Canonical disjoint NP-pairs of propositional proof systems

We prove that every disjoint NP-pair is polynomial-time, many-one equivalent to the canonical disjoint NP-pair of some propositional proof system. Therefore, the degree structure of the class of disjoint NP-pairs and of all canonical pairs is identical. We show that this degree structure is not superficial: Assuming there exist P-inseparable disjoint NP-pairs, countable distributive lattice can be embedded into every interval of polynomial NP- of disjoint pairs by maps that preserve the least and greatest element, respectively. As one consequence of this embedding, under the same assumption, there exist intermediate disjoint NP-pairs. That is, if is a P-separable disjoint NP-pair and is a P-inseparable disjoint NP-pair, then there exist P-inseparable, incomparable NP-pairs and whose degrees lie strictly between and . Furthermore, between any two disjoint NP-pairs that are comparable and inequivalent, such a diamond exists.