Commit 142aeef2 by Jean-Christophe Filliatre

### [gallery] cleaning up balance.mlw

parent 24fc5f16
 (* You are given 8 balls and a Roberval balance. All balls have the same weight, apart from one, which is lighter. Using the balance at most twice, determine the lighter ball. (** {1 Two puzzles involving a Roberval balance} Though this problem is not that difficult (though, you may want to think about it before reading any further), it is an interesting exercise in program specification. The index of the lighter ball is passed as a ghost argument to the program. Thus it cannot be used to compute the answer, but only to write the specification. Note: Ghost code is used to get elegant specifications. Jean-Christophe Filliâtre (CNRS), December 2013 Léon Gondelman (Université Paris-Sud), April 2014 *) module Balance module Roberval use export ref.Refint type outcome = Left | Equal | Right (** the side of the heaviest mass i.e. where the balance leans *) val ghost counter: ref int (** how many times can we use the balance *) val balance (left right: int) : outcome requires { !counter > 0 } ensures { match result with | Left -> left > right | Equal -> left = right | Right -> left < right end } writes { counter } ensures { !counter = old !counter - 1 } end (** You are given 8 balls and a Roberval balance. All balls have the same weight, apart from one, which is lighter. Using the balance at most twice, determine the lighter ball. use import int.Int Though this problem is not that difficult (though, you may want to think about it before reading any further), it is an interesting exercise in program specification. The index of the lighter ball is passed as a ghost argument to the program. Thus it cannot be used to compute the answer, but only to write the specification. *) module Puzzle8 use import Roberval use import array.Array (* All values in balls[lo..hi[ are equal to [w], apart from balls[lb] (** All values in [balls(lo..hi-1)] are equal to [w], apart from balls[lb] which is smaller. *) predicate spec (balls: array int) (lo hi: int) (lb w: int) = 0 <= lo <= lb < hi <= length balls && (forall i: int. lo <= i < hi -> i <> lb -> balls[i] = w) && balls[lb] < w (* Solve the problem for 3 balls, using exactly one weighing *) (** Solve the problem for 3 balls, using exactly one weighing. The solution [lb] is passed as a ghost argument. *) let solve3 (balls: array int) (lo: int) (ghost lb: int) (ghost w: int) : int requires { !counter >= 1 } requires { spec balls lo (lo + 3) lb w } ensures { result = lb } ensures { !counter = old !counter - 1 } = if balls[lo] < balls[lo+1] then lo else if balls[lo] > balls[lo+1] then lo+1 else lo+2 (* Solve the problem for 8 balls, using exactly two weighings *) match balance balls[lo] balls[lo+1] with | Right -> lo | Left -> lo+1 | Equal -> lo+2 end (** Solve the problem for 8 balls, using exactly two weighings. The solution [lb] is passed as a ghost argument. *) let solve8 (balls: array int) (ghost lb: int) (ghost w: int) : int requires { !counter = 2 } requires { spec balls 0 8 lb w } ensures { result = lb } = (* first, compare balls 0,1,2 with balls 3,4,5 *) if balls[0] + balls[1] + balls[2] < balls[3] + balls[4] + balls[5] then solve3 balls 0 lb w else if balls[0] + balls[1] + balls[2] > balls[3] + balls[4] + balls[5] then solve3 balls 3 lb w else (* 0,1,2 = 3,4,5 thus lb must be 6 or 7 *) if balls[6] < balls[7] then 6 else 7 match balance (balls[0] + balls[1] + balls[2]) (balls[3] + balls[4] + balls[5]) with | Right -> solve3 balls 0 lb w | Left -> solve3 balls 3 lb w (* 0,1,2 = 3,4,5 thus lb must be 6 or 7 *) | Equal -> match balance balls[6] balls[7] with Right -> 6 | _ -> 7 end end end (** You are given 12 balls, all of the same weight except one (for which you don't knwo whether it is lighter or heavier) Given a Roberval balance, you have to find the intruder, and determine whether it is lighter or heavier, using the balance at most three times. *) module Balance12 module Puzzle12 use import int.Int use import Roberval use import array.Array (* All values in balls[lo..hi[ are equal to [w], apart from balls[lb] which is of a different weight (lighter or heavier) . *) predicate spec (balls: array int) (lo hi j: int) (w: int) (b: bool) = 0 <= lo <= j < hi <= length balls && (forall i: int. lo <= i < hi -> i <> j -> balls[i] = w) && (if b then balls[j] < w else balls[j] > w) (* j : the index of the different ball, b : the status (lighter or heavier) of the different ball False True meaning heavier and True meaining lighter *) (** The index [j] of the intruder, its status [b] (whether it is lighter or heavier), and the weight [w] of the other balls are all passed as ghost arguments. *) let solve12 (balls: array int) (ghost w j: int) (ghost b: bool) : (int, bool) requires { spec balls 0 12 j w b } requires { !counter = 3 } requires { 0 <= j < 12 = length balls } requires { forall i: int. 0 <= i < 12 -> i <> j -> balls[i] = w } requires { if b then balls[j] < w else balls[j] > w } ensures { result = (j, b) } = if balls[0] + balls[1] + balls[2] + balls[3] = balls[4] + balls[5] + balls[6] + balls[7] then if balls[0] + balls[8] = balls[9] + balls[10] then if balls[0] < balls[11] then (11, False) else (11, True) else if balls[0] + balls[8] < balls[9] + balls[10] then if balls[9] = balls[10] then (8, True) else if balls[9] < balls[10] then (10, False) else (9, False) else if balls[9] = balls[10] then (8, False) else if balls[9] < balls[10] then (9, True) else (10, True) else if balls[0] + balls[1] + balls[2] + balls[3] < balls[4] + balls[5] + balls[6] + balls[7] then if balls[0] + balls[1] + balls[4] = balls[2] + balls[5] + balls[8] then if balls[6] = balls[7] then (3, True) else if balls[6] < balls[7] then (7, False) else (6, False) else if balls[0] + balls[1] + balls[4] < balls[2] + balls[5] + balls[8] then if balls[0] = balls[1] then (5, False) else if balls[0] < balls[1] then (0, True) else (1, True) else if balls[4] = balls[8] then (2, True) else (4, False) else if balls[0] + balls[1] + balls[4] = balls[2] + balls[5] + balls[8] then if balls[6] = balls[7] then (3, False) else if balls[6] < balls[7] then (6, True) else (7, True) else if balls[0] + balls[1] + balls[4] < balls[2] + balls[5] + balls[8] then if balls[2] = balls[5] then (4, True ) else if balls[2] < balls[5] then (5, False) else (2, False) else if balls[0] = balls[1] then (5, True) else if balls[0] < balls[1] then (1, False) else (0, False) match balance (balls[0] + balls[1] + balls[2] + balls[3]) (balls[4] + balls[5] + balls[6] + balls[7]) with | Equal -> (* 0,1,2,3 = 4,5,6,7 *) match balance (balls[0] + balls[8]) (balls[9] + balls[10]) with | Equal -> (* 0,8 = 9,10 *) match balance balls[0] balls[11] with | Right -> (11, False) | _ -> (11, True) end | Right -> (* 0,8 < 9,10 *) match balance balls[9] balls[10] with | Equal -> (8, True) | Right -> (10, False) | Left -> (9, False) end | Left -> (* 0,8 > 9,10 *) match balance balls[9] balls[10] with | Equal -> (8, False) | Right -> (9, True) | Left -> (10, True) end end | Right -> (* 0,1,2,3 < 4,5,6,7 *) match balance (balls[0] + balls[1] + balls[4]) (balls[2] + balls[5] + balls[8]) with | Equal -> (* 0,1,4 = 2,5,8 *) match balance balls[6] balls[7] with | Equal -> (3, True) | Right -> (7, False) | Left -> (6, False) end | Right -> (* 0,1,4 < 2,5,8 *) match balance balls[0] balls[1] with | Equal -> (5, False) | Right -> (0, True) | Left -> (1, True) end | Left -> (* 0,1,4 > 2,5,8 *) match balance balls[4] balls[8] with | Equal -> (2, True) | _ -> (4, False) end end | Left -> (* 0,1,2,3 > 4,5,6,7 *) match balance (balls[0] + balls[1] + balls[4]) (balls[2] + balls[5] + balls[8]) with | Equal -> (* 0,1,4 = 2,5,8 *) match balance balls[6] balls[7] with | Equal -> (3, False) | Right -> (6, True) | Left -> (7, True) end | Right -> (* 0,1,4 < 2,5,8 *) match balance balls[2] balls[5] with | Equal -> (4, True) | Right -> (5, False) | Left -> (2, False) end | Left -> (* 0,1,4 > 2,5,8 *) match balance balls[0] balls[1] with | Equal -> (5, True) | Right -> (1, False) | Left -> (0, False) end end end end \ No newline at end of file
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