The DPRM Theorem in Isabelle (Short Paper)

Formalizing a Diophantine Representation of the Set of Prime Numbers

“…Larchey-Wendling and Forster [8] formalized the DPRM theorem in Coq and Bayer et al [1] in Isabelle/HOL. Both formalizations develop register machines, Minsky machines, advanced properties of the Pell equation and prove that exponentiation is diophantine.…”

“…. The latter approach is used in the Coq [8] and Isabelle [1] formalizations. The similarity can be analysed by noticing the relation…”

“…Of course ψ is more general, while α has more properties. However, the condition [7]) is explicitly stated in the publication describing the Coq formalization [8] and the previous version of the Isabelle formalization [1] while [2] and in Mizar [14] 3 . Irrespective of the considered sequence, all formalizations prove that a = α b (c) or a = ψ b (c) can be represented using an (implicit) diophantine relation, stated as a combination of less complicated diophantine relations.…”

“…Additionally these sub-relation use additional explicit unknowns and may also have implicit ones in these relations. For example Bayer et al [1] express 3 < b ∧ a = α b (c) using 6 explicit unknowns and 15 relations including, e.g., 4 uses of equivalence ≡. Carneiro [2] uses 5 additional unknowns explicitly and several congruences.…”

“…. , x m ) = 0 (1) Eliminating the single universal quantifies from Equation 1, it becomes the condition defining a diophantine relation. However, the research undertaken by Robinson, Davis, and Putnam to eliminate this quantifier required almost 30 years.…”