Difference between type parameters and indices?

Question

I am new to dependent types and am confused about the difference between the two. It seems people usually say a type is parameterized by another type and indexe

Answer 1

Here is an example of a type paramerised by some value:

open import Data.Nat

infixr 4 _∷_

data ≤List (n : ℕ) : Set where
  []  : ≤List n
  _∷_ : {m : ℕ} -> m ≤ n -> ≤List n -> ≤List n

1≤3 : 1 ≤ 3
1≤3 = s≤s z≤n

3≤3 : 3 ≤ 3
3≤3 = s≤s (s≤s (s≤s z≤n))

example : ≤List 3
example = 3≤3 ∷ 1≤3 ∷ []

It's a type of lists with every element less or equal n. The general intuition is: if some property holds for every inhabitant of a type, then you can abstract it into parameter. There is a mechanical rule also: "The first index can be turned into a new parameter if each constructor has the same variable on the first index position (in the result type)." This quote is from *, you should read it.

• http://people.inf.elte.hu/divip/AgdaTutorial/Sets.Parameters_vs_Indices.html