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: [pairNs] = 0, [cons](x0, x1) = 2x0 + x1, [a__tail](x0) = 2x0 + 2, [s](x0) = 2x0, [0] = 0, [incr](x0) = 2x0, [a__repItems](x0) = 3x0, [tail](x0) = 2x0 + 2, [repItems](x0) = 3x0, [a__oddNs] = 0, [nil] = 0, [a__pairNs] = 0, [mark](x0) = x0, [zip](x0, x1) = x0 + x1, [a__zip](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + x1, [pair](x0, x1) = x0 + x1, [a__incr](x0) = 2x0, [oddNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 4X + 2XS >= 4X + 2XS = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS >= 0 = nil() a__take(s(N),cons(X,XS)) = 2N + 2X + XS >= N + 2X + XS = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = XS >= 0 = nil() a__zip(X,nil()) = X >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 2X + XS + 2Y + YS >= 2X + XS + 2Y + YS = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = 4X + 2XS + 2 >= XS = mark(XS) a__repItems(nil()) = 0 >= 0 = nil() a__repItems(cons(X,XS)) = 6X + 3XS >= 4X + 3XS = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = 2X >= 2X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = X1 + X2 >= X1 + X2 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = 2X + 2 >= 2X + 2 = a__tail(mark(X)) mark(repItems(X)) = 3X >= 3X = a__repItems(mark(X)) mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = X1 + X2 >= X1 + X2 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = 2X >= 2X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__zip(X1,X2) = X1 + X2 >= X1 + X2 = zip(X1,X2) a__tail(X) = 2X + 2 >= 2X + 2 = tail(X) a__repItems(X) = 3X >= 3X = 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__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__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) Matrix Interpretation Processor: dim=1 interpretation: [pairNs] = 0, [cons](x0, x1) = x0 + x1, [a__tail](x0) = x0, [s](x0) = x0, [0] = 0, [incr](x0) = x0, [a__repItems](x0) = 2x0, [tail](x0) = x0, [repItems](x0) = 2x0, [a__oddNs] = 0, [nil] = 4, [a__pairNs] = 0, [mark](x0) = x0, [zip](x0, x1) = 2x0 + x1, [a__zip](x0, x1) = 2x0 + x1, [take](x0, x1) = x0 + x1 + 5, [a__take](x0, x1) = x0 + x1 + 5, [pair](x0, x1) = 2x0 + x1, [a__incr](x0) = x0, [oddNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS >= X + XS = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS + 5 >= 4 = nil() a__take(s(N),cons(X,XS)) = N + X + XS + 5 >= N + X + XS + 5 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = XS + 8 >= 4 = nil() a__zip(X,nil()) = 2X + 4 >= 4 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 2X + 2XS + Y + YS >= 2X + 2XS + Y + YS = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(nil()) = 8 >= 4 = nil() a__repItems(cons(X,XS)) = 2X + 2XS >= 2X + 2XS = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = X >= X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(repItems(X)) = 2X >= 2X = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 4 >= 4 = nil() mark(pair(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = X >= X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = X1 + X2 + 5 >= X1 + X2 + 5 = take(X1,X2) a__zip(X1,X2) = 2X1 + X2 >= 2X1 + X2 = zip(X1,X2) a__tail(X) = X >= X = tail(X) a__repItems(X) = 2X >= 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__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(X,nil()) -> nil() 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(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) Matrix Interpretation Processor: dim=1 interpretation: [pairNs] = 0, [cons](x0, x1) = 2x0 + x1, [a__tail](x0) = 4x0, [s](x0) = x0, [0] = 0, [incr](x0) = 2x0, [a__repItems](x0) = 4x0, [tail](x0) = 4x0, [repItems](x0) = 4x0, [a__oddNs] = 0, [nil] = 2, [a__pairNs] = 0, [mark](x0) = 2x0, [zip](x0, x1) = 4x0 + 6x1, [a__zip](x0, x1) = 4x0 + 6x1, [take](x0, x1) = x0 + 4x1 + 4, [a__take](x0, x1) = x0 + 4x1 + 4, [pair](x0, x1) = 2x0 + x1, [a__incr](x0) = 2x0, [oddNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 4X + 2XS >= 4X + 2XS = cons(s(mark(X)),incr(XS)) a__take(s(N),cons(X,XS)) = N + 8X + 4XS + 4 >= N + 4X + 4XS + 4 = cons(mark(X),take(N,XS)) a__zip(X,nil()) = 4X + 12 >= 2 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 8X + 4XS + 12Y + 6YS >= 8X + 4XS + 4Y + 6YS = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) = 8X + 4XS >= 6X + 4XS = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = 4X >= 4X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = 2X1 + 8X2 + 8 >= 2X1 + 8X2 + 4 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 8X1 + 12X2 >= 8X1 + 12X2 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = 8X >= 8X = a__tail(mark(X)) mark(repItems(X)) = 8X >= 8X = a__repItems(mark(X)) mark(cons(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(nil()) = 4 >= 2 = nil() mark(pair(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = 2X >= 2X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = take(X1,X2) a__zip(X1,X2) = 4X1 + 6X2 >= 4X1 + 6X2 = zip(X1,X2) a__tail(X) = 4X >= 4X = tail(X) a__repItems(X) = 4X >= 4X = 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__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,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(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(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) Matrix Interpretation Processor: dim=1 interpretation: [pairNs] = 0, [cons](x0, x1) = x0 + x1, [a__tail](x0) = 4x0 + 3, [s](x0) = x0, [0] = 0, [incr](x0) = 2x0, [a__repItems](x0) = 3x0 + 4, [tail](x0) = 4x0 + 2, [repItems](x0) = 3x0 + 4, [a__oddNs] = 0, [a__pairNs] = 0, [mark](x0) = 2x0, [zip](x0, x1) = 4x0 + 2x1 + 1, [a__zip](x0, x1) = 4x0 + 2x1 + 2, [take](x0, x1) = 4x0 + x1, [a__take](x0, x1) = 4x0 + 2x1, [pair](x0, x1) = 2x0 + x1 + 1, [a__incr](x0) = 2x0, [oddNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 2X + 2XS >= 2X + 2XS = cons(s(mark(X)),incr(XS)) a__take(s(N),cons(X,XS)) = 4N + 2X + 2XS >= 4N + 2X + XS = cons(mark(X),take(N,XS)) a__zip(cons(X,XS),cons(Y,YS)) = 4X + 4XS + 2Y + 2YS + 2 >= 4X + 4XS + 2Y + 2YS + 2 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) = 3X + 3XS + 4 >= 3X + 3XS + 4 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = 4X >= 4X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(zip(X1,X2)) = 8X1 + 4X2 + 2 >= 8X1 + 4X2 + 2 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = 8X + 4 >= 8X + 3 = a__tail(mark(X)) mark(repItems(X)) = 6X + 8 >= 6X + 4 = a__repItems(mark(X)) mark(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(pair(X1,X2)) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 1 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = 2X >= 2X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = 4X1 + 2X2 >= 4X1 + X2 = take(X1,X2) a__zip(X1,X2) = 4X1 + 2X2 + 2 >= 4X1 + 2X2 + 1 = zip(X1,X2) a__tail(X) = 4X + 3 >= 4X + 2 = tail(X) a__repItems(X) = 3X + 4 >= 3X + 4 = 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__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,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(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__repItems(X) -> repItems(X) Matrix Interpretation Processor: dim=1 interpretation: [pairNs] = 0, [cons](x0, x1) = x0 + x1, [s](x0) = x0, [0] = 0, [incr](x0) = 4x0, [a__repItems](x0) = 6x0 + 6, [repItems](x0) = 4x0, [a__oddNs] = 0, [a__pairNs] = 0, [mark](x0) = 4x0, [zip](x0, x1) = 4x0 + 4x1 + 1, [a__zip](x0, x1) = 4x0 + 4x1 + 4, [take](x0, x1) = x0 + 4x1 + 1, [a__take](x0, x1) = 4x0 + 5x1 + 4, [pair](x0, x1) = x0 + x1 + 3, [a__incr](x0) = 4x0, [oddNs] = 0 orientation: a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 4X + 4XS >= 4X + 4XS = cons(s(mark(X)),incr(XS)) a__take(s(N),cons(X,XS)) = 4N + 5X + 5XS + 4 >= N + 4X + 4XS + 1 = cons(mark(X),take(N,XS)) a__zip(cons(X,XS),cons(Y,YS)) = 4X + 4XS + 4Y + 4YS + 4 >= 4X + 4XS + 4Y + 4YS + 4 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__repItems(cons(X,XS)) = 6X + 6XS + 6 >= 5X + 4XS = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = 16X >= 16X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(zip(X1,X2)) = 16X1 + 16X2 + 4 >= 16X1 + 16X2 + 4 = a__zip(mark(X1),mark(X2)) mark(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 4X >= 4X = s(mark(X)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = 4X >= 4X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = 4X1 + 5X2 + 4 >= X1 + 4X2 + 1 = take(X1,X2) a__repItems(X) = 6X + 6 >= 4X = 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)) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() Matrix Interpretation Processor: dim=3 interpretation: [0] [pairNs] = [1] [0], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 + [1] [0 1 1] [0 0 1] [0], [1 0 1] [s](x0) = [0 1 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [1] [a__oddNs] = [1] [0], [1] [a__pairNs] = [1] [0], [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [1 0 1] [zip](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 1] [1 0 1] [a__zip](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [pair](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 1] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [0] [oddNs] = [1] [0] orientation: [1] [0] a__pairNs() = [1] >= [1] = cons(0(),incr(oddNs())) [0] [0] [1] [1] a__oddNs() = [1] >= [1] = a__incr(a__pairNs()) [0] [0] [1 1 1] [1 0 1] [0] [1 1 1] [1 0 1] [0] a__incr(cons(X,XS)) = [0 1 1]X + [0 0 0]XS + [1] >= [0 1 0]X + [0 0 0]XS + [1] = cons(s(mark(X)),incr(XS)) [0 1 1] [0 0 1] [0] [0 1 0] [0 0 1] [0] [1 1 1] [1 0 1] [1 1 1] [1 0 1] [0] [1 1 0] [1 0 1] [1 1 0] [1 0 1] [0] a__zip(cons(X,XS),cons(Y,YS)) = [0 1 1]X + [0 0 0]XS + [0 0 0]Y + [0 0 0]YS + [1] >= [0 0 0]X + [0 0 0]XS + [0 0 0]Y + [0 0 0]YS + [1] = cons(pair(mark(X),mark(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 0] [0] [1] [1] mark(pairNs()) = [1] >= [1] = a__pairNs() [0] [0] [1 1 1] [1 1 1] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = a__incr(mark(X)) [0 0 1] [0 0 1] [1] [1] mark(oddNs()) = [1] >= [1] = a__oddNs() [0] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] mark(zip(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = a__zip(mark(X1),mark(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1] [1 1 0] [1 0 0] [0] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = cons(mark(X1),X2) [0 1 1] [0 0 1] [0] [0 1 1] [0 0 1] [0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 1] [1 1 1] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [1] [0] a__pairNs() = [1] >= [1] = pairNs() [0] [0] [1 0 1] [1 0 1] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [1] [0] a__oddNs() = [1] >= [1] = oddNs() [0] [0] problem: 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)) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=1 interpretation: [pairNs] = 4, [cons](x0, x1) = x0 + 3x1 + 2, [s](x0) = x0, [0] = 0, [incr](x0) = x0, [a__oddNs] = 1, [a__pairNs] = 1, [mark](x0) = x0, [zip](x0, x1) = 5x0 + 4x1 + 7, [a__zip](x0, x1) = 5x0 + 4x1 + 6, [pair](x0, x1) = x0 + x1 + 1, [a__incr](x0) = x0, [oddNs] = 2 orientation: a__oddNs() = 1 >= 1 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + 3XS + 2 >= X + 3XS + 2 = cons(s(mark(X)),incr(XS)) a__zip(cons(X,XS),cons(Y,YS)) = 5X + 15XS + 4Y + 12YS + 24 >= X + 15XS + Y + 12YS + 24 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) mark(pairNs()) = 4 >= 1 = a__pairNs() mark(incr(X)) = X >= X = a__incr(mark(X)) mark(oddNs()) = 2 >= 1 = a__oddNs() mark(zip(X1,X2)) = 5X1 + 4X2 + 7 >= 5X1 + 4X2 + 6 = a__zip(mark(X1),mark(X2)) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) a__incr(X) = X >= X = incr(X) problem: 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)) mark(incr(X)) -> a__incr(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 0] [s](x0) = [0 0 0]x0 [0 0 1] , [0] [0] = [0] [0], [1 0 0] [incr](x0) = [0 1 0]x0 [0 1 1] , [1] [a__oddNs] = [1] [1], [0] [a__pairNs] = [1] [0], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [zip](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [a__zip](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [pair](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [a__incr](x0) = [0 1 0]x0 [0 1 1] orientation: [1] [0] a__oddNs() = [1] >= [1] = a__incr(a__pairNs()) [1] [1] [1 1 1] [1 0 0] [1 0 1] [1 0 0] a__incr(cons(X,XS)) = [0 1 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = cons(s(mark(X)),incr(XS)) [0 1 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] a__zip(cons(X,XS),cons(Y,YS)) = [0 0 0]X + [0 0 0]XS + [0 0 0]Y + [0 0 0]YS + [0] >= [0 0 0]X + [0 0 0]XS + [0 0 0]Y + [0 0 0]YS = cons(pair(mark(X),mark(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 0] [1 1 1] [1 0 1] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = a__incr(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 1] [1 0 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 1 1] [0 1 1] problem: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(incr(X)) -> a__incr(mark(X)) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [1 0 1] [cons](x0, x1) = [0 0 1]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 1 0] [s](x0) = [0 0 0]x0 [0 0 1] , [0] [0] = [1] [1], [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [0] [0 1 1] [1], [1 0 1] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 1 0] [1] [a__incr](x0) = [0 1 0]x0 + [0] [0 1 1] [1] orientation: [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] a__incr(cons(X,XS)) = [0 0 1]X + [0 1 0]XS + [0] >= [0 0 1]X + [0 1 0]XS + [0] = cons(s(mark(X)),incr(XS)) [0 0 1] [0 1 0] [1] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] mark(incr(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__incr(mark(X)) [0 1 1] [1] [0 1 1] [1] [1] [0] mark(0()) = [1] >= [1] = 0() [1] [1] [1 1 1] [1 1 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 1] [0 0 1] [1 1 0] [1] [1 0 0] [0] a__incr(X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = incr(X) [0 1 1] [1] [0 1 1] [1] problem: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(incr(X)) -> a__incr(mark(X)) mark(s(X)) -> s(mark(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = 4x0 + x1 + 6, [s](x0) = x0 + 1, [incr](x0) = 2x0 + 3, [mark](x0) = 2x0 + 1, [a__incr](x0) = 2x0 + 5 orientation: a__incr(cons(X,XS)) = 8X + 2XS + 17 >= 8X + 2XS + 17 = cons(s(mark(X)),incr(XS)) mark(incr(X)) = 4X + 7 >= 4X + 7 = a__incr(mark(X)) mark(s(X)) = 2X + 3 >= 2X + 2 = s(mark(X)) problem: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) mark(incr(X)) -> a__incr(mark(X)) Matrix Interpretation Processor: dim=1 interpretation: [cons](x0, x1) = x0 + x1 + 2, [s](x0) = x0 + 4, [incr](x0) = 4x0 + 1, [mark](x0) = x0, [a__incr](x0) = 4x0 orientation: a__incr(cons(X,XS)) = 4X + 4XS + 8 >= X + 4XS + 7 = cons(s(mark(X)),incr(XS)) mark(incr(X)) = 4X + 1 >= 4X = a__incr(mark(X)) problem: Qed