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=3 interpretation: [1 0 0] [0] [tl](x0) = [1 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [0] [hd](x0) = [1 0 0]x0 + [0] [0 0 0] [1], [1] [nats] = [0] [1], [1 0 0] [0] [a__tl](x0) = [1 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [0] [mark](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [1 0 0] [0] [a__hd](x0) = [1 0 1]x0 + [0] [0 0 0] [1], [1 0 0] [0] [adx](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [incr](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [a__incr](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [1 1 1] [1 1 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [0] [0 0 0] [0 0 0] [1], [0] [zeros] = [0] [0], [0] [0] = [0] [0], [1 0 0] [0] [a__adx](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [0] [a__zeros] = [0] [1], [1] [a__nats] = [1] [1] orientation: [1] [0] a__nats() = [1] >= [1] = a__adx(a__zeros()) [1] [1] [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [1] [1] [1 1 1] [1 1 0] [0] [1 1 0] [1 0 0] [0] a__incr(cons(X,Y)) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = cons(s(X),incr(Y)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [1 1 0] [0] [1 1 1] [1 0 0] [0] a__adx(cons(X,Y)) = [0 0 0]X + [0 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 1] [1 1 0] [0] [1 0 0] [0] a__hd(cons(X,Y)) = [1 1 1]X + [1 1 0]Y + [1] >= [0 1 1]X + [0] = mark(X) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 1 0] [0] [1 0 0] [0] a__tl(cons(X,Y)) = [1 1 1]X + [1 1 1]Y + [1] >= [0 1 1]Y + [0] = mark(Y) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1] [1] mark(nats()) = [1] >= [1] = a__nats() [1] [1] [1 0 0] [0] [1 0 0] [0] mark(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__adx(mark(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [1] [1] [1 0 0] [0] [1 0 0] [0] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__incr(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(hd(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = a__hd(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(tl(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = a__tl(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 1 0] [0] [1 1 1] [1 1 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 1]X2 + [1] >= [0 0 0]X1 + [0 0 1]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] mark(0()) = [0] >= [0] = 0() [1] [0] [1 1 0] [0] [1 1 0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [1] [0 0 0] [1] [1] a__nats() = [1] >= [0] = nats() [1] [1] [1 0 0] [0] [1 0 0] [0] a__adx(X) = [0 0 1]X + [0] >= [0 0 0]X + [0] = adx(X) [0 0 0] [1] [0 0 0] [1] [0] [0] a__zeros() = [0] >= [0] = zeros() [1] [0] [1 0 0] [0] [1 0 0] [0] a__incr(X) = [0 0 1]X + [0] >= [0 0 0]X + [0] = incr(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] a__hd(X) = [1 0 1]X + [0] >= [1 0 0]X + [0] = hd(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] a__tl(X) = [1 1 0]X + [1] >= [1 1 0]X + [1] = tl(X) [0 0 1] [0] [0 0 1] [0] 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) 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) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1] [tl](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [hd](x0) = [0 1 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1 0 1] [1] [a__tl](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [mark](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [a__hd](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1] [adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [incr](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__incr](x0) = [1 0 1]x0 [0 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [1 0 0]x0 + [1 0 0]x1 [0 0 0] [0 0 0] , [1] [zeros] = [0] [0], [0] [0] = [0] [0], [1 0 0] [1] [a__adx](x0) = [1 0 0]x0 + [1] [0 0 0] [0], [1] [a__zeros] = [1] [0], [0] [a__nats] = [0] [0] orientation: [1] [1] a__zeros() = [1] >= [1] = cons(0(),zeros()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__incr(cons(X,Y)) = [1 0 0]X + [1 0 0]Y >= [1 0 0]X + [1 0 0]Y = cons(s(X),incr(Y)) [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] [1] a__adx(cons(X,Y)) = [1 0 0]X + [1 0 0]Y + [1] >= [1 0 0]X + [1 0 0]Y + [1] = a__incr(cons(X,adx(Y))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] a__hd(cons(X,Y)) = [1 0 0]X + [1 0 0]Y >= [1 0 0]X = mark(X) [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1] [1 0 0] a__tl(cons(X,Y)) = [1 0 0]X + [1 0 0]Y + [1] >= [1 0 0]Y = mark(Y) [0 0 0] [0 0 0] [0] [0 0 0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 0] [1] [1 0 0] [1] mark(adx(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = a__adx(mark(X)) [0 0 0] [0] [0 0 0] [0] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [0] [0] [1 0 0] [1 0 0] mark(incr(X)) = [1 0 0]X >= [1 0 0]X = a__incr(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(hd(X)) = [1 0 0]X >= [1 0 0]X = a__hd(mark(X)) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 0] [1] mark(tl(X)) = [1 0 1]X + [1] >= [1 0 0]X + [1] = a__tl(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [1 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 0] [1] [1 0 0] [1] a__adx(X) = [1 0 0]X + [1] >= [0 0 0]X + [0] = adx(X) [0 0 0] [0] [0 0 0] [0] [1] [1] a__zeros() = [1] >= [0] = zeros() [0] [0] [1 0 0] [1 0 0] a__incr(X) = [1 0 1]X >= [1 0 0]X = incr(X) [0 0 1] [0 0 0] [1 0 0] [1 0 0] a__hd(X) = [0 1 0]X >= [0 1 0]X = hd(X) [0 0 0] [0 0 0] [1 0 1] [1] [1 0 1] [1] a__tl(X) = [0 1 0]X + [1] >= [0 0 0]X + [0] = tl(X) [0 0 0] [0] [0 0 0] [0] 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(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) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tl](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 1] [0] [hd](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [0] [nats] = [0] [0], [1 0 0] [1] [a__tl](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 0 1] [0] [a__hd](x0) = [0 1 0]x0 + [1] [0 1 0] [1], [1 0 0] [adx](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [a__incr](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [0] [zeros] = [1] [0], [0] [0] = [0] [0], [1 0 0] [0] [a__adx](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1] [a__zeros] = [1] [0], [0] [a__nats] = [0] [0] orientation: [1] [0] a__zeros() = [1] >= [0] = cons(0(),zeros()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] a__incr(cons(X,Y)) = [0 1 1]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y = cons(s(X),incr(Y)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] a__adx(cons(X,Y)) = [0 1 1]X + [0 0 0]Y + [0] >= [0 1 1]X + [0 0 0]Y + [0] = a__incr(cons(X,adx(Y))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 1 0] [0] a__hd(cons(X,Y)) = [0 1 1]X + [0 0 0]Y + [1] >= [0 1 0]X + [0] = mark(X) [0 1 1] [0 0 0] [1] [0 0 1] [1] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [1] [0] [1 1 0] [0] [1 1 0] [0] mark(adx(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__adx(mark(X)) [0 0 0] [1] [0 0 0] [1] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [1] [0] [1 1 0] [0] [1 1 0] [0] mark(incr(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__incr(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [1] mark(hd(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__hd(mark(X)) [0 1 0] [1] [0 1 0] [1] [1 1 0] [1] [1 1 0] [1] mark(tl(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__tl(mark(X)) [0 0 0] [1] [0 0 0] [0] [1 1 1] [1 0 0] [0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 + [0] >= [0 1 1]X1 + [0 0 0]X2 = cons(X1,X2) [0 1 0] [0 0 0] [1] [0 1 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [1] [0] [1 0 0] [0] [1 0 0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [1] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 0] [0] [1 0 0] a__adx(X) = [0 1 0]X + [0] >= [0 1 0]X = adx(X) [0 0 0] [1] [0 0 0] [1] [0] a__zeros() = [1] >= [1] = zeros() [0] [0] [1 0 0] [0] [1 0 0] a__incr(X) = [0 1 0]X + [0] >= [0 1 0]X = incr(X) [0 0 0] [1] [0 0 0] [1 0 1] [0] [1 0 1] [0] a__hd(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = hd(X) [0 1 0] [1] [0 1 0] [0] [1 0 0] [1] [1 0 0] [0] a__tl(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = tl(X) [0 0 0] [0] [0 0 0] [0] problem: 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(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__incr(X) -> incr(X) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tl](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 1] [hd](x0) = [1 0 1]x0 [0 1 0] , [0] [nats] = [0] [0], [1 0 0] [a__tl](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [mark](x0) = [1 0 1]x0 [0 1 0] , [1 0 1] [a__hd](x0) = [1 0 1]x0 [0 1 0] , [0] [adx](x0) = x0 + [1] [1], [0] [incr](x0) = x0 + [1] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1] [a__incr](x0) = x0 + [1] [1], [1 1 1] [1 0 0] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [1] [a__adx](x0) = x0 + [1] [1], [0] [a__zeros] = [0] [0], [0] [a__nats] = [0] [0] orientation: [1 1 1] [1 0 0] [1] [1 0 0] [1 0 0] a__incr(cons(X,Y)) = [0 1 1]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y = cons(s(X),incr(Y)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [1] a__adx(cons(X,Y)) = [0 1 1]X + [0 0 0]Y + [1] >= [0 1 1]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 1] [1 0 0] [1 0 1] a__hd(cons(X,Y)) = [1 1 1]X + [1 0 0]Y >= [1 0 1]X = mark(X) [0 1 1] [0 0 0] [0 1 0] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 1] [1] [1 0 1] [1] mark(adx(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = a__adx(mark(X)) [0 1 0] [1] [0 1 0] [1] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 0 1] [1] [1 0 1] [1] mark(incr(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = a__incr(mark(X)) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1 1 1] mark(hd(X)) = [1 1 1]X >= [1 1 1]X = a__hd(mark(X)) [1 0 1] [1 0 1] [1 0 1] [0] [1 0 1] mark(tl(X)) = [1 0 1]X + [0] >= [0 0 0]X = a__tl(mark(X)) [0 0 0] [1] [0 0 0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] mark(cons(X1,X2)) = [1 1 1]X1 + [1 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = cons(X1,X2) [0 1 1] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [1 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1] [0] a__adx(X) = X + [1] >= X + [1] = adx(X) [1] [1] [1] [0] a__incr(X) = X + [1] >= X + [1] = incr(X) [1] [1] [1 0 1] [1 0 1] a__hd(X) = [1 0 1]X >= [1 0 1]X = hd(X) [0 1 0] [0 1 0] problem: a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) a__hd(cons(X,Y)) -> mark(X) 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__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tl](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [hd](x0) = [0 1 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1 0 0] [a__tl](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [mark](x0) = [0 1 0]x0 + [0] [0 1 0] [0], [1 1 0] [a__hd](x0) = [0 1 0]x0 [0 0 1] , [1 1 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 1 0] [0] [cons](x0, x1) = [0 1 1]x0 + [0 0 1]x1 + [1] [0 1 0] [0 0 0] [1], [0] [zeros] = [1] [0], [0] [0] = [0] [0], [1 0 1] [a__adx](x0) = [0 0 0]x0 [0 0 0] , [1] [a__zeros] = [0] [0], [1] [a__nats] = [0] [0] orientation: [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] a__adx(cons(X,Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y = a__incr(cons(X,adx(Y))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1] [1 0 0] [1] a__hd(cons(X,Y)) = [0 1 1]X + [0 0 1]Y + [1] >= [0 1 0]X + [0] = mark(X) [0 1 0] [0 0 0] [1] [0 1 0] [0] [1] [1] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [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)) [0 0 0] [0] [0 0 0] [0] [1] [1] mark(zeros()) = [1] >= [0] = a__zeros() [1] [0] [1 0 0] [1] [1 0 0] [1] mark(incr(X)) = [0 0 1]X + [0] >= [0 0 0]X + [0] = a__incr(mark(X)) [0 0 1] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] [1] mark(hd(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__hd(mark(X)) [0 1 0] [0] [0 1 0] [0] [1 0 0] [1] [1 0 0] [1] mark(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__tl(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [0] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 1]X2 + [1] >= [0 1 1]X1 + [0 0 1]X2 + [1] = cons(X1,X2) [0 1 1] [0 0 1] [1] [0 1 0] [0 0 0] [1] [1] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1] [1 0 0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] [1] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 1 0] [1 1 0] a__hd(X) = [0 1 0]X >= [0 1 0]X = hd(X) [0 0 1] [0 0 0] problem: a__hd(cons(X,Y)) -> mark(X) 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)) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tl](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [0] [hd](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [0] [nats] = [0] [1], [1 0 0] [0] [a__tl](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 1] [0] [a__hd](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 1] , [1 1 1] [incr](x0) = [1 1 1]x0 [0 0 0] , [1 1 0] [a__incr](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [1] [cons](x0, x1) = x0 + [0 0 0]x1 + [0] [0 0 0] [0], [1] [zeros] = [0] [0], [1 0 0] [a__adx](x0) = [0 0 0]x0 [0 0 1] , [0] [a__zeros] = [0] [1], [0] [a__nats] = [0] [0] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [0] a__hd(cons(X,Y)) = [0 1 0]X + [0 0 0]Y + [0] >= [0 1 0]X + [0] = mark(X) [0 0 0] [0 0 0] [1] [0 0 0] [1] [1] [0] mark(nats()) = [0] >= [0] = a__nats() [1] [0] [1 0 1] [0] [1 0 1] [0] mark(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__adx(mark(X)) [0 0 0] [1] [0 0 0] [1] [1] [0] mark(zeros()) = [0] >= [0] = a__zeros() [1] [1] [1 1 1] [0] [1 1 1] mark(incr(X)) = [1 1 1]X + [0] >= [1 0 1]X = a__incr(mark(X)) [0 0 0] [1] [0 0 0] [1 0 1] [1] [1 0 1] [1] mark(hd(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__hd(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] mark(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__tl(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 1] [0] a__hd(X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = hd(X) [0 0 0] [1] [0 0 0] [1] problem: mark(adx(X)) -> a__adx(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(hd(X)) -> a__hd(mark(X)) a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [hd](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [0] [a__hd](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 1 0] [1] [adx](x0) = [0 0 1]x0 + [0] [0 0 0] [0], [1 1 0] [0] [incr](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [1 0 0] [0] [a__incr](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [1 0 1] [a__adx](x0) = [0 0 1]x0 [0 0 0] orientation: [1 1 1] [1] [1 1 1] mark(adx(X)) = [0 0 1]X + [0] >= [0 0 1]X = a__adx(mark(X)) [0 0 0] [0] [0 0 0] [1 1 0] [1] [1 1 0] [0] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__incr(mark(X)) [0 1 0] [0] [0 1 0] [0] [1 1 0] [1] [1 1 0] [0] mark(hd(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__hd(mark(X)) [0 1 0] [0] [0 1 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 1 0] [0] [0 1 0] [0] problem: a__hd(X) -> hd(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [hd](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__hd](x0) = [0 0 0]x0 + [0] [0 0 1] [0] orientation: [1 0 0] [1] [1 0 0] a__hd(X) = [0 0 0]X + [0] >= [0 0 0]X = hd(X) [0 0 1] [0] [0 0 0] problem: Qed