--- Towers of Hanoi is a puzzle with three rods and n disks of decresing size.
---
--- The disks are stacked ontop of each other through the first rod.
--- The aim of the game is to move the stack of disks to another rod while
--- following these rules:
---
--- 1. Only one disk can be moved at a time
--- 2. You may only move a disk from the top of one of the stacks to the top of another stack
--- 3. No disk may be moved on top of a smaller disk
---
--- The function ;hanoi; computes the sequence of moves to solve puzzle.

module Hanoi;

open import Stdlib.Prelude;

--- Concatenation of ;String;s
infixr 5 ++str;
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";

--- Produce a singleton List
singleton : {A : Type} → A → List A;
singleton a ≔ a ∷ nil;

--- Produce a ;String; representation of a ;List ℕ;
showList : List ℕ → String;
showList xs ≔ "[" ++str intercalate "," (map natToStr xs) ++str "]";

--- A Peg represents a peg in the towers of Hanoi game
inductive Peg {
  left : Peg;
  middle : Peg;
  right : Peg;
};

showPeg : Peg → String;
showPeg left ≔ "left";
showPeg middle ≔ "middle";
showPeg right ≔ "right";


--- A Move represents a move between pegs
inductive Move {
  move : Peg → Peg → Move;
};

showMove : Move → String;
showMove (move from to) ≔ showPeg from ++str " -> " ++str showPeg to;

--- Produce a list of ;Move;s that solves the towers of Hanoi game
hanoi : ℕ → Peg → Peg → Peg → List Move;
hanoi zero _ _ _ ≔ nil;
hanoi (suc n) p1 p2 p3 ≔ hanoi n p1 p3 p2 ++ singleton (move p1 p2) ++ hanoi n p3 p2 p1;

main : IO;
main ≔ putStrLn (unlines (map showMove (hanoi 5 left middle right)));

end;