Budge: a programming language and a theorem prover

We present a simple programming language based on G"odel numbering and prime factorization, enhanced with explicit, scoped loops, allowing for easy program composition. Further, we will present a theorem prover that allows expressing and working with formal systems. The theorem prover is simple as it relies merely on a substitution rule and set equality to derive theorems. Finally, we will represent the programming language in the theorem prover. We will show the syntax and semantics of both, and then provide a few example programs and their evaluation.