Dependent type involving nat addition

Question

I need to define two versions of an operation with a slightly different definition. It is a series of compositions with Nat indices involved.

open import Data.Nat

data Hom : ℕ → ℕ → Set where
  id    : (m : ℕ) → Hom m m
  _∘_   : ∀ {m n k} → Hom n k → Hom m n → Hom m k
  p     : (n : ℕ) → Hom (suc n) n

p1 : (m n : ℕ) → Hom (m + n) n
p1 zero n    = id n
p1 (suc m) n = p1 m n ∘ p (m + n)

p2 : (m n : ℕ) → Hom (m + n) n
p2 zero n    = id n
p2 (suc m) n = {!!} -- p n ∘ p2 m (1 + n)

-- Goal: Hom (suc (m + n)) n
-- Have: Hom (m + suc n) n

I would like to define both p1 and p2 and be able to use them interchangeably. Is this doable?

Answer 1

You can define p2 by direct recursion (no subst or rewriting) over _+_ using the trick described here. Looks like this:

record Homable (H : ℕ → ℕ → Set) : Set where
  field
    id-able  : (m : ℕ) → H m m
    _∘-able_ : ∀ {m n k} → H n k → H m n → H m k
    p-able   : (n : ℕ) → H (suc n) n

suc-homable : ∀ {H} → Homable H → Homable (λ m n -> H (suc m) (suc n))
suc-homable homable = record
  { id-able  = λ m → id-able (suc m)
  ; _∘-able_ = _∘-able_
  ; p-able   = λ m → p-able (suc m)
  } where open Homable homable

p2-go : ∀ {H} → Homable H → (m : ℕ) → H m 0
p2-go homable  zero   = id-able 0 where
  open Homable homable
p2-go homable (suc m) = p-able 0 ∘-able p2-go (suc-homable homable) m where
  open Homable homable

plus-homable-hom : ∀ k → Homable (λ m n → Hom (m + k) (n + k))
plus-homable-hom k = record
  { id-able  = λ n → id (n + k)
  ; _∘-able_ = _∘_
  ; p-able   = λ n → p (n + k)
  }

p2 : (m n : ℕ) → Hom (m + n) n
p2 m n = p2-go (plus-homable-hom n) m

The cost is that you need to maintain those Homable records which is somewhat tedious, but to my experience proving things about functions defined this way is simpler than about functions defined in terms of subst or over _+′_ (unless you never want to coerce _+′_ to _+_, of course).

Answer 2

Well, the value you provide has a type that is equal to the type of the hole, but Agda does not see this fact. More formally, the two types are propositionally equal but not judgementally equal. The problem is caused by the index m + suc n, which is propositionally but not judgementally equal to suc m + n because of how addition is defined. One way to solve your problem is to manually explain to Agda that the two types are equal:

open import Data.Nat
open import Data.Nat.Properties
open import Relation.Binary.PropositionalEquality

data Hom : ℕ → ℕ → Set where
  id    : (m : ℕ) → Hom m m
  _∘_   : ∀ {m n k} → Hom n k → Hom m n → Hom m k
  p     : (n : ℕ) → Hom (suc n) n

p1 : (m n : ℕ) → Hom (m + n) n
p1 zero n    = id n
p1 (suc m) n = p1 m n ∘ p (m + n)

p2 : (m n : ℕ) → Hom (m + n) n
p2 zero n    = id n
p2 (suc m) n = subst (λ k → Hom k n) (+-suc m n) (p n ∘ p2 m (suc n))

However, this approach is not without downsides, as p2 (suc m) n is now not judgementally equal to your intended definition but to the expression above involving subst.

The problem seems essentially linked to what you're trying to do: IIUC, p1 and p2 are actually provably equal but defined using a different recursion structure. That's fine, but then the indices of your result type should follow the same recursion structure, i.e. you should define p2 using a different version of + that recurses in the appropriate way for p2:

_+′_ : ℕ → ℕ → ℕ
zero +′ n = n
suc m +′ n = m +′ suc n

p2′ : (m n : ℕ) → Hom (m +′ n) n
p2′ zero n = id n
p2′ (suc m) n = p n ∘ p2′ m (suc n)

However, this has another downside that the type of p1 and p2′ are no longer judgementally equal (but still propositionally equal though).

Another thing you can try is to use Agda's rewrite rules to give _+_ satisfy additional judgemental equalities, but this is dangerous as it may break some of Agda's desirable qualities as a logic. In this case, I suspect it's fine, but I'd have to check.

In summary, there are a number of things you can try but none is without downsides. Which is your best option depends on what you're trying to use this for.