## Subsequence

Checking whether a word is a subsequence of another word.

Authors: Jean-Christophe Filliâtre

Topics: Words / Algorithms

Tools: Why3

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# 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)

```