## White and black balls

A small puzzle from The Correctness Problem in Computer Science (Robert S. Boyer, J. Strother Moore, Academic Prees, 1982).

**Auteurs:** Jean-Christophe Filliâtre

**Catégories:** Historical examples / Divisibility

**Outils:** Why3

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A small puzzle from

The Correctness Problem in Computer Science Robert S. Boyer, J. Strother Moore Academic Prees, 1982

(suggested by Shankar at the Sixth Summer School on Formal Techniques)

Given a bag containing white and black balls, repeatedly - remove a pair from the bag - if they are of the same color, insert a white ball - if they are of difference colors, insert a black ball What is the color of the last ball?

Author: Jean-Christophe FilliĆ¢tre (CNRS)

module WhiteAndBlackBalls use int.Int use ref.Ref use number.Parity (* answer: the color of the last ball depends on the parity of the number of black balls, which is an invariant of the process *) let last_is_black (b0 w0: int) : bool requires { b0 >= 0 && w0 >= 0 && b0 + w0 >= 1 } ensures { result <-> odd b0 } = let ref b = b0 in let ref w = w0 in while b + w >= 2 do invariant { b >= 0 && w >= 0 && b + w >= 1 } invariant { odd b <-> odd b0 } variant { b + w } let x, y = any (x: int, y: int) ensures { 0 <= x && 0 <= y && x + y = 2 && x <= b && y <= w } in if x = 0 || y = 0 then begin (* same color -> insert a white *) b <- b - x; w <- w - y + 1 end else begin (* different color -> insert a black *) b <- b - x + 1; w <- w - y end done; b > 0 end

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# Why3 Proof Results for Project "white_and_black_balls"

## Theory "white_and_black_balls.WhiteAndBlackBalls": fully verified

Obligations | Alt-Ergo 2.0.0 |

VC for last_is_black | 9.30 |