I am learning Idris and as a personal exercise, I would like to implement a Primes type, consisting of all the prime numbers.
Is there a way in idris to define a new type starting from a type and a property, which will select all the elements of the starting type for which the property holds true? In my case, is there a way to define Primes as the set of Nat such that n <= p and n|p => n=1 or n=p?
If this is not possible, should I define prime numbers constructing them inductively using some kind of sieve?
I like copumpkin's Agda definition of Prime, which looks like this in Idris:
data Divides : Nat -> Nat -> Type where
MkDivides : (q : Nat) ->
n = q * S m ->
Divides (S m) n
data Prime : Nat -> Type where
MkPrime : GT p 1 ->
((d : Nat) -> Divides d p -> Either (d = 1) (d = p))
-> Prime p
Read as "if p is divisible by d, then d must be 1 or p" - a common definition for primality.
Proving this by hand for a number can be pretty tedious:
p2' : (d : Nat) -> Divides d 2 -> Either (d = 1) (d = 2)
p2' Z (MkDivides _ _) impossible
p2' (S Z) (MkDivides Z Refl) impossible
p2' (S Z) (MkDivides (S Z) Refl) impossible
p2' (S Z) (MkDivides (S (S Z)) Refl) = Left Refl
p2' (S Z) (MkDivides (S (S (S _))) Refl) impossible
p2' (S (S Z)) (MkDivides Z Refl) impossible
p2' (S (S Z)) (MkDivides (S Z) Refl) = Right Refl
p2' (S (S Z)) (MkDivides (S (S _)) Refl) impossible
p2' (S (S (S _))) (MkDivides Z Refl) impossible
p2' (S (S (S _))) (MkDivides (S _) Refl) impossible
p2 : Prime 2
p2 = MkPrime (LTESucc (LTESucc LTEZero)) p2'
It's also very involved to write a decision procedure for this. That'll be a big exercise! You'll probably find the rest of the definitions useful for that: