Subsequences

A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements (Wikipedia).

Checking whether a word is a subsequence of another word can be done in linear time (in the length of the second word) and is a nice example of an optimal greedy algorithm.

Author: Jean-Christophe Filliâtre (CNRS)

use int.Int
use map.Map
use seq.Seq

type char
type word = seq char

predicate subsequence (v u: word) (f: int -> int) =
  (* f maps v's characters to u's characters *)
  (forall i. 0 <= i < length v -> 0 <= f i < length u /\ v[i] = u[f i]) /\
  (* in a strictly increasing way *)
  (forall i j. 0 <= i < j < length v -> f i < f j)

val eq (x y: char) : bool ensures { result <-> x = y }

let is_subsequence (v u: word) : bool
  ensures { result <-> exists f. subsequence v u f }
= let rec aux i j (ghost f)
    requires { 0 <= i <= length v }
    requires { 0 <= j <= length u }
    requires { subsequence v[0..i] u[0..j] f }
    requires { forall f. subsequence v u f -> i < length v -> f i >= j }
    variant  { length u - j }
    ensures  { result <-> exists f. subsequence v u f }
  = i = length v ||
    j < length u && if eq v[i] u[j] then aux (i+1) (j+1) (Map.set f i j)
                                    else aux     i (j+1) f
  in
  aux 0 0 (fun i -> i)