YES Problem: a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) a__tl(cons(X,Y)) -> mark(Y) mark(nats()) -> a__nats() mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(tl(X)) -> a__tl(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [zeros] = 0, [tl](x0) = 4x0 + 2, [incr](x0) = x0, [a__zeros] = 0, [0] = 0, [cons](x0, x1) = 2x0 + 2x1, [a__hd](x0) = 2x0, [a__nats] = 1, [s](x0) = x0, [hd](x0) = 2x0, [a__tl](x0) = 4x0 + 2, [mark](x0) = 2x0, [adx](x0) = x0, [nats] = 1, [a__incr](x0) = x0, [a__adx](x0) = x0 orientation: a__nats() = 1 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__incr(cons(X,Y)) = 2X + 2Y >= 2X + 2Y = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 2X + 2Y >= 2X + 2Y = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = 4X + 4Y >= 2X = mark(X) a__tl(cons(X,Y)) = 8X + 8Y + 2 >= 2Y = mark(Y) mark(nats()) = 2 >= 1 = a__nats() mark(adx(X)) = 2X >= 2X = a__adx(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(incr(X)) = 2X >= 2X = a__incr(mark(X)) mark(hd(X)) = 4X >= 4X = a__hd(mark(X)) mark(tl(X)) = 8X + 4 >= 8X + 2 = a__tl(mark(X)) mark(cons(X1,X2)) = 4X1 + 4X2 >= 2X1 + 2X2 = cons(X1,X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = 2X >= X = s(X) a__nats() = 1 >= 1 = nats() a__adx(X) = X >= X = adx(X) a__zeros() = 0 >= 0 = zeros() a__incr(X) = X >= X = incr(X) a__hd(X) = 2X >= 2X = hd(X) a__tl(X) = 4X + 2 >= 4X + 2 = tl(X) problem: a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__nats() -> nats() a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) a__tl(X) -> tl(X) Matrix Interpretation Processor: dim=1 interpretation: [zeros] = 0, [tl](x0) = 2x0, [incr](x0) = x0, [a__zeros] = 0, [0] = 0, [cons](x0, x1) = 2x0 + x1, [a__hd](x0) = 2x0 + 3, [a__nats] = 1, [s](x0) = x0, [hd](x0) = 2x0 + 3, [a__tl](x0) = 4x0 + 2, [mark](x0) = x0, [adx](x0) = 2x0 + 1, [nats] = 0, [a__incr](x0) = x0, [a__adx](x0) = 2x0 + 1 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__incr(cons(X,Y)) = 2X + Y >= 2X + Y = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 4X + 2Y + 1 >= 2X + 2Y + 1 = a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) = 4X + 2Y + 3 >= X = mark(X) mark(adx(X)) = 2X + 1 >= 2X + 1 = a__adx(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(incr(X)) = X >= X = a__incr(mark(X)) mark(hd(X)) = 2X + 3 >= 2X + 3 = a__hd(mark(X)) mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(X1,X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(X) a__nats() = 1 >= 0 = nats() a__adx(X) = 2X + 1 >= 2X + 1 = adx(X) a__zeros() = 0 >= 0 = zeros() a__incr(X) = X >= X = incr(X) a__hd(X) = 2X + 3 >= 2X + 3 = hd(X) a__tl(X) = 4X + 2 >= 2X = tl(X) problem: a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(zeros()) -> a__zeros() mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) mark(cons(X1,X2)) -> cons(X1,X2) mark(0()) -> 0() mark(s(X)) -> s(X) a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=1 interpretation: [zeros] = 0, [incr](x0) = x0, [a__zeros] = 0, [0] = 0, [cons](x0, x1) = 4x0 + x1, [a__hd](x0) = x0, [s](x0) = x0, [hd](x0) = x0, [mark](x0) = x0 + 2, [adx](x0) = x0, [a__incr](x0) = x0, [a__adx](x0) = x0 orientation: a__zeros() = 0 >= 0 = cons(0(),zeros()) a__incr(cons(X,Y)) = 4X + Y >= 4X + Y = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 4X + Y >= 4X + Y = a__incr(cons(X,adx(Y))) mark(adx(X)) = X + 2 >= X + 2 = a__adx(mark(X)) mark(zeros()) = 2 >= 0 = a__zeros() mark(incr(X)) = X + 2 >= X + 2 = a__incr(mark(X)) mark(hd(X)) = X + 2 >= X + 2 = a__hd(mark(X)) mark(cons(X1,X2)) = 4X1 + X2 + 2 >= 4X1 + X2 = cons(X1,X2) mark(0()) = 2 >= 0 = 0() mark(s(X)) = X + 2 >= X = s(X) a__adx(X) = X >= X = adx(X) a__zeros() = 0 >= 0 = zeros() a__incr(X) = X >= X = incr(X) a__hd(X) = X >= X = hd(X) problem: a__zeros() -> cons(0(),zeros()) a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) a__adx(X) -> adx(X) a__zeros() -> zeros() a__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=1 interpretation: [zeros] = 0, [incr](x0) = x0, [a__zeros] = 6, [0] = 2, [cons](x0, x1) = x0 + x1, [a__hd](x0) = 2x0, [s](x0) = x0, [hd](x0) = 2x0, [mark](x0) = 2x0, [adx](x0) = 2x0, [a__incr](x0) = x0, [a__adx](x0) = 2x0 orientation: a__zeros() = 6 >= 2 = cons(0(),zeros()) a__incr(cons(X,Y)) = X + Y >= X + Y = cons(s(X),incr(Y)) a__adx(cons(X,Y)) = 2X + 2Y >= X + 2Y = a__incr(cons(X,adx(Y))) mark(adx(X)) = 4X >= 4X = a__adx(mark(X)) mark(incr(X)) = 2X >= 2X = a__incr(mark(X)) mark(hd(X)) = 4X >= 4X = a__hd(mark(X)) a__adx(X) = 2X >= 2X = adx(X) a__zeros() = 6 >= 0 = zeros() a__incr(X) = X >= X = incr(X) a__hd(X) = 2X >= 2X = hd(X) problem: a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) a__adx(X) -> adx(X) a__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [incr](x0) = [0 1 0]x0 [0 1 0] , [1 0 0] [1 0 1] [0] [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [1] [0 0 1] [0 0 0] [0], [1 1 0] [a__hd](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [hd](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [1] [mark](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [1 1 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [a__incr](x0) = [0 1 0]x0 [0 1 1] , [1 1 0] [a__adx](x0) = [0 1 0]x0 [0 1 1] orientation: [1 0 0] [1 1 1] [1] [1 0 0] [1 1 0] [0] a__incr(cons(X,Y)) = [0 0 0]X + [0 1 0]Y + [1] >= [0 0 0]X + [0 1 0]Y + [1] = cons(s(X),incr(Y)) [0 0 1] [0 1 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 1] [1] [1 0 0] [1 1 0] [1] a__adx(cons(X,Y)) = [0 0 0]X + [0 1 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = a__incr(cons(X,adx(Y))) [0 0 1] [0 1 0] [1] [0 0 1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] mark(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__adx(mark(X)) [1 1 0] [0] [1 0 0] [0] [1 1 0] [1] [1 1 0] [1] mark(incr(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__incr(mark(X)) [1 0 0] [0] [1 0 0] [0] [1 1 0] [1] [1 1 0] [1] mark(hd(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__hd(mark(X)) [1 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] a__adx(X) = [0 1 0]X >= [0 0 0]X = adx(X) [0 1 1] [0 0 0] [1 1 0] [1 0 0] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 1 1] [0 1 0] [1 1 0] [1 0 0] a__hd(X) = [0 1 0]X >= [0 1 0]X = hd(X) [0 0 0] [0 0 0] problem: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) a__adx(X) -> adx(X) a__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=1 interpretation: [incr](x0) = x0, [cons](x0, x1) = x0 + 4x1, [a__hd](x0) = 4x0 + 4, [hd](x0) = 4x0 + 4, [mark](x0) = 4x0 + 4, [adx](x0) = x0 + 1, [a__incr](x0) = x0, [a__adx](x0) = x0 + 4 orientation: a__adx(cons(X,Y)) = X + 4Y + 4 >= X + 4Y + 4 = a__incr(cons(X,adx(Y))) mark(adx(X)) = 4X + 8 >= 4X + 8 = a__adx(mark(X)) mark(incr(X)) = 4X + 4 >= 4X + 4 = a__incr(mark(X)) mark(hd(X)) = 16X + 20 >= 16X + 20 = a__hd(mark(X)) a__adx(X) = X + 4 >= X + 1 = adx(X) a__incr(X) = X >= X = incr(X) a__hd(X) = 4X + 4 >= 4X + 4 = hd(X) problem: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) a__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 1] [1 0 0] [1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [0] [a__hd](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [hd](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 1 0] [0] [mark](x0) = [1 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [adx](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [0] [a__incr](x0) = [0 1 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [a__adx](x0) = [1 0 0]x0 + [0] [0 0 0] [1] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] a__adx(cons(X,Y)) = [1 0 1]X + [1 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = a__incr(cons(X,adx(Y))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [0] [1 1 0] [0] mark(adx(X)) = [1 1 0]X + [0] >= [1 1 0]X + [0] = a__adx(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [1] mark(incr(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = a__incr(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1] [1 1 0] [0] mark(hd(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = a__hd(mark(X)) [0 0 0] [1] [0 0 0] [0] [1 0 1] [0] [1 0 0] [0] a__incr(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = incr(X) [0 0 0] [1] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] a__hd(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = hd(X) [0 0 0] [0] [0 0 0] [0] problem: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) a__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=1 interpretation: [incr](x0) = x0, [cons](x0, x1) = x0 + x1 + 5, [a__hd](x0) = x0 + 1, [hd](x0) = x0, [mark](x0) = x0 + 2, [adx](x0) = x0 + 3, [a__incr](x0) = x0, [a__adx](x0) = x0 + 3 orientation: a__adx(cons(X,Y)) = X + Y + 8 >= X + Y + 8 = a__incr(cons(X,adx(Y))) mark(adx(X)) = X + 5 >= X + 5 = a__adx(mark(X)) mark(incr(X)) = X + 2 >= X + 2 = a__incr(mark(X)) a__incr(X) = X >= X = incr(X) a__hd(X) = X + 1 >= X = hd(X) problem: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=1 interpretation: [incr](x0) = x0, [cons](x0, x1) = x0 + 3x1 + 7, [mark](x0) = 2x0 + 2, [adx](x0) = 3x0 + 4, [a__incr](x0) = x0, [a__adx](x0) = 3x0 orientation: a__adx(cons(X,Y)) = 3X + 9Y + 21 >= X + 9Y + 19 = a__incr(cons(X,adx(Y))) mark(adx(X)) = 6X + 10 >= 6X + 6 = a__adx(mark(X)) mark(incr(X)) = 2X + 2 >= 2X + 2 = a__incr(mark(X)) a__incr(X) = X >= X = incr(X) problem: mark(incr(X)) -> a__incr(mark(X)) a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [incr](x0) = x0 + [1] [0], [1 1 0] [0] [mark](x0) = [1 1 0]x0 + [0] [0 0 0] [1], [0] [a__incr](x0) = x0 + [1] [0] orientation: [1 1 0] [1] [1 1 0] [0] mark(incr(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = a__incr(mark(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] a__incr(X) = X + [1] >= X + [1] = incr(X) [0] [0] problem: a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__incr](x0) = [0 0 0]x0 + [0] [0 0 1] [0] orientation: [1 0 0] [1] [1 0 0] a__incr(X) = [0 0 0]X + [0] >= [0 0 0]X = incr(X) [0 0 1] [0] [0 0 0] problem: Qed