YES Problem: active(nats()) -> mark(adx(zeros())) active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) adx(mark(X)) -> adx(X) adx(active(X)) -> adx(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) hd(mark(X)) -> hd(X) hd(active(X)) -> hd(X) tl(mark(X)) -> tl(X) tl(active(X)) -> tl(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [adx](x0) = [0 0 1]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [0] [tl](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1] [nats] = [0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [hd](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 1] , [0] [zeros] = [0] [1] orientation: [1] [0] active(nats()) = [0] >= [0] = mark(adx(zeros())) [0] [0] [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = mark(cons(s(X),incr(Y))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(adx(cons(X,Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = mark(incr(cons(X,adx(Y)))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] active(hd(cons(X,Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X = mark(X) [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [0] [1 0 0] active(tl(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y = mark(Y) [0 0 1] [0 0 1] [1] [0 0 1] [1] [1] mark(nats()) = [0] >= [0] = active(nats()) [0] [0] [1 0 0] [1 0 0] mark(adx(X)) = [0 0 0]X >= [0 0 0]X = active(adx(mark(X))) [0 0 0] [0 0 0] [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(X1,X2)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 0] [1 0 0] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = active(incr(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(X)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(hd(X)) = [0 0 0]X >= [0 0 0]X = active(hd(mark(X))) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] mark(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(tl(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] adx(mark(X)) = [0 0 1]X >= [0 0 1]X = adx(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] adx(active(X)) = [0 0 1]X >= [0 0 1]X = adx(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] s(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = s(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] hd(mark(X)) = [0 0 0]X >= [0 0 0]X = hd(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] hd(active(X)) = [0 0 0]X >= [0 0 0]X = hd(X) [0 0 1] [0 0 1] [1 0 0] [0] [1 0 0] [0] tl(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(X) [0 0 1] [1] [0 0 1] [1] [1 0 0] [0] [1 0 0] [0] tl(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(X) [0 0 1] [1] [0 0 1] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) active(tl(cons(X,Y))) -> mark(Y) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) adx(mark(X)) -> adx(X) adx(active(X)) -> adx(X) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) hd(mark(X)) -> hd(X) hd(active(X)) -> hd(X) tl(mark(X)) -> tl(X) tl(active(X)) -> tl(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [active](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1] [adx](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [0] = [0] [0], [1 0 0] [1] [tl](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [0] [nats] = [0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [hd](x0) = [0 0 0]x0 [1 0 0] , [1 0 1] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [0] >= [0] = mark(cons(0(),zeros())) [1] [1] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 0] [0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(cons(s(X),incr(Y))) [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 0] [1] active(adx(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(incr(cons(X,adx(Y)))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 1] [1 0 1] [0] [1 0 0] [0] active(hd(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0] = mark(X) [1 0 1] [1 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 1] [1] [1 0 0] [0] active(tl(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y + [0] = mark(Y) [1 0 1] [1 0 1] [1] [0 0 1] [1] [0] [0] mark(nats()) = [0] >= [0] = active(nats()) [1] [1] [1 0 0] [1] [1 0 0] [1] mark(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(adx(mark(X))) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [1] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(X1,X2)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [1 0 0] [0] [1 0 0] [0] mark(incr(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(incr(mark(X))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(s(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(hd(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(hd(mark(X))) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] mark(tl(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(tl(mark(X))) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] adx(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] adx(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [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] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [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] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1 0 0] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] hd(mark(X)) = [0 0 0]X >= [0 0 0]X = hd(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] hd(active(X)) = [0 0 0]X >= [0 0 0]X = hd(X) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1] tl(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(X) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1] [1 0 0] [1] tl(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(X) [1 0 0] [0] [1 0 0] [0] problem: active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) adx(mark(X)) -> adx(X) adx(active(X)) -> adx(X) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) hd(mark(X)) -> hd(X) hd(active(X)) -> hd(X) tl(mark(X)) -> tl(X) tl(active(X)) -> tl(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [active](x0) = x0 + [1] [0], [1 1 0] [0] [adx](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [0] = [0] [0], [1 1 0] [0] [tl](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [nats] = [0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [0] [hd](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 0 0] [1 1 0] [cons](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 1 1] [0 0 0] , [0] [zeros] = [0] [0] orientation: [0] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [0] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(cons(s(X),incr(Y))) [0 1 1] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [1 1 0] [0] [1 1 1] [1 1 0] [0] active(adx(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(incr(cons(X,adx(Y)))) [0 1 1] [0 0 0] [1] [0 1 1] [0 0 0] [0] [1 1 1] [1 1 0] [0] [1 0 1] [0] active(hd(cons(X,Y))) = [0 1 1]X + [0 0 0]Y + [1] >= [0 1 0]X + [1] = mark(X) [0 1 1] [0 0 0] [1] [0 0 1] [0] [0] [0] mark(nats()) = [1] >= [1] = active(nats()) [0] [0] [1 1 1] [1] [1 1 1] [1] mark(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(adx(mark(X))) [0 0 1] [1] [0 0 1] [1] [0] [0] mark(zeros()) = [1] >= [1] = active(zeros()) [0] [0] [1 1 1] [1 1 0] [0] [1 0 0] [1 1 0] [0] mark(cons(X1,X2)) = [0 1 1]X1 + [0 0 0]X2 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1] = active(cons(X1,X2)) [0 1 1] [0 0 0] [0] [0 1 1] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 1] [0] [1 0 1] [0] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(incr(mark(X))) [0 0 1] [0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [1] mark(hd(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = active(hd(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 1] [1] mark(tl(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(tl(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 0] [0] adx(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(X) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [0] adx(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = adx(X) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 1] [1 0 1] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 0] [0] hd(mark(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = hd(X) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [0] hd(active(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = hd(X) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 1 0] [0] tl(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(X) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1] [1 1 0] [0] tl(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tl(X) [0 0 1] [1] [0 0 1] [1] problem: active(zeros()) -> mark(cons(0(),zeros())) active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [0] [adx](x0) = x0 + [0] [1], [0] [0] = [0] [0], [1 1 0] [0] [tl](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [0] [nats] = [0] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [0] [hd](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 1]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [zeros] = [1] [1] orientation: [1] [0] active(zeros()) = [1] >= [1] = mark(cons(0(),zeros())) [1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(cons(s(X),incr(Y))) [0 0 1] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] active(adx(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(incr(cons(X,adx(Y)))) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 1] [0] active(hd(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [1] = mark(X) [0 0 1] [0 0 0] [1] [0 0 1] [0] [0] [0] mark(nats()) = [1] >= [1] = active(nats()) [0] [0] [1 0 1] [1] [1 0 1] [1] mark(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(adx(mark(X))) [0 0 1] [1] [0 0 1] [1] [1] [1] mark(zeros()) = [1] >= [1] = active(zeros()) [1] [1] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(cons(X1,X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 1] [0] [1 0 1] [0] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(incr(mark(X))) [0 0 1] [0] [0 0 1] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 0 1] [1] mark(hd(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(hd(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 1 1] [1] [1 0 1] [1] mark(tl(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(tl(mark(X))) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 1] [1 1 0] [1 0 0] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 1] [0 0 1] [1 0 1] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] problem: active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) -> mark(X) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [s](x0) = x0, [active](x0) = x0, [adx](x0) = x0, [0] = 0, [tl](x0) = 4x0, [nats] = 2, [incr](x0) = x0, [hd](x0) = x0 + 2, [cons](x0, x1) = x0 + x1, [zeros] = 0 orientation: active(incr(cons(X,Y))) = X + Y >= X + Y = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = X + Y >= X + Y = mark(incr(cons(X,adx(Y)))) active(hd(cons(X,Y))) = X + Y + 2 >= X = mark(X) mark(nats()) = 2 >= 2 = active(nats()) mark(adx(X)) = X >= X = active(adx(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(X1,X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = X >= X = active(incr(mark(X))) mark(s(X)) = X >= X = active(s(X)) mark(hd(X)) = X + 2 >= X + 2 = active(hd(mark(X))) mark(tl(X)) = 4X >= 4X = active(tl(mark(X))) incr(mark(X)) = X >= X = incr(X) incr(active(X)) = X >= X = incr(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) problem: active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = x0, [s](x0) = x0, [active](x0) = x0, [adx](x0) = 2x0 + 1, [0] = 0, [tl](x0) = 4x0, [nats] = 0, [incr](x0) = x0, [hd](x0) = 2x0 + 2, [cons](x0, x1) = 2x0 + 4x1 + 4, [zeros] = 4 orientation: active(incr(cons(X,Y))) = 2X + 4Y + 4 >= 2X + 4Y + 4 = mark(cons(s(X),incr(Y))) active(adx(cons(X,Y))) = 4X + 8Y + 9 >= 2X + 8Y + 8 = mark(incr(cons(X,adx(Y)))) mark(nats()) = 0 >= 0 = active(nats()) mark(adx(X)) = 2X + 1 >= 2X + 1 = active(adx(mark(X))) mark(zeros()) = 4 >= 4 = active(zeros()) mark(cons(X1,X2)) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 4 = active(cons(X1,X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = X >= X = active(incr(mark(X))) mark(s(X)) = X >= X = active(s(X)) mark(hd(X)) = 2X + 2 >= 2X + 2 = active(hd(mark(X))) mark(tl(X)) = 4X >= 4X = active(tl(mark(X))) incr(mark(X)) = X >= X = incr(X) incr(active(X)) = X >= X = incr(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) problem: active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [active](x0) = [0 0 0]x0 [0 0 1] , [adx](x0) = x0 , [0] [0] = [0] [0], [1 1 0] [tl](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0], [1 1 0] [incr](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [hd](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + x1 + [0] [0 0 0] [1], [0] [zeros] = [0] [0] orientation: [1 0 0] [1 1 1] [1] [1 0 0] [1 1 1] [0] active(incr(cons(X,Y))) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(cons(s(X),incr(Y))) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [0] [0] mark(nats()) = [0] >= [0] = active(nats()) [0] [0] [1 1 0] [1 1 0] mark(adx(X)) = [0 0 0]X >= [0 0 0]X = active(adx(mark(X))) [0 0 1] [0 0 1] [0] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [0] [0] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(X1,X2)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 1] [1 1 1] mark(incr(X)) = [0 0 0]X >= [0 0 0]X = active(incr(mark(X))) [0 0 1] [0 0 1] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(hd(X)) = [0 0 0]X >= [0 0 0]X = active(hd(mark(X))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(tl(X)) = [0 0 0]X >= [0 0 0]X = active(tl(mark(X))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] incr(mark(X)) = [0 0 1]X >= [0 0 1]X = incr(X) [0 0 1] [0 0 1] [1 1 0] [1 1 0] incr(active(X)) = [0 0 1]X >= [0 0 1]X = incr(X) [0 0 1] [0 0 1] [1 1 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] problem: mark(nats()) -> active(nats()) mark(adx(X)) -> active(adx(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [active](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [adx](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [1], [1 0 0] [tl](x0) = [1 0 0]x0 [0 0 0] , [0] [nats] = [1] [0], [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [hd](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 1] [0] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [zeros] = [0] [0] orientation: [1] [0] mark(nats()) = [1] >= [0] = active(nats()) [0] [0] [1 0 0] [1] [1 0 0] [1] mark(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(adx(mark(X))) [0 0 0] [0] [0 0 0] [0] [1] [0] mark(zeros()) = [0] >= [0] = active(zeros()) [0] [0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [1] mark(cons(X1,X2)) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 0 0] [1] [1 0 0] [1] mark(incr(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(incr(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(s(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(hd(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = active(hd(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(tl(X)) = [1 0 0]X + [0] >= [0 0 0]X + [0] = active(tl(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] incr(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = incr(X) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 0] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [0] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] [1 0 1] [0] [1 0 0] [0] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [1] [0 0 0] [1] problem: mark(adx(X)) -> active(adx(mark(X))) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(active(X)) -> incr(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 6x0, [s](x0) = x0, [active](x0) = x0, [adx](x0) = 3x0, [0] = 0, [tl](x0) = 2x0, [incr](x0) = x0 + 1, [hd](x0) = 4x0, [cons](x0, x1) = x0 + 3x1 + 3 orientation: mark(adx(X)) = 18X >= 18X = active(adx(mark(X))) mark(cons(X1,X2)) = 6X1 + 18X2 + 18 >= X1 + 3X2 + 3 = active(cons(X1,X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = 6X + 6 >= 6X + 1 = active(incr(mark(X))) mark(s(X)) = 6X >= X = active(s(X)) mark(hd(X)) = 24X >= 24X = active(hd(mark(X))) mark(tl(X)) = 12X >= 12X = active(tl(mark(X))) incr(active(X)) = X + 1 >= X + 1 = incr(X) s(active(X)) = X >= X = s(X) problem: mark(adx(X)) -> active(adx(mark(X))) mark(0()) -> active(0()) mark(s(X)) -> active(s(X)) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(active(X)) -> incr(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 2x0, [s](x0) = 4x0 + 4, [active](x0) = x0, [adx](x0) = 4x0, [0] = 0, [tl](x0) = x0, [incr](x0) = x0 + 5, [hd](x0) = x0 orientation: mark(adx(X)) = 8X >= 8X = active(adx(mark(X))) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = 8X + 8 >= 4X + 4 = active(s(X)) mark(hd(X)) = 2X >= 2X = active(hd(mark(X))) mark(tl(X)) = 2X >= 2X = active(tl(mark(X))) incr(active(X)) = X + 5 >= X + 5 = incr(X) s(active(X)) = 4X + 4 >= 4X + 4 = s(X) problem: mark(adx(X)) -> active(adx(mark(X))) mark(0()) -> active(0()) mark(hd(X)) -> active(hd(mark(X))) mark(tl(X)) -> active(tl(mark(X))) incr(active(X)) -> incr(X) s(active(X)) -> s(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [mark](x0) = [0 0 1]x0 [0 0 0] , [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [active](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0] [adx](x0) = x0 + [0] [1], [0] [0] = [0] [1], [0] [tl](x0) = x0 + [0] [1], [1 1 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [hd](x0) = x0 + [0] [1] orientation: [1 1 1] [1] [1 1 1] [0] mark(adx(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = active(adx(mark(X))) [0 0 0] [0] [0 0 0] [0] [1] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 1 1] [1] [1 1 1] [0] mark(hd(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = active(hd(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 1] [0] mark(tl(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = active(tl(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1] [1 1 0] incr(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = incr(X) [0 0 0] [0] [0 0 0] [1 1 0] [1] [1 1 0] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] problem: Qed