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We present an extension to the mathlib 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.doi:10.4230/lipics.itp.2019.12 dblp:conf/itp/Carneiro19 fatcat:h3py5mpwxbfvzgu5gjjsi7qc34