NO Problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) zip(nil(),XS) -> nil() zip(X,nil()) -> nil() zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) tail(cons(X,XS)) -> activate(XS) repItems(nil()) -> nil() repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) repItems(X) -> n__repItems(X) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__repItems(X)) -> repItems(X) activate(X) -> X Proof: Matrix Interpretation Processor: dim=1 interpretation: [n__cons](x0, x1) = x0 + x1, [n__repItems](x0) = 2x0 + 2, [repItems](x0) = 2x0 + 2, [tail](x0) = x0 + 4, [n__zip](x0, x1) = x0 + x1 + 2, [pair](x0, x1) = x0 + x1, [zip](x0, x1) = x0 + x1 + 2, [n__take](x0, x1) = 2x0 + 4x1, [nil] = 0, [take](x0, x1) = 2x0 + 4x1, [activate](x0) = x0, [s](x0) = x0, [incr](x0) = 4x0, [cons](x0, x1) = x0 + x1, [n__incr](x0) = 4x0, [oddNs] = 0, [0] = 0, [pairNs] = 0 orientation: pairNs() = 0 >= 0 = cons(0(),n__incr(oddNs())) oddNs() = 0 >= 0 = incr(pairNs()) incr(cons(X,XS)) = 4X + 4XS >= X + 4XS = cons(s(X),n__incr(activate(XS))) take(0(),XS) = 4XS >= 0 = nil() take(s(N),cons(X,XS)) = 2N + 4X + 4XS >= 2N + X + 4XS = cons(X,n__take(N,activate(XS))) zip(nil(),XS) = XS + 2 >= 0 = nil() zip(X,nil()) = X + 2 >= 0 = nil() zip(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS + 2 >= X + XS + Y + YS + 2 = cons(pair(X,Y),n__zip(activate(XS),activate(YS))) tail(cons(X,XS)) = X + XS + 4 >= XS = activate(XS) repItems(nil()) = 2 >= 0 = nil() repItems(cons(X,XS)) = 2X + 2XS + 2 >= 2X + 2XS + 2 = cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) = 4X >= 4X = n__incr(X) take(X1,X2) = 2X1 + 4X2 >= 2X1 + 4X2 = n__take(X1,X2) zip(X1,X2) = X1 + X2 + 2 >= X1 + X2 + 2 = n__zip(X1,X2) cons(X1,X2) = X1 + X2 >= X1 + X2 = n__cons(X1,X2) repItems(X) = 2X + 2 >= 2X + 2 = n__repItems(X) activate(n__incr(X)) = 4X >= 4X = incr(X) activate(n__take(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = take(X1,X2) activate(n__zip(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = zip(X1,X2) activate(n__cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) activate(n__repItems(X)) = 2X + 2 >= 2X + 2 = repItems(X) activate(X) = X >= X = X problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) repItems(X) -> n__repItems(X) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__repItems(X)) -> repItems(X) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [n__cons](x0, x1) = 5x0 + x1, [n__repItems](x0) = 2x0 + 1, [repItems](x0) = 4x0 + 2, [n__zip](x0, x1) = x0 + x1, [pair](x0, x1) = x0 + 2x1, [zip](x0, x1) = 2x0 + 2x1, [n__take](x0, x1) = x0 + x1, [nil] = 0, [take](x0, x1) = x0 + 2x1, [activate](x0) = 2x0, [s](x0) = 4x0, [incr](x0) = 6x0, [cons](x0, x1) = 5x0 + 2x1, [n__incr](x0) = 3x0, [oddNs] = 0, [0] = 0, [pairNs] = 0 orientation: pairNs() = 0 >= 0 = cons(0(),n__incr(oddNs())) oddNs() = 0 >= 0 = incr(pairNs()) incr(cons(X,XS)) = 30X + 12XS >= 20X + 12XS = cons(s(X),n__incr(activate(XS))) take(0(),XS) = 2XS >= 0 = nil() take(s(N),cons(X,XS)) = 4N + 10X + 4XS >= 2N + 5X + 4XS = cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) = 10X + 4XS + 10Y + 4YS >= 5X + 4XS + 10Y + 4YS = cons(pair(X,Y),n__zip(activate(XS),activate(YS))) repItems(cons(X,XS)) = 20X + 8XS + 2 >= 15X + 8XS + 2 = cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) = 6X >= 3X = n__incr(X) take(X1,X2) = X1 + 2X2 >= X1 + X2 = n__take(X1,X2) zip(X1,X2) = 2X1 + 2X2 >= X1 + X2 = n__zip(X1,X2) cons(X1,X2) = 5X1 + 2X2 >= 5X1 + X2 = n__cons(X1,X2) repItems(X) = 4X + 2 >= 2X + 1 = n__repItems(X) activate(n__incr(X)) = 6X >= 6X = incr(X) activate(n__take(X1,X2)) = 2X1 + 2X2 >= X1 + 2X2 = take(X1,X2) activate(n__zip(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = zip(X1,X2) activate(n__cons(X1,X2)) = 10X1 + 2X2 >= 5X1 + 2X2 = cons(X1,X2) activate(n__repItems(X)) = 4X + 2 >= 4X + 2 = repItems(X) activate(X) = 2X >= X = X problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(n__repItems(X)) -> repItems(X) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [n__cons](x0, x1) = x0 + x1, [n__repItems](x0) = x0 + 5, [repItems](x0) = 4x0 + 5, [n__zip](x0, x1) = x0 + x1, [pair](x0, x1) = x0 + 4x1, [zip](x0, x1) = 4x0 + 4x1, [n__take](x0, x1) = x0 + x1, [nil] = 0, [take](x0, x1) = 2x0 + 4x1, [activate](x0) = 4x0, [s](x0) = 4x0, [incr](x0) = 4x0, [cons](x0, x1) = x0 + x1, [n__incr](x0) = x0, [oddNs] = 0, [0] = 0, [pairNs] = 0 orientation: pairNs() = 0 >= 0 = cons(0(),n__incr(oddNs())) oddNs() = 0 >= 0 = incr(pairNs()) incr(cons(X,XS)) = 4X + 4XS >= 4X + 4XS = cons(s(X),n__incr(activate(XS))) take(0(),XS) = 4XS >= 0 = nil() take(s(N),cons(X,XS)) = 8N + 4X + 4XS >= N + X + 4XS = cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) = 4X + 4XS + 4Y + 4YS >= X + 4XS + 4Y + 4YS = cons(pair(X,Y),n__zip(activate(XS),activate(YS))) repItems(cons(X,XS)) = 4X + 4XS + 5 >= 2X + 4XS + 5 = cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) = 4X >= X = n__incr(X) take(X1,X2) = 2X1 + 4X2 >= X1 + X2 = n__take(X1,X2) zip(X1,X2) = 4X1 + 4X2 >= X1 + X2 = n__zip(X1,X2) cons(X1,X2) = X1 + X2 >= X1 + X2 = n__cons(X1,X2) activate(n__incr(X)) = 4X >= 4X = incr(X) activate(n__take(X1,X2)) = 4X1 + 4X2 >= 2X1 + 4X2 = take(X1,X2) activate(n__zip(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = zip(X1,X2) activate(n__cons(X1,X2)) = 4X1 + 4X2 >= X1 + X2 = cons(X1,X2) activate(n__repItems(X)) = 4X + 20 >= 4X + 5 = repItems(X) activate(X) = 4X >= X = X problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [n__cons](x0, x1) = 2x0 + 2x1, [n__repItems](x0) = x0, [repItems](x0) = 4x0 + 4, [n__zip](x0, x1) = x0 + x1, [pair](x0, x1) = x0 + 2x1, [zip](x0, x1) = 2x0 + 2x1, [n__take](x0, x1) = x0 + x1, [nil] = 0, [take](x0, x1) = 2x0 + 2x1, [activate](x0) = 2x0, [s](x0) = 2x0, [incr](x0) = 2x0, [cons](x0, x1) = 4x0 + 4x1, [n__incr](x0) = x0, [oddNs] = 0, [0] = 0, [pairNs] = 0 orientation: pairNs() = 0 >= 0 = cons(0(),n__incr(oddNs())) oddNs() = 0 >= 0 = incr(pairNs()) incr(cons(X,XS)) = 8X + 8XS >= 8X + 8XS = cons(s(X),n__incr(activate(XS))) take(0(),XS) = 2XS >= 0 = nil() take(s(N),cons(X,XS)) = 4N + 8X + 8XS >= 4N + 4X + 8XS = cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) = 8X + 8XS + 8Y + 8YS >= 4X + 8XS + 8Y + 8YS = cons(pair(X,Y),n__zip(activate(XS),activate(YS))) repItems(cons(X,XS)) = 16X + 16XS + 4 >= 12X + 16XS = cons(X,n__cons(X,n__repItems(activate(XS)))) incr(X) = 2X >= X = n__incr(X) take(X1,X2) = 2X1 + 2X2 >= X1 + X2 = n__take(X1,X2) zip(X1,X2) = 2X1 + 2X2 >= X1 + X2 = n__zip(X1,X2) cons(X1,X2) = 4X1 + 4X2 >= 2X1 + 2X2 = n__cons(X1,X2) activate(n__incr(X)) = 2X >= 2X = incr(X) activate(n__take(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = take(X1,X2) activate(n__zip(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = zip(X1,X2) activate(n__cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(X1,X2) activate(X) = 2X >= X = X problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [n__cons](x0, x1) = [0 1 0]x0 + [0 0 1]x1 [0 0 0] [0 1 0] , [1 0 0] [1 0 0] [0] [n__zip](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [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] [1 0 0] [0] [zip](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 1] [1 0 0] [n__take](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [0 1 1] , [0] [nil] = [0] [0], [1 0 1] [1 0 0] [take](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [0 1 1] , [activate](x0) = x0 , [1 1 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 1] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 1]x1 [0 0 0] [0 1 0] , [1 1 1] [n__incr](x0) = [0 0 0]x0 [0 0 0] , [0] [oddNs] = [0] [0], [0] [0] = [0] [0], [0] [pairNs] = [0] [0] orientation: [0] [0] pairNs() = [0] >= [0] = cons(0(),n__incr(oddNs())) [0] [0] [0] [0] oddNs() = [0] >= [0] = incr(pairNs()) [0] [0] [1 1 0] [1 1 1] [1 1 0] [1 1 1] incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = cons(s(X),n__incr(activate(XS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] take(0(),XS) = [0 1 1]XS >= [0] = nil() [0 1 1] [0] [1 1 1] [1 0 0] [1 0 0] [1] [1 0 1] [1 0 0] [1 0 0] take(s(N),cons(X,XS)) = [0 0 0]N + [0 1 0]X + [0 1 1]XS + [0] >= [0 0 0]N + [0 1 0]X + [0 1 1]XS = cons(X,n__take(N,activate(XS))) [0 0 0] [0 1 0] [0 1 1] [0] [0 0 0] [0 0 0] [0 1 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [0] zip(cons(X,XS),cons(Y,YS)) = [0 0 0]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(X,Y),n__zip(activate(XS),activate(YS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] [1 1 1] [1 1 1] incr(X) = [0 0 0]X >= [0 0 0]X = n__incr(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] take(X1,X2) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = n__take(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] zip(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = n__zip(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,X2) = [0 1 0]X1 + [0 0 1]X2 >= [0 1 0]X1 + [0 0 1]X2 = n__cons(X1,X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 1 1] [1 1 1] activate(n__incr(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] activate(n__take(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = take(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] activate(n__zip(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = zip(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] activate(n__cons(X1,X2)) = [0 1 0]X1 + [0 0 1]X2 >= [0 1 0]X1 + [0 0 1]X2 = cons(X1,X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] activate(X) = X >= X = X problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) take(0(),XS) -> nil() zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=1 interpretation: [n__cons](x0, x1) = x0 + x1, [n__zip](x0, x1) = x0 + x1, [pair](x0, x1) = x0 + x1, [zip](x0, x1) = x0 + x1, [n__take](x0, x1) = x0 + 2x1 + 2, [nil] = 0, [take](x0, x1) = x0 + 2x1 + 2, [activate](x0) = x0, [s](x0) = x0, [incr](x0) = 2x0, [cons](x0, x1) = x0 + x1, [n__incr](x0) = 2x0, [oddNs] = 0, [0] = 0, [pairNs] = 0 orientation: pairNs() = 0 >= 0 = cons(0(),n__incr(oddNs())) oddNs() = 0 >= 0 = incr(pairNs()) incr(cons(X,XS)) = 2X + 2XS >= X + 2XS = cons(s(X),n__incr(activate(XS))) take(0(),XS) = 2XS + 2 >= 0 = nil() zip(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS >= X + XS + Y + YS = cons(pair(X,Y),n__zip(activate(XS),activate(YS))) incr(X) = 2X >= 2X = n__incr(X) take(X1,X2) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = n__take(X1,X2) zip(X1,X2) = X1 + X2 >= X1 + X2 = n__zip(X1,X2) cons(X1,X2) = X1 + X2 >= X1 + X2 = n__cons(X1,X2) activate(n__incr(X)) = 2X >= 2X = incr(X) activate(n__take(X1,X2)) = X1 + 2X2 + 2 >= X1 + 2X2 + 2 = take(X1,X2) activate(n__zip(X1,X2)) = X1 + X2 >= X1 + X2 = zip(X1,X2) activate(n__cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) activate(X) = X >= X = X problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) zip(X1,X2) -> n__zip(X1,X2) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [n__cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [0 0 0] [0 1 0] , [1 0 0] [1 0 0] [0] [n__zip](x0, x1) = [0 0 1]x0 + [0 0 1]x1 + [1] [0 1 0] [0 1 0] [1], [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] [1 0 1] [1] [zip](x0, x1) = [0 0 1]x0 + [0 0 1]x1 + [1] [0 1 0] [0 1 0] [1], [1 0 0] [1 0 0] [0] [n__take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1 0 0] [0] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [activate](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [incr](x0) = [0 1 1]x0 [0 1 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [0 0 0] [0 1 0] , [1 0 0] [n__incr](x0) = [0 1 1]x0 [0 1 1] , [0] [oddNs] = [0] [0], [0] [0] = [0] [0], [0] [pairNs] = [0] [0] orientation: [0] [0] pairNs() = [0] >= [0] = cons(0(),n__incr(oddNs())) [0] [0] [0] [0] oddNs() = [0] >= [0] = incr(pairNs()) [0] [0] [1 0 0] [1 1 1] [1 0 0] [1 1 0] incr(cons(X,XS)) = [0 0 0]X + [0 1 1]XS >= [0 0 0]X + [0 1 1]XS = cons(s(X),n__incr(activate(XS))) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 0 0] [1 1 0] [0] zip(cons(X,XS),cons(Y,YS)) = [0 0 0]X + [0 1 0]XS + [0 0 0]Y + [0 1 0]YS + [1] >= [0 0 0]X + [0 1 0]XS + [0 0 0]Y + [0 1 0]YS + [1] = cons(pair(X,Y),n__zip(activate(XS),activate(YS))) [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [0 0 0] [0 0 1] [1] [1 1 1] [1 0 0] incr(X) = [0 1 1]X >= [0 1 1]X = n__incr(X) [0 1 1] [0 1 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] take(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = n__take(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 0] [0] zip(X1,X2) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [1] = n__zip(X1,X2) [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,X2) = [0 0 0]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 1]X2 = n__cons(X1,X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 1 1] [1 1 1] activate(n__incr(X)) = [0 1 1]X >= [0 1 1]X = incr(X) [0 1 1] [0 1 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] activate(n__take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] activate(n__zip(X1,X2)) = [0 0 1]X1 + [0 0 1]X2 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [1] = zip(X1,X2) [0 1 0] [0 1 0] [1] [0 1 0] [0 1 0] [1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] activate(n__cons(X1,X2)) = [0 0 0]X1 + [0 0 1]X2 >= [0 0 0]X1 + [0 0 1]X2 = cons(X1,X2) [0 0 0] [0 1 0] [0 0 0] [0 1 0] [1 1 0] activate(X) = [0 1 0]X >= X = X [0 0 1] problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) incr(X) -> n__incr(X) take(X1,X2) -> n__take(X1,X2) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(n__cons(X1,X2)) -> cons(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [0] [n__cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [1], [1 1 0] [1 0 0] [n__zip](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [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] [0] [n__take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1 0 0] [1] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 1] [activate](x0) = [1 1 1]x0 [0 1 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [incr](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [1 0 1] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [1], [1 0 0] [0] [n__incr](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [oddNs] = [1] [1], [0] [0] = [0] [0], [0] [pairNs] = [1] [1] orientation: [0] [0] pairNs() = [1] >= [0] = cons(0(),n__incr(oddNs())) [1] [1] [0] [0] oddNs() = [1] >= [1] = incr(pairNs()) [1] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS + [1] >= [0 0 0]X + [0 0 0]XS + [0] = cons(s(X),n__incr(activate(XS))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] incr(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = n__incr(X) [0 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [0] take(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = n__take(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] cons(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = n__cons(X1,X2) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [1 0 0] [0] [1 0 0] [0] activate(n__incr(X)) = [1 0 0]X + [1] >= [0 0 0]X + [1] = incr(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] activate(n__take(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [1 0 0] [1 0 0] activate(n__zip(X1,X2)) = [1 1 0]X1 + [1 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = zip(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [0] activate(n__cons(X1,X2)) = [1 0 0]X1 + [1 0 1]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [1 0 1] activate(X) = [1 1 1]X >= X = X [0 1 1] problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) incr(X) -> n__incr(X) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zip(X1,X2)) -> zip(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [n__cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [n__zip](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [zip](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [n__take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [activate](x0) = [0 1 0]x0 [0 1 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 1 0]x0 [1 0 1] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 1 0] [0], [1 0 0] [n__incr](x0) = [0 1 0]x0 [1 0 1] , [0] [oddNs] = [1] [1], [0] [0] = [0] [0], [0] [pairNs] = [1] [1] orientation: [0] [0] pairNs() = [1] >= [1] = cons(0(),n__incr(oddNs())) [1] [1] [0] [0] oddNs() = [1] >= [1] = incr(pairNs()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] incr(cons(X,XS)) = [0 0 0]X + [0 0 0]XS + [1] >= [0 0 0]X + [0 0 0]XS + [1] = cons(s(X),n__incr(activate(XS))) [1 0 0] [1 1 0] [0] [0 0 0] [0 1 0] [0] [1 0 0] [1 0 0] incr(X) = [0 1 0]X >= [0 1 0]X = n__incr(X) [1 0 1] [1 0 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] cons(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = n__cons(X1,X2) [0 0 0] [0 1 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] activate(n__incr(X)) = [0 1 0]X >= [0 1 0]X = incr(X) [1 1 1] [1 0 1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] activate(n__take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = take(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] activate(n__zip(X1,X2)) = [0 0 0]X1 + [0 0 1]X2 + [0] >= [0 0 0]X1 + [0 0 1]X2 = zip(X1,X2) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 0] [1 0 0] activate(X) = [0 1 0]X >= X = X [0 1 1] problem: pairNs() -> cons(0(),n__incr(oddNs())) oddNs() -> incr(pairNs()) incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS))) incr(X) -> n__incr(X) cons(X1,X2) -> n__cons(X1,X2) activate(n__incr(X)) -> incr(X) activate(X) -> X Unfolding Processor: loop length: 2 terms: pairNs() cons(0(),n__incr(oddNs())) context: cons(0(),n__incr(incr([]))) substitution: Qed