437 research outputs found
Conversion of HOL Light proofs into Metamath
We present an algorithm for converting proofs from the OpenTheory interchange
format, which can be translated to and from any of the HOL family of proof
languages (HOL4, HOL Light, ProofPower, and Isabelle), into the ZFC-based
Metamath language. This task is divided into two steps: the translation of an
OpenTheory proof into a Metamath HOL formalization, ,
followed by the embedding of the HOL formalization into the main ZFC
foundations of the main Metamath library, . This
process provides a means to link the simplicity of the Metamath foundations to
the intense automation efforts which have borne fruit in HOL Light, allowing
the production of complete Metamath proofs of theorems in HOL Light, while also
proving that HOL Light is consistent, relative to Metamath's ZFC
axiomatization.Comment: 14 pages, 2 figures, accepted to Journal of Formalized Reasonin
Formalizing Computability Theory via Partial Recursive Functions
We present an extension to the library of the Lean theorem
prover formalizing the foundations of computability theory. We use primitive
recursive functions and partial recursive functions as the main objects of
study, and we use a constructive encoding of partial functions such that they
are executable when the programs in question provably halt. Main theorems
include the construction of a universal partial recursive function and a proof
of the undecidability of the halting problem. Type class inference provides a
transparent way to supply G\"{o}del numberings where needed and encapsulate the
encoding details.Comment: 16 pages, accepted to ITP 201
The billiard inside an ellipse deformed by the curvature flow
The billiard dynamics inside an ellipse is integrable. It has zero
topological entropy, four separatrices in the phase space, and a continuous
family of convex caustics: the confocal ellipses. We prove that the curvature
flow destroys the integrability, increases the topological entropy, splits the
separatrices in a transverse way, and breaks all resonant convex caustics.Comment: 13 pages, 1 figur
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