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open import Data.Unit
open import Data.Bool
open import Data.Nat
open import Data.Product
module _ where
{-
This odd approach gives you 2 choices when defining an indexed type:
1. Use a McBride-Dagand encoding so your datatype
"need not store its indices".
2. Use a standard "propositional" encoding, whose semantics
is defined without an external equality type.
Using either choice, we still get "general" indexed types
in the Dybjer-Setzer sense. That is, the
initial algebra constructor can only be matched against
when the index constraints are met. In contrast,
"restricted" indexed types are encoded as a parameterized type
with explicit propositional equality arguments, and can
always match against the initial algebra before an equality
argument (i.e. restricted types are a subset of all
Agda types).
-}
----------------------------------------------------------------------
data Desc (I : Set) : Set₁ where
: I Desc I
: I Desc I Desc I
: (A : Set) (D : A Desc I) Desc I
⟦_⟧ : {I : Set} (D : Desc I) (X : I I Set) Set
⟦ `ι i ⟧ X =
-- take diagonal of X
⟦ `δ i D ⟧ X = X i i × ⟦ D ⟧ X
⟦ `σ A D ⟧ X = Σ A λ a ⟦ D a ⟧ X
idx : {I : Set} (D : Desc I) (X : I I Set) (xs : ⟦ D ⟧ X) I
idx (`ι i) X tt = i
idx (`δ i D) X (x , xs) = idx D X xs
idx (`σ A D) X (a , xs) = idx (D a) X xs
-- Combine McBride-Dagand indexed descriptions
-- where we compute a description from an index
-- with Dybjer-Setzer "general" type families
-- where we compute an index from arguments.
-- Key: definition of ⟦_⟧ by diagonal in inductive case
data μ {I : Set} (R : I Desc I) (i : I) : I Set where
init : (xs : ⟦ R i ⟧ (μ R)) μ R i (idx (R i) (μ R) xs)
----------------------------------------------------------------------
module CompVec where
-- compute desc from index
VecD : Set Desc ℕ
VecD A zero = `ι zero
VecD A (suc n) = `σ A λ _ `δ n (`ι (suc n))
-- take diagonal of μ (VecD A)
Vec : Set Set
Vec A n = μ (VecD A) n n
nil : (A : Set) Vec A zero
nil A = init tt
cons : (A : Set) (n : ℕ) A Vec A n Vec A (suc n)
cons A n a xs = init (a , xs , tt)
----------------------------------------------------------------------
module PropVec where
-- ignore the index
-- but do not need separate propositional equality (≡)
VecD : Set Desc ℕ
VecD A _ = `σ Bool λ where
true `ι zero
false `σ ℕ λ n `σ A λ _ `δ n (`ι (suc n))
-- take diagonal of μ (VecD A)
Vec : Set Set
Vec A n = μ (VecD A) n n
nil : (A : Set) Vec A zero
nil A = init (true , tt)
cons : (A : Set) (n : ℕ) A Vec A n Vec A (suc n)
cons A n a xs = init (false , n , a , xs , tt)
----------------------------------------------------------------------
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