module PolymorphismHoles;

type Pair (A : Type) (B : Type) :=
  mkPair : A → B → Pair A B;

type Nat :=
  | zero : Nat
  | suc : Nat → Nat;

type List (A : Type) :=
  | nil : List A
  | cons : A → List A → List A;

type Bool :=
  | false : Bool
  | true : Bool;

id : (A : Type) → A → A
  | _ a := a;

terminating
undefined : (A : Type) → A
  | A := undefined A;

add : Nat → Nat → Nat
  | zero b := b
  | (suc a) b := suc (add a b);

fst : (A : Type) → (B : Type) → Pair A B → A
  | _ _ (mkPair a b) := a;

p : Pair Bool Bool := mkPair true false;

swap : (A : Type) → (B : Type) → Pair A B → Pair B A
  | A B (mkPair a b) := mkPair b a;

curry
  : (A : Type)
    → (B : Type)
    → (C : Type)
    → (Pair A B → C)
    → A
    → B
    → C
  | A B C f a b := f (mkPair a b);

ap : (A : Type) → (B : Type) → (A → B) → A → B
  | A B f a := f a;

headDef : (A : Type) → A → List A → A
  | _ d nil := d
  | A _ (cons h _) := h;

ite : (A : Type) → Bool → A → A → A
  | _ true tt _ := tt
  | _ false _ ff := ff;

filter : (A : Type) → (A → Bool) → List A → List A
  | _ f nil := nil
  | _ f (cons x xs) :=
    ite _ (f x) (cons x (filter _ f xs)) (filter _ f xs);

map : (A : Type) → (B : Type) → (A → B) → List _ → List _
  | _ _ f nil := nil
  | _ _ f (cons x xs) := cons (f x) (map _ _ f xs);

zip
  : (A : Type)
    → (B : Type)
    → List A
    → List B
    → List (Pair A B)
  | A _ nil _ := nil
  | _ _ _ nil := nil
  | _ _ (cons a as) (cons b bs) := nil;

zipWith
  : (A : Type)
    → (B : Type)
    → (C : Type)
    → (A → B → C)
    → List A
    → List B
    → List C
  | _ _ C f nil _ := nil
  | _ _ C f _ nil := nil
  | _ _ _ f (cons a as) (cons b bs) :=
    cons (f a b) (zipWith _ _ _ f as bs);

rankn : ((A : Type) → A → A) → Bool → Nat → Pair Bool Nat
  | f b n := mkPair (f _ b) (f _ n);

-- currying
trankn : Pair Bool Nat := rankn id false zero;

l1 : List Nat := cons zero nil;

pairEval : (A : Type) → (B : Type) → Pair (A → B) A → B
  | _ _ (mkPair f x) := f x;

main : Nat :=
  headDef
    _
    (pairEval _ _ (mkPair (add zero) zero))
    (zipWith _ _ _ add l1 l1);