module Polymorphism;

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;
id _ a := a;

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

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

nil' : (E : Type) → List E;
nil' A := nil;

-- currying
nil'' : (E : Type) → List E;
nil'' E := nil;

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

p : Pair Bool Bool;
p := mkPair true false;

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

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

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

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

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

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

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

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

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

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

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

l1 : List Nat;
l1 := cons zero nil;

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

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

end;