Abstract
Equational reasoning is one of the key features of pure functional languages such as Haskell. To date, however, such reasoning always took place externally to Haskell, either manually on paper, or mechanised in a theorem prover. This article shows how equational reasoning can be performed directly and seamlessly within Haskell itself, and be checked using Liquid Haskell. In particular, language learners --- to whom external theorem provers are out of reach --- can benefit from having their proofs mechanically checked. Concretely, we show how the equational proofs and derivations from Graham's textbook can be recast as proofs in Haskell (spoiler: they look essentially the same).
Highlights
IntroductionWe mechanize equational reasoning in Haskell using an expressive type system
Advocates for pure functional languages such as Haskell have long argued that a key benefit of these languages is the ability to reason equationally, using basic mathematics to reason about, verify, and derive programs
To demonstrate the applicability of our approach, we present a series of examples1 that replay equational reasoning, program optimizations, and program calculations from the literature
Summary
We mechanize equational reasoning in Haskell using an expressive type system. Doing so remains faithful to the traditional techniques for verifying and deriving programs, while enjoying the added benefit of being mechanically checked for correctness This approach is well-suited to beginners because the language of proofs is Haskell itself. Even though these proofs can be performed in external tools such as Agda, Coq, Dafny, Isabelle or Lean, equational reasoning using Liquid Haskell is unique in that the proofs are literally just Haskell functions. It can be used by any Haskell programmer or learner
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