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Equality up to spaces

Check that two strings are equal up to space characters.

Authors: Jean-Christophe Filliâtre

Topics: Strings

Tools: Why3

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Check that two strings are equal up to space characters.

This problem was suggested by Rustan Leino (Amazon Web Services). The trick is in finding a loop invariant that makes the proof easy.

Author: Jean-Christophe FilliĆ¢tre (CNRS)

use int.Int
use seq.Seq
use seq.FreeMonoid

type char
val constant space: char
val function eq (x y: char): bool ensures { result <-> x = y }
type char_string = seq char

The specification uses a function remove_spaces. It is recursively defined over a string, from left to right.

let rec function remove_spaces (s: char_string) : char_string
  variant { length s }
=      if length s = 0  then s
  else if eq s[0] space then remove_spaces s[1..]
                        else cons s[0] (remove_spaces s[1..])

Code and proof.

It would be natural to have a loop invariant such as

remove_spaces x[..i] = remove_spaces y[..j]

Unfortunately, since remove_spaces is defined recursively from left to right, we would have hard time proving such an invariant. So instead we use an invariant which refers to the *suffixes* of x and y.

let rec compare_up_to_spaces (x y: char_string) : bool
  ensures { result <-> remove_spaces x = remove_spaces y }
= let n = length x in
  let m = length y in
  let ref i = 0 in
  let ref j = 0 in
  while i < n || j < m do
    invariant { 0 <= i <= n }
    invariant { 0 <= j <= m }
    invariant { remove_spaces x      = remove_spaces y
            <-> remove_spaces x[i..] = remove_spaces y[j..] }
    variant   { n - i + m - j }
    if i < n && eq x[i] space then begin
      assert { remove_spaces x[i..] == remove_spaces x[i+1..] };
      i <- i + 1
    end else if j < m && eq y[j] space then begin
      assert { remove_spaces y[j..] == remove_spaces y[j+1..] };
      j <- j + 1
    end else begin
      assert { i < n ->
               remove_spaces x[i..] == cons x[i] (remove_spaces x[i+1..]) };
      assert { j < m ->
               remove_spaces y[j..] == cons y[j] (remove_spaces y[j+1..]) };
      if i = n || j = m then return false;
      if not (eq x[i] y[j]) then return false;
      i <- i + 1;
      j <- j + 1
  return true

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Why3 Proof Results for Project "equality_up_to_spaces"

Theory "equality_up_to_spaces.Top": fully verified

ObligationsAlt-Ergo 2.3.3CVC4 1.8Z3 4.8.10
VC for remove_spaces0.010.090.04
VC for compare_up_to_spaces---------
loop invariant init0.000.070.02
loop invariant init0.000.080.03
loop invariant init0.030.920.82
loop variant decrease0.010.070.03
loop invariant preservation0.010.070.04
loop invariant preservation0.010.060.02
loop invariant preservation0.010.07---
loop variant decrease0.010.070.03
loop invariant preservation0.010.050.02
loop invariant preservation0.010.060.04
loop invariant preservation0.010.06---
loop variant decrease0.010.070.04
loop invariant preservation0.010.080.04
loop invariant preservation0.010.070.04
loop invariant preservation0.020.09---