YES Problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__nats() -> a__adx(a__zeros()) a__zeros() -> cons(0(),zeros()) a__head(cons(X,L)) -> mark(X) a__tail(cons(X,L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [head](x0) = x0 + 1, [a__nats] = 1, [a__incr](x0) = x0, [mark](x0) = x0, [tail](x0) = 3x0 + 2, [incr](x0) = x0, [0] = 0, [nil] = 0, [adx](x0) = x0, [nats] = 1, [a__head](x0) = x0 + 1, [zeros] = 0, [a__zeros] = 0, [a__tail](x0) = 3x0 + 2, [a__adx](x0) = x0, [cons](x0, x1) = 4x0 + x1 orientation: a__incr(nil()) = 0 >= 0 = nil() a__incr(cons(X,L)) = L + 4X >= L + 4X = cons(s(mark(X)),incr(L)) a__adx(nil()) = 0 >= 0 = nil() a__adx(cons(X,L)) = L + 4X >= L + 4X = a__incr(cons(mark(X),adx(L))) a__nats() = 1 >= 0 = a__adx(a__zeros()) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__head(cons(X,L)) = L + 4X + 1 >= X = mark(X) a__tail(cons(X,L)) = 3L + 12X + 2 >= L = mark(L) mark(incr(X)) = X >= X = a__incr(mark(X)) mark(adx(X)) = X >= X = a__adx(mark(X)) mark(nats()) = 1 >= 1 = a__nats() mark(zeros()) = 0 >= 0 = a__zeros() mark(head(X)) = X + 1 >= X + 1 = a__head(mark(X)) mark(tail(X)) = 3X + 2 >= 3X + 2 = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X >= X = incr(X) a__adx(X) = X >= X = adx(X) a__nats() = 1 >= 1 = nats() a__zeros() = 0 >= 0 = zeros() a__head(X) = X + 1 >= X + 1 = head(X) a__tail(X) = 3X + 2 >= 3X + 2 = tail(X) problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(nil()) -> nil() a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [a__nats] = [0] [0], [1 0 0] [a__incr](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [mark](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [0] [nil] = [0] [0], [1 0 0] [1] [adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [nats] = [0] [0], [1 1 0] [a__head](x0) = [0 0 0]x0 [0 0 1] , [0] [zeros] = [0] [0], [0] [a__zeros] = [0] [0], [1 1 0] [a__tail](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [1] [a__adx](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 0] orientation: [0] [0] a__incr(nil()) = [0] >= [0] = nil() [0] [0] [1 1 1] [1 0 0] [1 0 0] [1 0 0] a__incr(cons(X,L)) = [0 0 0]L + [0 0 0]X >= [0 0 0]L + [0 0 0]X = cons(s(mark(X)),incr(L)) [1 1 1] [1 0 0] [1 0 0] [0 0 0] [1] [0] a__adx(nil()) = [0] >= [0] = nil() [1] [0] [1 1 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] a__adx(cons(X,L)) = [0 0 0]L + [0 0 0]X + [0] >= [0 0 0]L + [0 0 0]X + [0] = a__incr(cons(mark(X),adx(L))) [1 1 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1 0 0] [1 0 0] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = a__incr(mark(X)) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1] mark(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__adx(mark(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1 1 0] [1 0 0] mark(head(X)) = [0 0 0]X >= [0 0 0]X = a__head(mark(X)) [1 1 0] [1 0 0] [1 0 0] [1 0 0] mark(tail(X)) = [0 0 0]X >= [0 0 0]X = a__tail(mark(X)) [1 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(mark(X1),X2) [1 0 0] [1 1 1] [0 0 0] [1 0 0] [1 1 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [1 1 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [1 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] a__adx(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(X) [1 0 0] [1] [0 0 0] [0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 1 0] [1 1 0] a__head(X) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 1] [0 0 0] [1 1 0] [1 0 0] a__tail(X) = [0 0 0]X >= [0 0 0]X = tail(X) [0 1 0] [0 0 0] problem: a__incr(nil()) -> nil() a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [a__nats] = [0] [0], [1 1 0] [a__incr](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [incr](x0) = [0 1 0]x0 [0 0 0] , [0] [0] = [0] [0], [0] [nil] = [1] [0], [1 0 0] [1] [adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [nats] = [0] [0], [1 0 0] [a__head](x0) = [0 0 1]x0 [0 0 1] , [0] [zeros] = [1] [0], [0] [a__zeros] = [1] [0], [1 0 0] [a__tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] orientation: [1] [0] a__incr(nil()) = [1] >= [1] = nil() [0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] a__incr(cons(X,L)) = [0 1 0]L + [0 0 0]X >= [0 1 0]L + [0 0 0]X = cons(s(mark(X)),incr(L)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1] [1 0 0] [1 1 0] [1] a__adx(cons(X,L)) = [0 0 0]L + [0 0 0]X + [0] >= [0 0 0]L + [0 0 0]X + [0] = a__incr(cons(mark(X),adx(L))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] a__zeros() = [1] >= [1] = cons(0(),zeros()) [0] [0] [1 1 0] [1 1 0] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = a__incr(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 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] [0] [0] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [0] [0] mark(zeros()) = [1] >= [1] = a__zeros() [0] [0] [1 0 0] [1 0 0] mark(head(X)) = [0 0 0]X >= [0 0 0]X = a__head(mark(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(tail(X)) = [0 0 0]X >= [0 0 0]X = a__tail(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [1] >= [1] = nil() [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = cons(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] a__adx(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(X) [0 0 0] [0] [0 0 0] [0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [0] [0] a__zeros() = [1] >= [1] = zeros() [0] [0] [1 0 0] [1 0 0] a__head(X) = [0 0 1]X >= [0 0 0]X = head(X) [0 0 1] [0 0 0] [1 0 0] [1 0 0] a__tail(X) = [0 0 0]X >= [0 0 0]X = tail(X) [0 0 0] [0 0 0] problem: a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) a__zeros() -> cons(0(),zeros()) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__zeros() -> zeros() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [s](x0) = x0 , [1 0 0] [head](x0) = [0 0 0]x0 [0 0 1] , [1] [a__nats] = [1] [0], [1 1 0] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [1 0 1] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [0] [tail](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 1 0] [incr](x0) = [0 1 0]x0 [0 0 1] , [0] [0] = [0] [0], [0] [nil] = [0] [0], [1 0 1] [adx](x0) = [0 1 0]x0 [0 1 1] , [1] [nats] = [1] [0], [1 0 0] [a__head](x0) = [0 0 0]x0 [0 0 1] , [0] [zeros] = [1] [1], [1] [a__zeros] = [1] [1], [1 0 0] [0] [a__tail](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 1 1] [a__adx](x0) = [0 1 0]x0 [0 1 1] , [1 0 0] [cons](x0, x1) = [0 0 1]x0 + x1 [0 0 1] orientation: [1 1 0] [1 0 1] [1 1 0] [1 0 1] a__incr(cons(X,L)) = [0 1 0]L + [0 0 1]X >= [0 1 0]L + [0 0 1]X = cons(s(mark(X)),incr(L)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 1 1] [1 0 2] [1 1 1] [1 0 2] a__adx(cons(X,L)) = [0 1 0]L + [0 0 1]X >= [0 1 0]L + [0 0 1]X = a__incr(cons(mark(X),adx(L))) [0 1 1] [0 0 2] [0 1 1] [0 0 1] [1] [0] a__zeros() = [1] >= [1] = cons(0(),zeros()) [1] [1] [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 2] [1 1 2] mark(adx(X)) = [0 1 0]X >= [0 1 0]X = a__adx(mark(X)) [0 1 1] [0 1 1] [1] [1] mark(nats()) = [1] >= [1] = a__nats() [0] [0] [1] [1] mark(zeros()) = [1] >= [1] = a__zeros() [1] [1] [1 0 1] [1 0 1] mark(head(X)) = [0 0 0]X >= [0 0 0]X = a__head(mark(X)) [0 0 1] [0 0 1] [1 1 1] [0] [1 0 1] [0] mark(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__tail(mark(X)) [0 1 1] [0] [0 1 1] [0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 0 1] [1 0 1] [1 0 1] mark(cons(X1,X2)) = [0 0 1]X1 + [0 1 0]X2 >= [0 0 1]X1 + X2 = cons(mark(X1),X2) [0 0 1] [0 0 1] [0 0 1] [1 0 1] [1 0 1] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 0 1] [0 0 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [1 1 1] [1 0 1] a__adx(X) = [0 1 0]X >= [0 1 0]X = adx(X) [0 1 1] [0 1 1] [1] [1] a__nats() = [1] >= [1] = nats() [0] [0] [1] [0] a__zeros() = [1] >= [1] = zeros() [1] [1] [1 0 0] [1 0 0] a__head(X) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] a__tail(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(X) [0 1 1] [0] [0 1 1] [0] problem: a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(zeros()) -> a__zeros() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0, [head](x0) = x0, [a__nats] = 0, [a__incr](x0) = x0 + 2, [mark](x0) = x0, [tail](x0) = x0, [incr](x0) = x0 + 2, [0] = 5, [nil] = 1, [adx](x0) = 2x0, [nats] = 0, [a__head](x0) = x0, [zeros] = 7, [a__zeros] = 0, [a__tail](x0) = x0, [a__adx](x0) = 2x0, [cons](x0, x1) = 4x0 + x1 + 2 orientation: a__incr(cons(X,L)) = L + 4X + 4 >= L + 4X + 4 = cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) = 2L + 8X + 4 >= 2L + 4X + 4 = a__incr(cons(mark(X),adx(L))) mark(incr(X)) = X + 2 >= X + 2 = a__incr(mark(X)) mark(adx(X)) = 2X >= 2X = a__adx(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(zeros()) = 7 >= 0 = a__zeros() mark(head(X)) = X >= X = a__head(mark(X)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(cons(X1,X2)) = 4X1 + X2 + 2 >= 4X1 + X2 + 2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 5 >= 5 = 0() a__incr(X) = X + 2 >= X + 2 = incr(X) a__adx(X) = 2X >= 2X = adx(X) a__nats() = 0 >= 0 = nats() a__head(X) = X >= X = head(X) a__tail(X) = X >= X = tail(X) problem: a__incr(cons(X,L)) -> cons(s(mark(X)),incr(L)) a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1] [a__nats] = [0] [0], [1 1 0] [a__incr](x0) = [0 1 0]x0 [0 0 1] , [mark](x0) = x0 , [1 1 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [incr](x0) = [0 1 0]x0 [0 0 1] , [0] [0] = [0] [0], [0] [nil] = [0] [0], [1 0 1] [adx](x0) = [0 1 0]x0 [0 0 1] , [1] [nats] = [0] [0], [1 0 0] [1] [a__head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 1 0] [a__tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [a__adx](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [1] [0 1 1] [0 1 1] [1] orientation: [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] a__incr(cons(X,L)) = [0 1 0]L + [0 0 0]X + [1] >= [0 1 0]L + [0 0 0]X + [1] = cons(s(mark(X)),incr(L)) [0 1 1] [0 1 1] [1] [0 1 1] [0 0 0] [1] [1 1 1] [1 1 1] [1] [1 1 1] [1 0 0] [1] a__adx(cons(X,L)) = [0 1 0]L + [0 0 0]X + [1] >= [0 1 0]L + [0 0 0]X + [1] = a__incr(cons(mark(X),adx(L))) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 1 0] [1 1 0] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = a__incr(mark(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] mark(adx(X)) = [0 1 0]X >= [0 1 0]X = a__adx(mark(X)) [0 0 1] [0 0 1] [1] [1] mark(nats()) = [0] >= [0] = a__nats() [0] [0] [1 0 0] [1] [1 0 0] [1] mark(head(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__head(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] mark(tail(X)) = [0 0 0]X >= [0 0 0]X = a__tail(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [1] = cons(mark(X1),X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = s(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 1 0] [1 1 0] a__incr(X) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [1 0 1] [1 0 1] a__adx(X) = [0 1 0]X >= [0 1 0]X = adx(X) [0 0 1] [0 0 1] [1] [1] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 0] [1] [1 0 0] [1] a__head(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] a__tail(X) = [0 0 0]X >= [0 0 0]X = tail(X) [0 0 0] [0 0 0] problem: a__adx(cons(X,L)) -> a__incr(cons(mark(X),adx(L))) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = x0 + 2, [head](x0) = x0 + 1, [a__nats] = 0, [a__incr](x0) = x0 + 2, [mark](x0) = x0, [tail](x0) = x0, [incr](x0) = x0 + 2, [0] = 0, [nil] = 0, [adx](x0) = 2x0, [nats] = 0, [a__head](x0) = x0 + 1, [a__tail](x0) = x0, [a__adx](x0) = 2x0, [cons](x0, x1) = 2x0 + x1 + 5 orientation: a__adx(cons(X,L)) = 2L + 4X + 10 >= 2L + 2X + 7 = a__incr(cons(mark(X),adx(L))) mark(incr(X)) = X + 2 >= X + 2 = a__incr(mark(X)) mark(adx(X)) = 2X >= 2X = a__adx(mark(X)) mark(nats()) = 0 >= 0 = a__nats() mark(head(X)) = X + 1 >= X + 1 = a__head(mark(X)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = 2X1 + X2 + 5 >= 2X1 + X2 + 5 = cons(mark(X1),X2) mark(s(X)) = X + 2 >= X + 2 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__incr(X) = X + 2 >= X + 2 = incr(X) a__adx(X) = 2X >= 2X = adx(X) a__nats() = 0 >= 0 = nats() a__head(X) = X + 1 >= X + 1 = head(X) a__tail(X) = X >= X = tail(X) problem: mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [s](x0) = x0 + [0] [1], [0] [head](x0) = x0 + [1] [0], [1] [a__nats] = [1] [1], [0] [a__incr](x0) = x0 + [0] [1], [1 0 1] [mark](x0) = [0 1 0]x0 [0 1 1] , [0] [tail](x0) = x0 + [0] [1], [0] [incr](x0) = x0 + [0] [1], [0] [0] = [0] [0], [0] [nil] = [0] [0], [0] [adx](x0) = x0 + [1] [0], [1] [nats] = [1] [0], [0] [a__head](x0) = x0 + [1] [1], [1] [a__tail](x0) = x0 + [0] [1], [0] [a__adx](x0) = x0 + [1] [0], [1 0 0] [1] [cons](x0, x1) = x0 + [1 0 0]x1 + [0] [0 0 0] [0] orientation: [1 0 1] [1] [1 0 1] [0] 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 1] [0] [1 0 1] [0] mark(adx(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__adx(mark(X)) [0 1 1] [1] [0 1 1] [0] [1] [1] mark(nats()) = [1] >= [1] = a__nats() [1] [1] [1 0 1] [0] [1 0 1] [0] mark(head(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__head(mark(X)) [0 1 1] [1] [0 1 1] [1] [1 0 1] [1] [1 0 1] [1] mark(tail(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__tail(mark(X)) [0 1 1] [1] [0 1 1] [1] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] mark(cons(X1,X2)) = [0 1 0]X1 + [1 0 0]X2 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0] = cons(mark(X1),X2) [0 1 1] [1 0 0] [0] [0 1 1] [0 0 0] [0] [1 0 1] [1] [1 0 1] [0] mark(s(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = s(mark(X)) [0 1 1] [1] [0 1 1] [1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] a__incr(X) = X + [0] >= X + [0] = incr(X) [1] [1] [0] [0] a__adx(X) = X + [1] >= X + [1] = adx(X) [0] [0] [1] [1] a__nats() = [1] >= [1] = nats() [1] [0] [0] [0] a__head(X) = X + [1] >= X + [1] = head(X) [1] [0] [1] [0] a__tail(X) = X + [0] >= X + [0] = tail(X) [1] [1] problem: mark(adx(X)) -> a__adx(mark(X)) mark(nats()) -> a__nats() mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [head](x0) = x0 , [1] [a__nats] = [0] [0], [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [mark](x0) = [0 1 0]x0 [1 0 1] , [1 0 1] [0] [tail](x0) = [0 0 0]x0 + [0] [0 1 0] [1], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [1], [0] [nil] = [1] [0], [0] [adx](x0) = x0 + [0] [1], [1] [nats] = [0] [0], [a__head](x0) = x0 , [1 0 0] [0] [a__tail](x0) = [0 0 0]x0 + [0] [0 1 1] [1], [0] [a__adx](x0) = x0 + [0] [1], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 1 1] [0 0 0] [1] orientation: [1 1 1] [1] [1 1 1] [0] mark(adx(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = a__adx(mark(X)) [1 0 1] [1] [1 0 1] [1] [1] [1] mark(nats()) = [0] >= [0] = a__nats() [1] [0] [1 1 1] [1 1 1] mark(head(X)) = [0 1 0]X >= [0 1 0]X = a__head(mark(X)) [1 0 1] [1 0 1] [1 1 1] [1] [1 1 1] [0] mark(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__tail(mark(X)) [1 1 1] [1] [1 1 1] [1] [1] [0] mark(nil()) = [1] >= [1] = nil() [0] [0] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(mark(X1),X2) [1 1 1] [1 0 0] [1] [1 1 1] [0 0 0] [1] [1] [0] mark(0()) = [0] >= [0] = 0() [1] [1] [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [0] [0] a__adx(X) = X + [0] >= X + [0] = adx(X) [1] [1] [1] [1] a__nats() = [0] >= [0] = nats() [0] [0] a__head(X) = X >= X = head(X) problem: mark(nats()) -> a__nats() mark(head(X)) -> a__head(mark(X)) a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [head](x0) = x0 + [1] [0], [0] [a__nats] = [0] [1], [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [1], [0] [a__head](x0) = x0 + [1] [0], [1 0 0] [a__adx](x0) = [0 0 0]x0 [0 0 0] orientation: [1] [0] mark(nats()) = [0] >= [0] = a__nats() [1] [1] [1 1 1] [1] [1 1 1] [0] mark(head(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__head(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__adx(X) = [0 0 0]X >= [0 0 0]X = adx(X) [0 0 0] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [1] [1] [0] [0] a__head(X) = X + [1] >= X + [1] = head(X) [0] [0] problem: a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() a__head(X) -> head(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [head](x0) = [0 0 0]x0 [0 0 0] , [0] [a__nats] = [0] [0], [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1 0 0] [1] [a__head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [a__adx](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__adx(X) = [0 0 0]X >= [0 0 0]X = adx(X) [0 0 0] [0 0 0] [0] [0] a__nats() = [0] >= [0] = nats() [0] [0] [1 0 0] [1] [1 0 0] a__head(X) = [0 0 0]X + [0] >= [0 0 0]X = head(X) [0 0 0] [0] [0 0 0] problem: a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats() -> nats() Matrix Interpretation Processor: dim=3 interpretation: [1] [a__nats] = [0] [0], [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1 0 0] [a__adx](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__adx(X) = [0 0 0]X >= [0 0 0]X = adx(X) [0 0 0] [0 0 0] [1] [0] a__nats() = [0] >= [0] = nats() [0] [0] problem: a__incr(X) -> incr(X) a__adx(X) -> adx(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [a__incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [a__adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0] orientation: [1 0 0] [1 0 0] a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] a__adx(X) = [0 0 0]X + [0] >= [0 0 0]X = adx(X) [0 0 0] [0] [0 0 0] problem: a__incr(X) -> incr(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [a__incr](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 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