--- Pascal's triangle is the arrangement of binomial coefficients in a triangular array.
--- The rows of the triangle are staggered so that each number can be computed as the

--- sum of the numbers to the left and right in the previous row.
--- The function ;pascal; produces the triangle to a given depth.
module PascalsTriangle;
  open import Stdlib.Prelude;

  zipWith : {A : Type}  {B : Type}  {C : Type}  (A  B  C)  List A  List B  List C;
  zipWith f nil _ := nil;
  zipWith f _ nil := nil;
  zipWith f (x  xs) (y  ys) := f x y  zipWith f xs ys;

  --- Return a list of repeated applications of a given function
  iterate : {A : Type}  Nat  (A  A)  A  List A;
  iterate zero _ _ := nil;
  iterate (suc n) f a := a  iterate n f (f a);

  --- Produce a singleton List
  singleton : {A : Type}  A  List A;
  singleton a := a  nil;

  infixr 5 ++str;
  --- Concatenation of ;String;s
  axiom ++str : String  String  String;

  compile ++str {
    c  "concat";

  --- Concatenates a list of strings
  --- ;concat (("a"  nil)  "b"  nil); evaluates to ;"a"  "b"  nil;
  concat : List String  String;
  concat := foldl (++str) "";

  intercalate : String  List String  String;
  intercalate sep xs := concat (intersperse sep xs);

  --- Joins a list of strings with the newline character
  unlines : List String  String;
  unlines := intercalate "\n";

  showList : List Nat  String;
  showList xs := "[" ++str intercalate "," (map natToStr xs) ++str "]";

  --- Compute the next row of Pascal's triangle
  pascalNextRow : List Nat  List Nat;
  pascalNextRow row := zipWith (+) (singleton zero ++ row) (row ++ singleton zero);

  --- Produce Pascal's triangle to a given depth
  pascal : Nat  List (List Nat);
  pascal rows := iterate rows pascalNextRow (singleton one);

  main : IO;
  main := putStrLn (unlines (map showList (pascal 10)));