mb64 / cubical-free-monads

Verified free monads in cubical agda

Verified (cubical) free monads

An experiment in Cubical Agda.

Free monads are not legit monads

Free monads aren't real monads according to the monad laws. For example, for all monads m >>= return should equal m. However, in free monads, the data structure Bind m Return is clearly not equal to m.

In a sense, free monads are simply deferring the responsibility of satisfying the monad laws to the interpreter that actually runs Bind.

They can be in Cubical Agda

This is where Cubical Agda and Higher Inductive Types come in. With HITs, you can define you own equalities, and functions that pattern match on them have to prove that they respect that equality.

FreeReader, in FreeReader.agda, defines all of the monad laws as higher inductive paths:

-- The Reader monad, as a free monad
data FreeReader (R : Set) : Set → Set₁ where
  Pure : A → FreeReader R A
  Bind : FreeReader R A → (A → FreeReader R B) → FreeReader R B
  Ask : FreeReader R R

  -- Monad laws
  LeftId : ∀ {A B}
    → (a : A)
    → (f : A → FreeReader R B)
    → Bind (Pure a) f ≡ f a
  RightId : ∀ {A}
    → (m : FreeReader R A)
    → Bind m Pure ≡ m
  Assoc : ∀ {A B C}
    → (m : FreeReader R A)
    → (f : A → FreeReader R B)
    → (g : B → FreeReader R C)
    → Bind (Bind m f) g ≡ Bind m (λ x → Bind (f x) g)

Then, it can be proven to follow the functor, applicative functor, and monad laws.

(This is a lot better than without cubical type theory: usually, it's impossible to even prove the normal Reader monad is a functor, since that requires function extensionality.)

File layout

• Class.agda has the Funtor, Applicative, and Monad classes, including all their laws
• FreeReader.agda has FreeReader R's instances, including all the proofs that they're legit
• Interpreter.agda has the runFree function, which runs FreeReader, and proves that it respects the monad laws

Non-ASCII Unicode used

Char   Emacs
 →      \r
 λ      \lambda
 ≡      \==
 ₁      \_1
 ₂      \_2
 ∘      \o
 ∀      \forall
 ∙      \.
 ⟨      \<
 ⟩      \>
 ℓ      \ell
 ′      \'1
 ″      \'2