# System FR as Foundations for Stainless

## TL;DR

This paper introduces System FR, a formal foundation and verifier for higher-order functional programs with generics and recursive types, ensuring safety and termination through formal proofs and practical implementation.

## Contribution

It formalizes System FR, a calculus supporting polymorphism, dependent types, and recursion, and implements a verifier integrated with Stainless for verifying complex Scala programs.

## Key findings

- Verified 14,000 lines of Scala code including algorithms and data structures.
- Proved soundness and safety of the verifier using Coq.
- Demonstrated efficiency on real-world functional programs.

## Abstract

We present the design, implementation, and foundation of a verifier for higher-order functional programs with generics and recursive data types. Our system supports proving safety and termination using preconditions, postconditions and assertions. It supports writing proof hints using assertions and recursive calls. To formalize the soundness of the system we introduce System FR, a calculus supporting System F polymorphism, dependent refinement types, and recursive types (including recursion through contravariant positions of function types). Through the use of sized types, System FR supports reasoning about termination of lazy data structures such as streams. We formalize a reducibility argument using the Coq proof assistant and prove the soundness of a type-checker with respect to call-by-value semantics, ensuring type safety and normalization for typeable programs. Our program verifier is implemented as an alternative verification-condition generator for the Stainless tool, which relies on existing SMT-based solver backend for automation. We demonstrate the efficiency of our approach by verifying a collection of higher-order functional programs comprising around 14000 lines of polymorphic higher-order Scala code, including graph search algorithms, basic number theory, monad laws, functional data structures, and assignments from popular Functional Programming MOOCs.