 + 6, [s](x0) = 5x0 + 4, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1 + 4, [incr](x0) = 5x0, [oddNs] = 0, [0] = 0, [active](x0) = x0, [pairNs] = 4 orientation: active(pairNs()) = 4 >= 4 = mark(cons(0(),incr(oddNs()))) active(incr(cons(X,XS))) = 20X + 5XS + 20 >= 20X + 5XS + 20 = mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) = 16X + 4XS + 4Y + YS + 24 >= 16X + 4XS + 4Y + YS + 24 = mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) = 8X + 2XS + 10 >= 8X + 2XS + 10 = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = cons(active(X1),X2) active(incr(X)) = 5X >= 5X = incr(active(X)) active(s(X)) = 5X + 4 >= 5X + 4 = s(active(X)) active(zip(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = zip(active(X1),X2) active(zip(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = zip(X1,active(X2)) active(pair(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = pair(active(X1),X2) active(pair(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = pair(X1,active(X2)) active(tail(X)) = 2X >= 2X = tail(active(X)) active(repItems(X)) = 2X + 2 >= 2X + 2 = repItems(active(X)) cons(mark(X1),X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(cons(X1,X2)) incr(mark(X)) = 5X >= 5X = mark(incr(X)) s(mark(X)) = 5X + 4 >= 5X + 4 = mark(s(X)) take(mark(X1),X2) = X1 + X2 + 6 >= X1 + X2 + 6 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = mark(take(X1,X2)) zip(mark(X1),X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(zip(X1,X2)) zip(X1,mark(X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(zip(X1,X2)) pair(mark(X1),X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(pair(X1,X2)) tail(mark(X)) = 2X >= 2X = mark(tail(X)) repItems(mark(X)) = 2X + 2 >= 2X + 2 = mark(repItems(X)) proper(cons(X1,X2)) = 8X1 + 2X2 + 8 >= 8X1 + 2X2 + 4 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(incr(X)) = 10X >= 10X = incr(proper(X)) proper(s(X)) = 10X + 8 >= 10X + 4 = s(proper(X)) proper(zip(X1,X2)) = 8X1 + 2X2 + 8 >= 8X1 + 2X2 + 4 = zip(proper(X1),proper(X2)) proper(pair(X1,X2)) = 8X1 + 2X2 + 8 >= 8X1 + 2X2 + 4 = pair(proper(X1),proper(X2)) proper(tail(X)) = 4X >= 4X = tail(proper(X)) proper(repItems(X)) = 4X + 4 >= 4X + 2 = repItems(proper(X)) incr(ok(X)) = 5X >= 5X = ok(incr(X)) s(ok(X)) = 5X + 4 >= 5X + 4 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = ok(take(X1,X2)) repItems(ok(X)) = 2X + 2 >= 2X + 2 = ok(repItems(X)) problem: active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(s(X)) -> s(active(X)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) proper(0()) -> ok(0()) proper(incr(X)) -> incr(proper(X)) proper(tail(X)) -> tail(proper(X)) incr(ok(X)) -> ok(incr(X)) s(ok(X)) -> ok(s(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) repItems(ok(X)) -> ok(repItems(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = x0 + 5, [proper](x0) = 4x0, [repItems](x0) = 4x0, [tail](x0) = 2x0 + 4, [pair](x0, x1) = x0 + x1, [zip](x0, x1) = 4x0 + x1, [take](x0, x1) = x0 + 4x1, [s](x0) = 4x0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0 + 4, [oddNs] = 0, [0] = 4, [active](x0) = 4x0, [pairNs] = 6 orientation: active(pairNs()) = 24 >= 8 = mark(cons(0(),incr(oddNs()))) active(incr(cons(X,XS))) = 4X + 4XS + 16 >= 4X + XS + 4 = mark(cons(s(X),incr(XS))) active(zip(cons(X,XS),cons(Y,YS))) = 16X + 16XS + 4Y + 4YS >= X + 4XS + Y + YS = mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) = 16X + 16XS >= 2X + 4XS = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = cons(active(X1),X2) active(incr(X)) = 4X + 16 >= 4X + 4 = incr(active(X)) active(s(X)) = 16X >= 16X = s(active(X)) active(zip(X1,X2)) = 16X1 + 4X2 >= 16X1 + X2 = zip(active(X1),X2) active(zip(X1,X2)) = 16X1 + 4X2 >= 4X1 + 4X2 = zip(X1,active(X2)) active(pair(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = pair(active(X1),X2) active(pair(X1,X2)) = 4X1 + 4X2 >= X1 + 4X2 = pair(X1,active(X2)) active(tail(X)) = 8X + 16 >= 8X + 4 = tail(active(X)) active(repItems(X)) = 16X >= 16X = repItems(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) incr(mark(X)) = X + 4 >= X + 4 = mark(incr(X)) s(mark(X)) = 4X >= 4X = mark(s(X)) take(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 4X2 >= X1 + 4X2 = mark(take(X1,X2)) zip(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(zip(X1,X2)) zip(X1,mark(X2)) = 4X1 + X2 >= 4X1 + X2 = mark(zip(X1,X2)) pair(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(pair(X1,X2)) tail(mark(X)) = 2X + 4 >= 2X + 4 = mark(tail(X)) repItems(mark(X)) = 4X >= 4X = mark(repItems(X)) proper(0()) = 16 >= 9 = ok(0()) proper(incr(X)) = 4X + 16 >= 4X + 4 = incr(proper(X)) proper(tail(X)) = 8X + 16 >= 8X + 4 = tail(proper(X)) incr(ok(X)) = X + 9 >= X + 9 = ok(incr(X)) s(ok(X)) = 4X + 20 >= 4X + 5 = ok(s(X)) take(ok(X1),ok(X2)) = X1 + 4X2 + 25 >= X1 + 4X2 + 5 = ok(take(X1,X2)) repItems(ok(X)) = 4X + 20 >= 4X + 5 = ok(repItems(X)) problem: active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) -> cons(active(X1),X2) active(s(X)) -> s(active(X)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(X1,X2)) -> zip(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) s(mark(X)) -> mark(s(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) tail(mark(X)) -> mark(tail(X)) repItems(mark(X)) -> mark(repItems(X)) incr(ok(X)) -> ok(incr(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0 + 4, [repItems](x0) = 4x0 + 6, [tail](x0) = 4x0, [pair](x0, x1) = 2x0 + x1 + 5, [zip](x0, x1) = x0 + 2x1 + 4, [take](x0, x1) = 6x0 + 4x1 + 2, [s](x0) = x0 + 7, [mark](x0) = x0 + 2, [cons](x0, x1) = x0 + x1 + 1, [incr](x0) = 4x0 + 4, [active](x0) = 2x0 + 2 orientation: active(zip(cons(X,XS),cons(Y,YS))) = 2X + 2XS + 4Y + 4YS + 16 >= 2X + XS + Y + 2YS + 12 = mark(cons(pair(X,Y),zip(XS,YS))) active(repItems(cons(X,XS))) = 8X + 8XS + 22 >= 2X + 4XS + 10 = mark(cons(X,cons(X,repItems(XS)))) active(cons(X1,X2)) = 2X1 + 2X2 + 4 >= 2X1 + X2 + 3 = cons(active(X1),X2) active(s(X)) = 2X + 16 >= 2X + 9 = s(active(X)) active(zip(X1,X2)) = 2X1 + 4X2 + 10 >= 2X1 + 2X2 + 6 = zip(active(X1),X2) active(zip(X1,X2)) = 2X1 + 4X2 + 10 >= X1 + 4X2 + 8 = zip(X1,active(X2)) active(pair(X1,X2)) = 4X1 + 2X2 + 12 >= 4X1 + X2 + 9 = pair(active(X1),X2) active(pair(X1,X2)) = 4X1 + 2X2 + 12 >= 2X1 + 2X2 + 7 = pair(X1,active(X2)) active(repItems(X)) = 8X + 14 >= 8X + 14 = repItems(active(X)) cons(mark(X1),X2) = X1 + X2 + 3 >= X1 + X2 + 3 = mark(cons(X1,X2)) incr(mark(X)) = 4X + 12 >= 4X + 6 = mark(incr(X)) s(mark(X)) = X + 9 >= X + 9 = mark(s(X)) take(mark(X1),X2) = 6X1 + 4X2 + 14 >= 6X1 + 4X2 + 4 = mark(take(X1,X2)) take(X1,mark(X2)) = 6X1 + 4X2 + 10 >= 6X1 + 4X2 + 4 = mark(take(X1,X2)) zip(mark(X1),X2) = X1 + 2X2 + 6 >= X1 + 2X2 + 6 = mark(zip(X1,X2)) zip(X1,mark(X2)) = X1 + 2X2 + 8 >= X1 + 2X2 + 6 = mark(zip(X1,X2)) pair(mark(X1),X2) = 2X1 + X2 + 9 >= 2X1 + X2 + 7 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 2X1 + X2 + 7 >= 2X1 + X2 + 7 = mark(pair(X1,X2)) tail(mark(X)) = 4X + 8 >= 4X + 2 = mark(tail(X)) repItems(mark(X)) = 4X + 14 >= 4X + 8 = mark(repItems(X)) incr(ok(X)) = 16X + 20 >= 16X + 20 = ok(incr(X)) problem: active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) zip(mark(X1),X2) -> mark(zip(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) incr(ok(X)) -> ok(incr(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 3x0 + 1, [repItems](x0) = 4x0 + 2, [pair](x0, x1) = 4x0 + x1 + 3, [zip](x0, x1) = 2x0 + 4x1, [s](x0) = x0 + 3, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [incr](x0) = 7x0 + 3, [active](x0) = 7x0 + 4 orientation: active(repItems(X)) = 28X + 18 >= 28X + 18 = repItems(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) s(mark(X)) = X + 4 >= X + 4 = mark(s(X)) zip(mark(X1),X2) = 2X1 + 4X2 + 2 >= 2X1 + 4X2 + 1 = mark(zip(X1,X2)) pair(X1,mark(X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = mark(pair(X1,X2)) incr(ok(X)) = 21X + 10 >= 21X + 10 = ok(incr(X)) problem: active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) incr(ok(X)) -> ok(incr(X)) Matrix Interpretation Processor: dim=1 interpretation: [ok](x0) = 4x0 + 4, [repItems](x0) = 4x0 + 4, [pair](x0, x1) = x0 + x1, [s](x0) = x0, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [incr](x0) = 4x0 + 3, [active](x0) = x0 orientation: active(repItems(X)) = 4X + 4 >= 4X + 4 = repItems(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) pair(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(pair(X1,X2)) incr(ok(X)) = 16X + 19 >= 16X + 16 = ok(incr(X)) problem: active(repItems(X)) -> repItems(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) Matrix Interpretation Processor: dim=1 interpretation: [repItems](x0) = x0 + 3, [pair](x0, x1) = x0 + 4x1 + 5, [s](x0) = 4x0 + 2, [mark](x0) = x0, [cons](x0, x1) = 4x0 + x1, [active](x0) = 4x0 + 3 orientation: active(repItems(X)) = 4X + 15 >= 4X + 6 = repItems(active(X)) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = mark(cons(X1,X2)) s(mark(X)) = 4X + 2 >= 4X + 2 = mark(s(X)) pair(X1,mark(X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = mark(pair(X1,X2)) problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) Matrix Interpretation Processor: dim=1 interpretation: [pair](x0, x1) = 2x0 + 2x1 + 1, [s](x0) = x0 + 2, [mark](x0) = x0 + 7, [cons](x0, x1) = x0 + 4x1 + 4 orientation: cons(mark(X1),X2) = X1 + 4X2 + 11 >= X1 + 4X2 + 11 = mark(cons(X1,X2)) s(mark(X)) = X + 9 >= X + 9 = mark(s(X)) pair(X1,mark(X2)) = 2X1 + 2X2 + 15 >= 2X1 + 2X2 + 8 = mark(pair(X1,X2)) problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) s(mark(X)) -> mark(s(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [s](x0) = [0 1 0]x0 [0 0 1] , [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] orientation: [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(s(X)) [0 0 1] [1] [0 0 1] [1] problem: Qed