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Towers of Hanoi

Classical recursive algorithm


Authors: Andrei Paskevich

Topics: Ghost code

Tools: Why3

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module Hanoi
  (* a simple version where the disks are natural numbers from 1 to n *)

  use int.Int
  use list.List
  use list.Length
  use list.SortedInt

  type tower = {
    mutable rod : list int;
  } invariant {
    sorted rod
  }

  function prepend (n: int) (s: list int) : list int

  axiom prepend_def_zero :
    forall s: list int, n: int.
      n <= 0 -> prepend n s = s

  axiom prepend_def_succ :
    forall s: list int, n: int.
      n > 0 -> prepend n s = prepend (n - 1) (Cons n s)

  let move (a b: tower) (ghost n: int) (ghost s: list int)
    requires { a.rod = Cons n s }
    requires { match b.rod with Nil -> true | Cons x _ -> x > n end }
    ensures  { a.rod = s }
    ensures  { b.rod = Cons n (old b.rod) }
  = match a.rod with
    | Cons x r ->
        a.rod <- r;
        b.rod <- Cons x b.rod
    | Nil -> absurd
    end

  let rec hanoi_rec (a b c: tower) (n: int) (ghost s: list int)
    requires { a.rod = prepend n s }
    requires { match b.rod with Nil -> true | Cons x _ -> x > n end }
    requires { match c.rod with Nil -> true | Cons x _ -> x > n end }
    variant  { n }
    ensures  { a.rod = s }
    ensures  { b.rod = prepend n (old b.rod) }
    ensures  { c.rod = old c.rod }
  = if n > 0 then begin
      let ghost t = c.rod in
      hanoi_rec a c b (n-1) (Cons n s);
      move a b n s;
      hanoi_rec c b a (n-1) t
    end

  let tower_of_hanoi (a b c: tower)
    requires { a.rod = prepend (length a.rod) Nil }
    requires { b.rod = c.rod = Nil }
    ensures  { b.rod = old a.rod }
    ensures  { a.rod = c.rod = Nil }
  = hanoi_rec a b c (length a.rod) Nil

end

module Tower_of_Hanoi
  (* a generalized version where the disks are arbitrary integers *)

  use int.Int
  use list.List
  use list.Length

  use list.RevAppend
  clone list.RevSorted with type t = int, predicate le = (<),
  goal Transitive.Trans

  type tower = {
    mutable rod : list int;
  } invariant {
    Incr.sorted rod
  }

  let move (a b: tower) (ghost x: int) (ghost s: list int)
    requires { a.rod = Cons x s }
    requires { match b.rod with Nil -> true | Cons y _ -> x < y end }
    ensures  { a.rod = s }
    ensures  { b.rod = Cons x (old b.rod) }
  = match a.rod with
    | Cons x r ->
        a.rod <- r;
        b.rod <- Cons x b.rod
    | Nil -> absurd
    end

  let rec hanoi_rec (a b c: tower) (n: int) (ghost p s: list int)
    requires { length p = n /\ Decr.sorted p }
    requires { a.rod = rev_append p s }
    requires { compat p b.rod }
    requires { compat p c.rod }
    variant  { n }
    ensures  { a.rod = s }
    ensures  { b.rod = rev_append p (old b.rod) }
    ensures  { c.rod = old c.rod }
  = if n > 0 then begin
      let ghost t = c.rod in
      let x,r = match p with Cons x r -> x,r | Nil -> absurd end in
      hanoi_rec a c b (n-1) r (Cons x s);
      move a b x s;
      hanoi_rec c b a (n-1) r t
    end

  let tower_of_hanoi (a b c: tower)
    requires { b.rod = c.rod = Nil }
    ensures  { b.rod = old a.rod }
    ensures  { a.rod = c.rod = Nil }
  = hanoi_rec a b c (length a.rod) (ghost rev_append a.rod Nil) Nil

end

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Why3 Proof Results for Project "tower_of_hanoi"

Theory "tower_of_hanoi.Hanoi": fully verified

ObligationsAlt-Ergo 2.0.0
VC for tower0.00
VC for move---
split_goal_right
type invariant0.01
type invariant0.02
postcondition0.01
postcondition0.01
unreachable point0.00
VC for hanoi_rec---
split_goal_right
variant decrease0.00
precondition0.01
precondition0.02
precondition0.01
precondition0.00
precondition0.01
variant decrease0.00
precondition0.00
precondition0.01
precondition0.86
postcondition0.01
postcondition0.01
postcondition0.01
postcondition0.00
postcondition0.00
postcondition0.00
VC for tower_of_hanoi---
split_goal_right
precondition0.00
precondition0.01
precondition0.00
postcondition0.00
postcondition0.00

Theory "tower_of_hanoi.Tower_of_Hanoi": fully verified

ObligationsAlt-Ergo 2.0.0Alt-Ergo 2.3.0
RevSorted.Transitive.Trans0.00---
VC for tower0.00---
VC for move------
split_goal_right
type invariant0.01---
type invariant0.03---
postcondition0.01---
postcondition0.02---
unreachable point0.01---
VC for hanoi_rec------
split_goal_right
unreachable point0.01---
variant decrease0.01---
precondition0.01---
precondition0.01---
precondition---1.27
precondition---1.10
precondition0.01---
precondition0.05---
variant decrease0.01---
precondition0.02---
precondition0.01---
precondition0.05---
precondition---1.78
postcondition0.01---
postcondition0.01---
postcondition0.01---
postcondition0.01---
postcondition0.03---
postcondition0.01---
VC for tower_of_hanoi------
split_goal_right
precondition0.02---
precondition0.01---
precondition0.02---
precondition0.02---
postcondition0.02---
postcondition0.01---