YES Problem: active(zeros()) -> mark(cons(0(),zeros())) active(tail(cons(X,XS))) -> mark(XS) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [top](x0) = 4x0, [ok](x0) = x0, [proper](x0) = x0, [tail](x0) = x0 + 1, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 3 orientation: active(zeros()) = 3 >= 3 = mark(cons(0(),zeros())) active(tail(cons(X,XS))) = X + XS + 1 >= XS = mark(XS) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2)) tail(mark(X)) = X + 1 >= X + 1 = mark(tail(X)) proper(zeros()) = 3 >= 3 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) top(mark(X)) = 4X >= 4X = top(proper(X)) top(ok(X)) = 4X >= 4X = top(active(X)) problem: active(zeros()) -> mark(cons(0(),zeros())) active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [proper](x0) = [0 0 0]x0 + [1] [0 0 1] [0], [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [0] = [0] [0], [1 0 1] [active](x0) = [1 0 1]x0 [0 0 0] , [0] [zeros] = [0] [1] orientation: [1] [0] active(zeros()) = [1] >= [0] = mark(cons(0(),zeros())) [0] [0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] active(cons(X1,X2)) = [1 0 1]X1 + [1 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] active(tail(X)) = [1 0 1]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] tail(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(tail(X)) [0 0 0] [0 0 0] [0] [0] proper(zeros()) = [1] >= [0] = ok(zeros()) [1] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [0] [0] [1 0 0] [0] [1 0 0] proper(tail(X)) = [0 0 0]X + [1] >= [0 0 0]X = tail(proper(X)) [0 0 1] [0] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [0 0 0]x0 [1 0 1] , [1 0 0] [ok](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [1] [0] orientation: [1 0 1] [1 0 0] [1 0 0] [1 0 0] active(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(active(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] active(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(active(X)) [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [1] [1 0 0] [1] tail(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(tail(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] proper(zeros()) = [1] >= [0] = ok(zeros()) [0] [0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] proper(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 0 0] [0 0 0] [1 1 1] [1] [1 1 0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(proper(X)) [1 1 1] [1] [1 0 0] [1 0 1] [1 0 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [1 0 1] [1 0 0] problem: active(cons(X1,X2)) -> cons(active(X1),X2) active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [top](x0) = [0 0 0]x0 + [0] [1 0 0] [0], [1 1 1] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 1] [0] [proper](x0) = [1 1 1]x0 + [0] [0 0 0] [1], [tail](x0) = x0 , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 1] [0] [cons](x0, x1) = [0 1 1]x0 + [0 1 0]x1 + [0] [0 0 0] [0 0 0] [1], [0] [0] = [0] [0], [1 1 1] [active](x0) = [1 1 0]x0 [0 0 1] , [1] [zeros] = [0] [0] orientation: [1 1 1] [1 1 1] [1] [1 1 1] [1 0 1] [0] active(cons(X1,X2)) = [1 1 1]X1 + [1 1 1]X2 + [0] >= [1 1 1]X1 + [0 1 0]X2 + [0] = cons(active(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [1 1 1] active(tail(X)) = [1 1 0]X >= [1 1 0]X = tail(active(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = mark(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] tail(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(tail(X)) [0 0 0] [0] [0 0 0] [0] [1] [1] proper(zeros()) = [1] >= [0] = ok(zeros()) [1] [1] [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] proper(cons(X1,X2)) = [1 1 1]X1 + [1 1 1]X2 + [1] >= [1 1 1]X1 + [1 1 1]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 1 1] [0] [1 1 1] [0] proper(tail(X)) = [1 1 1]X + [0] >= [1 1 1]X + [0] = tail(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 1 1] [1] [1 1 1] [1 1 1] [1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] tail(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = ok(tail(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [1] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [1 1 1] [0] [1 1 1] [0] problem: active(tail(X)) -> tail(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [top](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [ok](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 1 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 0 1] [0] [tail](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [1 0 0] [1] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 1] [0 0 0] [0], [0] [0] = [0] [0], [1 1 0] [active](x0) = [0 1 0]x0 [0 0 1] , [0] [zeros] = [0] [0] orientation: [1 1 1] [1] [1 1 1] [0] active(tail(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = tail(active(X)) [0 0 1] [0] [0 0 1] [0] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [1] cons(mark(X1),X2) = [0 0 0]X1 + [0 1 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [0] [0 0 1] [0 0 0] [0] [1 0 1] [1] [1 0 1] [1] tail(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = mark(tail(X)) [0 0 1] [0] [0 0 1] [0] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [1] [1] [1 1 1] [1 1 0] [1] [1 1 0] [1 1 0] [1] proper(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [1] [1 1 1] [1] [1 1 0] [1] proper(tail(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = tail(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 1 1] [1 1 0] [1] [1 1 1] [1 1 0] [1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 1] [1] [1 1 1] [1] tail(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(tail(X)) [0 0 1] [1] [0 0 1] [1] [1 1 0] [1 1 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [top](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [0] [ok](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [1 0 0] [proper](x0) = [0 0 1]x0 [1 0 0] , [1 0 0] [1] [tail](x0) = [0 0 1]x0 + [1] [1 0 0] [1], [mark](x0) = x0 , [1 0 0] [1 0 0] [1] [cons](x0, x1) = [1 0 0]x0 + [0 0 1]x1 + [1] [1 0 0] [1 0 0] [1], [0] [0] = [0] [1], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 0] , [0] [zeros] = [0] [1] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(mark(X1),X2) = [1 0 0]X1 + [0 0 1]X2 + [1] >= [1 0 0]X1 + [0 0 1]X2 + [1] = mark(cons(X1,X2)) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] tail(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = mark(tail(X)) [1 0 0] [1] [1 0 0] [1] [0] [0] proper(zeros()) = [1] >= [1] = ok(zeros()) [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] proper(cons(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [1 0 0]X2 + [1] = cons(proper(X1),proper(X2)) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [0] [0] proper(0()) = [1] >= [1] = ok(0()) [0] [0] [1 0 0] [1] [1 0 0] [1] proper(tail(X)) = [1 0 0]X + [1] >= [1 0 0]X + [1] = tail(proper(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] cons(ok(X1),ok(X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = ok(cons(X1,X2)) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] tail(ok(X)) = [1 0 0]X + [1] >= [0 0 0]X + [1] = ok(tail(X)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] [1 0 0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(active(X)) [1 0 0] [0] [1 0 0] problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) tail(mark(X)) -> mark(tail(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 0 1]x0 [0 0 1] , [1 0 1] [tail](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [0] [mark](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] [0] = [0] [1], [1] [zeros] = [0] [0] orientation: [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(mark(X1),X2) = [0 0 1]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 1] [1] [1 0 1] [0] tail(mark(X)) = [0 0 1]X + [1] >= [0 0 0]X + [0] = mark(tail(X)) [0 0 1] [1] [0 0 1] [1] [1] [1] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 1]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [1] [1] [1 0 1] [1 0 1] proper(tail(X)) = [0 0 1]X >= [0 0 1]X = tail(proper(X)) [0 0 1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [0 0 1]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [1 0 1] tail(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(tail(X)) [0 0 1] [0 0 1] problem: cons(mark(X1),X2) -> mark(cons(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [proper](x0) = [0 1 1]x0 [0 0 0] , [1 0 1] [tail](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1], [1 0 1] [1 0 0] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [0] = [1] [0], [0] [zeros] = [0] [0] orientation: [1 0 1] [1 0 0] [1] [1 0 1] [1 0 0] [0] cons(mark(X1),X2) = [1 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = mark(cons(X1,X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [0] [0] proper(zeros()) = [0] >= [0] = ok(zeros()) [0] [0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] proper(cons(X1,X2)) = [1 0 1]X1 + [0 0 0]X2 >= [1 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(0()) = [1] >= [0] = ok(0()) [0] [0] [1 0 1] [1 0 0] proper(tail(X)) = [0 0 0]X >= [0 0 0]X = tail(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 0] cons(ok(X1),ok(X2)) = [1 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [1 0 1] tail(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(tail(X)) [0 0 0] [0 0 0] problem: proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(tail(X)) -> tail(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [ok](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [proper](x0) = [1 1 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [tail](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 1] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [1] [0] = [0] [0], [1] [zeros] = [0] [0] orientation: [1] [1] proper(zeros()) = [1] >= [0] = ok(zeros()) [1] [0] [1 1 0] [1 1 1] [1] [1 1 0] [1 1 0] [1] proper(cons(X1,X2)) = [1 1 0]X1 + [1 1 1]X2 + [1] >= [1 1 0]X1 + [1 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [1] [1] proper(0()) = [1] >= [0] = ok(0()) [1] [0] [1 1 0] [1] [1 1 0] [0] proper(tail(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = tail(proper(X)) [0 0 0] [1] [0 0 0] [0] [1 0 1] [1 0 1] [0] [1 0 0] [1 0 1] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 1] [0] [1 0 0] tail(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X = ok(tail(X)) [0 0 0] [0] [0 0 0] problem: proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [1 0 1]x0 [0 0 1] , [1 1 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [0] [tail](x0) = x0 + [1] [1], [1 0 0] [1 0 1] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0], [0] [zeros] = [1] [0] orientation: [1] [0] proper(zeros()) = [1] >= [0] = ok(zeros()) [1] [0] [1 1 0] [1 1 1] [1] [1 1 0] [1 1 0] [1] proper(cons(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [1] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] cons(ok(X1),ok(X2)) = [1 0 1]X1 + [1 0 1]X2 + [1] >= [1 0 0]X1 + [1 0 1]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] tail(ok(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = ok(tail(X)) [0 0 1] [1] [0 0 1] [1] problem: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [ok](x0) = [1 0 0]x0 [0 0 0] , [1 1 0] [proper](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [tail](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 1 0]x0 + [0 1 1]x1 + [1] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0] orientation: [1 1 0] [1 1 1] [1] [1 1 0] [1 1 0] [0] proper(cons(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = cons(proper(X1),proper(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] proper(0()) = [0] >= [0] = ok(0()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] cons(ok(X1),ok(X2)) = [1 0 0]X1 + [1 0 0]X2 + [1] >= [1 0 0]X1 + [1 0 0]X2 = ok(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] tail(ok(X)) = [1 0 0]X >= [1 0 0]X = ok(tail(X)) [0 0 0] [0 0 0] problem: proper(0()) -> ok(0()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) tail(ok(X)) -> ok(tail(X)) Matrix Interpretation Processor: dim=3 interpretation: [0] [ok](x0) = x0 + [0] [1], [1 0 0] [1] [proper](x0) = [0 0 0]x0 + [1] [0 1 0] [0], [1 0 1] [tail](x0) = [0 0 1]x0 [0 1 1] , [1 0 0] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 1] , [0] [0] = [1] [0] orientation: [1] [0] proper(0()) = [1] >= [1] = ok(0()) [1] [1] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 1] [0] cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = ok(cons(X1,X2)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 1] [1] [1 0 1] [1] [1 0 1] [0] tail(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [0] = ok(tail(X)) [0 1 1] [1] [0 1 1] [1] problem: Qed