YES Problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) -> mark(X) active(tail(cons(X,XS))) -> mark(XS) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(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) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Proof: Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 2x0, [head](x0) = x0 + 1, [s](x0) = x0, [odds] = 2, [pairs] = 2, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [0] = 0, [active](x0) = x0, [nats] = 0 orientation: active(nats()) = 0 >= 0 = mark(cons(0(),incr(nats()))) active(pairs()) = 2 >= 2 = mark(cons(0(),incr(odds()))) active(odds()) = 2 >= 2 = mark(incr(pairs())) active(incr(cons(X,XS))) = X + XS >= X + XS = mark(cons(s(X),incr(XS))) active(head(cons(X,XS))) = X + XS + 1 >= X = mark(X) active(tail(cons(X,XS))) = 2X + 2XS >= XS = mark(XS) mark(nats()) = 0 >= 0 = active(nats()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(incr(X)) = X >= X = active(incr(mark(X))) mark(pairs()) = 2 >= 2 = active(pairs()) mark(odds()) = 2 >= 2 = active(odds()) mark(s(X)) = X >= X = active(s(mark(X))) mark(head(X)) = X + 1 >= X + 1 = active(head(mark(X))) mark(tail(X)) = 2X >= 2X = active(tail(mark(X))) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) 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) head(mark(X)) = X + 1 >= X + 1 = head(X) head(active(X)) = X + 1 >= X + 1 = head(X) tail(mark(X)) = 2X >= 2X = tail(X) tail(active(X)) = 2X >= 2X = tail(X) problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(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) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [tail](x0) = [1 0 0]x0 + [1] [1 0 0] [0], [1 0 0] [head](x0) = [1 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1] [odds] = [0] [0], [1] [pairs] = [0] [0], [1 0 0] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 1 1] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] , [1] [nats] = [0] [0] orientation: [1] [1] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [0] [0] [1] [1] active(pairs()) = [0] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [1] [1] active(odds()) = [0] >= [0] = mark(incr(pairs())) [0] [0] [1 1 1] [1 1 1] [1 0 0] [1 0 0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1] [1 0 0] active(tail(cons(X,XS))) = [1 1 1]X + [1 1 1]XS + [0] >= [0 0 1]XS = mark(XS) [1 1 1] [1 1 1] [1] [0 1 0] [1] [1] mark(nats()) = [0] >= [0] = active(nats()) [0] [0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [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] [1] mark(pairs()) = [0] >= [0] = active(pairs()) [0] [0] [1] [1] mark(odds()) = [0] >= [0] = active(odds()) [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(head(X)) = [0 0 0]X >= [0 0 0]X = active(head(mark(X))) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1] mark(tail(X)) = [1 0 0]X + [0] >= [1 0 0]X + [0] = active(tail(mark(X))) [1 0 0] [1] [1 0 0] [1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] 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 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] 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 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] 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 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] 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 0] [0 0 0] [0 0 0] [0 0 0] [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] head(mark(X)) = [1 0 0]X >= [1 0 0]X = head(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] head(active(X)) = [1 0 0]X >= [1 0 0]X = head(X) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] tail(mark(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = tail(X) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1] [1 0 0] [1] tail(active(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = tail(X) [1 0 0] [0] [1 0 0] [0] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(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) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1] [head](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1] [odds] = [0] [0], [1] [pairs] = [0] [0], [0] [mark](x0) = x0 + [1] [0], [1 0 0] [1 1 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 1] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 0 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1] [nats] = [0] [0] orientation: [1] [1] active(nats()) = [1] >= [1] = mark(cons(0(),incr(nats()))) [0] [0] [1] [1] active(pairs()) = [1] >= [1] = mark(cons(0(),incr(odds()))) [0] [0] [1] [1] active(odds()) = [1] >= [1] = mark(incr(pairs())) [0] [0] [1 0 1] [1 1 1] [0] [1 0 1] [1 0 1] [0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS + [1] >= [0 0 0]X + [0 0 0]XS + [1] = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(nats()) = [1] >= [1] = active(nats()) [0] [0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 1] [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(mark(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 0] [0] [0 0 0] [0] [1] [1] mark(pairs()) = [1] >= [1] = active(pairs()) [0] [0] [1] [1] mark(odds()) = [1] >= [1] = active(odds()) [0] [0] [1 0 1] [0] [1 0 1] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(head(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] mark(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(tail(mark(X))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] 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 0] [0 0 1] [0 0 0] [1 0 0] [1 1 1] [1] [1 0 0] [1 1 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 1] [0 0 0] [0] [0 0 1] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 1 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 1 1] [1] [1 0 0] [1 1 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 1] [0 0 0] [0] [0 0 1] [0 0 0] [1 0 1] [1 0 1] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 1 1] [1 0 1] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 1] [1 0 1] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] head(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] head(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = head(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0] [1 0 1] [0] tail(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 0 1] [0] tail(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [1] [0 0 0] [1] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(odds()) -> active(odds()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [head](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [0] [odds] = [1] [0], [0] [pairs] = [0] [0], [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 0 0]x1 [0 1 0] [0 0 1] , [1 0 1] [incr](x0) = [0 1 0]x0 [0 0 1] , [0] [0] = [0] [0], [active](x0) = x0 , [0] [nats] = [0] [0] orientation: [0] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [0] [0] [0] [0] active(pairs()) = [0] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [0] [0] active(odds()) = [1] >= [0] = mark(incr(pairs())) [0] [0] [1 1 0] [1 0 1] [1 1 0] [1 0 1] active(incr(cons(X,XS))) = [0 1 0]X + [0 0 0]XS >= [0 1 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [0] [0] mark(nats()) = [0] >= [0] = active(nats()) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 1] [1 1 1] mark(incr(X)) = [0 1 0]X >= [0 1 0]X = active(incr(mark(X))) [0 0 1] [0 0 1] [0] [0] mark(pairs()) = [0] >= [0] = active(pairs()) [0] [0] [1] [0] mark(odds()) = [1] >= [1] = active(odds()) [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 1 0]X >= [0 1 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(head(X)) = [0 1 0]X >= [0 1 0]X = active(head(mark(X))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] mark(tail(X)) = [0 1 0]X >= [0 1 0]X = active(tail(mark(X))) [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 1 0]X1 + [0 0 0]X2 >= [0 1 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 1 0] [0 0 1] [0 1 0] [0 0 1] [1 1 1] [1 0 1] incr(mark(X)) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [1 0 1] [1 0 1] incr(active(X)) = [0 1 0]X >= [0 1 0]X = incr(X) [0 0 1] [0 0 1] [1 1 0] [1 0 0] s(mark(X)) = [0 1 0]X >= [0 1 0]X = s(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] s(active(X)) = [0 1 0]X >= [0 1 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] head(mark(X)) = [0 1 0]X >= [0 1 0]X = head(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] head(active(X)) = [0 1 0]X >= [0 1 0]X = head(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] tail(mark(X)) = [0 1 0]X >= [0 1 0]X = tail(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] tail(active(X)) = [0 1 0]X >= [0 1 0]X = tail(X) [0 0 0] [0 0 0] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(pairs()) -> mark(cons(0(),incr(odds()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [head](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [odds] = [1] [0], [1] [pairs] = [0] [1], [1 0 0] [mark](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 0] [0 0 1] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 0] , [0] [nats] = [0] [0] orientation: [0] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [0] [0] [1] [0] active(pairs()) = [0] >= [0] = mark(cons(0(),incr(odds()))) [0] [0] [1] [1] active(odds()) = [0] >= [0] = mark(incr(pairs())) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS >= [0 0 0]X + [0 0 0]XS = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(nats()) = [0] >= [0] = active(nats()) [0] [0] [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(mark(X1),X2)) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [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] [1] mark(pairs()) = [0] >= [0] = active(pairs()) [1] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] mark(head(X)) = [0 0 0]X >= [0 0 0]X = active(head(mark(X))) [1 0 0] [0 0 0] [1 0 0] [1 0 0] mark(tail(X)) = [0 0 0]X >= [0 0 0]X = active(tail(mark(X))) [1 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 0] [0 0 1] [0 0 0] [0 0 1] [1 1 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 0] [0 0 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 1 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 1 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] head(mark(X)) = [0 0 0]X >= [0 0 0]X = head(X) [1 0 0] [1 0 0] [1 1 0] [1 0 0] head(active(X)) = [0 0 0]X >= [0 0 0]X = head(X) [1 1 0] [1 0 0] [1 0 0] [1 0 0] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = tail(X) [1 0 0] [1 0 0] [1 1 0] [1 0 0] tail(active(X)) = [0 0 0]X >= [0 0 0]X = tail(X) [1 1 0] [1 0 0] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(odds()) -> mark(incr(pairs())) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [head](x0) = [0 0 0]x0 [1 1 0] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1] [odds] = [0] [0], [0] [pairs] = [0] [0], [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [0], [1 1 0] [0] [active](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [nats] = [0] [0] orientation: [0] [0] active(nats()) = [1] >= [1] = mark(cons(0(),incr(nats()))) [0] [0] [1] [0] active(odds()) = [1] >= [1] = mark(incr(pairs())) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS + [1] >= [0 0 0]X + [0 0 0]XS + [1] = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(nats()) = [1] >= [1] = active(nats()) [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 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [1 0 0] [0] [1 0 0] [0] mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(incr(mark(X))) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(pairs()) = [1] >= [1] = active(pairs()) [0] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(head(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] mark(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(tail(mark(X))) [0 0 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 0] [0 0 0] [0 0 0] [0 0 0] [1 1 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 0] [0 0 0] [0 0 0] [0 0 0] [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 1 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 1 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] [0] [1 0 0] head(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = head(X) [1 1 0] [1] [1 1 0] [1 1 0] [0] [1 0 0] head(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = head(X) [1 1 0] [1] [1 1 0] [1 0 0] [1 0 0] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = tail(X) [1 0 0] [1 0 0] [1 1 0] [1 0 0] tail(active(X)) = [0 0 0]X >= [0 0 0]X = tail(X) [1 1 0] [1 0 0] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(incr(X)) -> active(incr(mark(X))) mark(pairs()) -> active(pairs()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [head](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [s](x0) = [0 0 0]x0 [0 0 1] , [0] [pairs] = [0] [0], [1 0 1] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 1]x0 + [0 1 0]x1 [0 0 1] [0 0 0] , [1 1 0] [0] [incr](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [0] [0] = [0] [0], [active](x0) = x0 , [0] [nats] = [0] [0] orientation: [0] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [0] [0] [1 0 1] [1 1 0] [0] [1 0 1] [1 1 0] active(incr(cons(X,XS))) = [0 0 1]X + [0 1 0]XS + [0] >= [0 0 1]X + [0 1 0]XS = mark(cons(s(X),incr(XS))) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [0] [0] mark(nats()) = [0] >= [0] = active(nats()) [0] [0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] mark(cons(X1,X2)) = [0 0 1]X1 + [0 1 0]X2 >= [0 0 1]X1 + [0 1 0]X2 = active(cons(mark(X1),X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 1] [1] [1 1 1] [0] mark(incr(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = active(incr(mark(X))) [0 0 1] [1] [0 0 1] [1] [0] [0] mark(pairs()) = [0] >= [0] = active(pairs()) [0] [0] [1 0 1] [1 0 1] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [0 0 1] [0 0 1] [1 0 1] [1 0 1] mark(head(X)) = [0 0 0]X >= [0 0 0]X = active(head(mark(X))) [0 0 1] [0 0 1] [1 0 1] [1 0 1] mark(tail(X)) = [0 0 0]X >= [0 0 0]X = active(tail(mark(X))) [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 1]X1 + [0 1 0]X2 >= [0 0 1]X1 + [0 1 0]X2 = cons(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 1]X1 + [0 1 0]X2 >= [0 0 1]X1 + [0 1 0]X2 = cons(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 1] [0] [1 1 0] [0] incr(mark(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = incr(X) [0 0 1] [1] [0 0 1] [1] [1 1 0] [0] [1 1 0] [0] incr(active(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = incr(X) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 1] [0 0 1] [1 0 1] [1 0 0] head(mark(X)) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] head(active(X)) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 1] [0 0 1] [1 0 1] [1 0 0] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = tail(X) [0 0 1] [0 0 1] [1 0 0] [1 0 0] tail(active(X)) = [0 0 0]X >= [0 0 0]X = tail(X) [0 0 1] [0 0 1] problem: active(nats()) -> mark(cons(0(),incr(nats()))) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(pairs()) -> active(pairs()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [tail](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [head](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1] [pairs] = [0] [0], [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 0 0] [1 0 1] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 1] [0] [incr](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [0] = [1] [0], [1 0 0] [active](x0) = [0 1 0]x0 [0 1 1] , [0] [nats] = [1] [0] orientation: [0] [0] active(nats()) = [1] >= [1] = mark(cons(0(),incr(nats()))) [1] [1] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [0] active(incr(cons(X,XS))) = [0 0 0]X + [0 0 0]XS + [1] >= [0 0 0]X + [0 0 0]XS + [1] = mark(cons(s(X),incr(XS))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] mark(nats()) = [1] >= [1] = active(nats()) [1] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(mark(X1),X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] mark(0()) = [1] >= [1] = active(0()) [1] [1] [1] [1] mark(pairs()) = [1] >= [0] = active(pairs()) [0] [0] [1 0 0] [0] [1 0 0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(head(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(head(mark(X))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] mark(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = active(tail(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(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 1] [0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [0] [1 0 1] [0] incr(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = incr(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [0] [1 0 1] [0] incr(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = incr(X) [0 0 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] [0] [1 0 0] [0] head(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = head(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] head(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = head(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] tail(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] tail(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = tail(X) [0 0 0] [0] [0 0 0] [0] problem: active(nats()) -> mark(cons(0(),incr(nats()))) mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(pairs()) -> active(pairs()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [tail](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 1 0] [head](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [pairs] = [0] [0], [1 1 0] [mark](x0) = [0 0 0]x0 [1 0 1] , [1 1 0] [1 1 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [incr](x0) = [0 0 0]x0 [0 0 0] , [0] [0] = [0] [1], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 1] , [0] [nats] = [1] [1] orientation: [1] [0] active(nats()) = [0] >= [0] = mark(cons(0(),incr(nats()))) [1] [1] [1] [1] mark(nats()) = [0] >= [0] = active(nats()) [1] [1] [1 1 0] [1 1 0] [1 1 0] [1 1 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(mark(X1),X2)) [1 1 1] [1 1 0] [1 0 1] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [0] [0] mark(pairs()) = [0] >= [0] = active(pairs()) [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(mark(X))) [1 1 0] [0 0 0] [1 1 0] [1 1 0] mark(head(X)) = [0 0 0]X >= [0 0 0]X = active(head(mark(X))) [1 1 0] [0 0 0] [1 1 0] [1] [1 1 0] [1] mark(tail(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(tail(mark(X))) [1 1 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [1 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 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 0] [0 0 1] [0 0 0] [1 1 0] [1 0 0] incr(mark(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] incr(active(X)) = [0 0 0]X >= [0 0 0]X = incr(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] head(mark(X)) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] head(active(X)) = [0 0 0]X >= [0 0 0]X = head(X) [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] [0] tail(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] tail(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(X) [0 0 0] [0] [0 0 0] [0] problem: mark(nats()) -> active(nats()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(pairs()) -> active(pairs()) mark(s(X)) -> active(s(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(tail(X)) -> active(tail(mark(X))) cons(mark(X1),X2) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) incr(mark(X)) -> incr(X) incr(active(X)) -> incr(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=1 interpretation: [tail](x0) = 2x0 + 4, [head](x0) = 2x0 + 3, [s](x0) = 4x0 + 6, [pairs] = 0, [mark](x0) = 2x0 + 2, [cons](x0, x1) = 2x0 + x1 + 5, [incr](x0) = 2x0, [0] = 1, [active](x0) = x0, [nats] = 0 orientation: mark(nats()) = 2 >= 0 = active(nats()) mark(cons(X1,X2)) = 4X1 + 2X2 + 12 >= 4X1 + X2 + 9 = active(cons(mark(X1),X2)) mark(0()) = 4 >= 1 = active(0()) mark(pairs()) = 2 >= 0 = active(pairs()) mark(s(X)) = 8X + 14 >= 8X + 14 = active(s(mark(X))) mark(head(X)) = 4X + 8 >= 4X + 7 = active(head(mark(X))) mark(tail(X)) = 4X + 10 >= 4X + 8 = active(tail(mark(X))) cons(mark(X1),X2) = 4X1 + X2 + 9 >= 2X1 + X2 + 5 = cons(X1,X2) cons(active(X1),X2) = 2X1 + X2 + 5 >= 2X1 + X2 + 5 = cons(X1,X2) incr(mark(X)) = 4X + 4 >= 2X = incr(X) incr(active(X)) = 2X >= 2X = incr(X) s(mark(X)) = 8X + 14 >= 4X + 6 = s(X) s(active(X)) = 4X + 6 >= 4X + 6 = s(X) head(mark(X)) = 4X + 7 >= 2X + 3 = head(X) head(active(X)) = 2X + 3 >= 2X + 3 = head(X) tail(mark(X)) = 4X + 8 >= 2X + 4 = tail(X) tail(active(X)) = 2X + 4 >= 2X + 4 = tail(X) problem: mark(s(X)) -> active(s(mark(X))) cons(active(X1),X2) -> cons(X1,X2) incr(active(X)) -> incr(X) s(active(X)) -> s(X) head(active(X)) -> head(X) tail(active(X)) -> tail(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [tail](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [1 0 0] [0] [head](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [s](x0) = [0 0 0]x0 + [0] [0 1 1] [1], [1 0 1] [mark](x0) = [0 0 1]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [1 1 0] [0 0 0] , [1 0 0] [0] [incr](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 0] [1] [active](x0) = [0 1 1]x0 + [0] [0 0 0] [0] orientation: [1 1 1] [1] [1 0 1] [1] mark(s(X)) = [0 1 1]X + [1] >= [0 0 1]X + [1] = active(s(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [1 1 1] [0 0 0] [1] [1 1 0] [0 0 0] [1 0 0] [1] [1 0 0] [0] incr(active(X)) = [0 1 1]X + [1] >= [0 1 0]X + [1] = incr(X) [0 1 1] [0] [0 1 0] [0] [1 0 0] [1] [1 0 0] [0] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1] [1 0 0] [0] head(active(X)) = [0 1 1]X + [1] >= [0 1 0]X + [1] = head(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [0] tail(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = tail(X) [0 1 1] [0] [0 1 0] [0] problem: mark(s(X)) -> active(s(mark(X))) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1] [1 0 1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X = active(s(mark(X))) [0 0 0] [0] [0 0 0] problem: Qed