module ind-ind where

open import Data.Unit


mutual
    data Type : Set where
        ℕ : Type
        Bool : Type
        Eq : (A : Type) → Term A → Term A → Type

    data Term : Type → Set where
        zero : Term ℕ
        succ : Term ℕ → Term ℕ
        true : Term Bool
        false : Term Bool

-- Eq is usuful to define things like refl
refl : (A : Type) → (x : Term A) → Type
refl A x = Eq A x x 

-- The BaseOrInd type family encodes information about whether an element of type Term t 
-- is a "base" or "inductive" case, defaulting to ⊤ for all possibilities.
BaseOrInd : {t : Type} → Term t → Set
BaseOrInd zero = ⊤
BaseOrInd (succ b) = BaseOrInd b
BaseOrInd true = ⊤
BaseOrInd false = ⊤