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|
(** String search
Author: Jean-Christophe Filliâtre (CNRS)
from a SPARK demo by Claire Dross (Adacore) *)
module Spec
use int.Int
use string.Char
use string.String
(* ***** all this to be moved to String? ***** *)
predicate (==) (x y: string) = eq_string x y
predicate matches (pat text: string) (p: int) =
0 <= p <= length text - length pat /\
substring text p (length pat) == pat
(* ***** *)
end
module Occurs
use int.Int
use mach.int.Int63
use string.String
use string.Char
use string.OCaml
use Spec
let partial occurs (pat text: string) (p: int63) : bool
requires { 0 <= p <= length text - length pat }
ensures { result <-> matches pat text p }
= let (ghost n) = length text in
let m = length pat in
for i = 0 to m - 1 do
invariant { substring text p i == substring pat 0 i }
assert { p + i <= n };
if not (eq_char text[p + i] pat[i]) then return false
done;
true
end
module Naive
use int.Int
use mach.int.Int63
use string.String
use string.Char
use string.OCaml
use Spec
use Occurs
let partial search1 (pat text: string) : int63
requires { length pat <= length text }
ensures { -1 <= result <= length text - length pat }
ensures { if result = -1 then
forall j. not (matches pat text j)
else
matches pat text result }
=
let m = length pat in
let n = length text in
for i = 0 to n - m do
invariant { forall j. 0 <= j < i -> substring text j m <> pat }
if occurs pat text i then return i;
done;
-1
let partial search2 (pat text: string) : int63
requires { length pat <= length text }
ensures { -1 <= result <= length text - length pat }
ensures { if result = -1 then
forall j. not (matches pat text j)
else
matches pat text result }
=
let m = length pat in
let n = length text in
for i = 0 to n - m do
invariant { forall j. 0 <= j < i -> substring text j m <> pat }
if sub text i m = pat then return i;
done;
-1
end
module BadShiftTable
use int.Int
use mach.int.Int63
use string.String
use string.Char
use string.OCaml
use Spec
use Occurs
clone fmap.MapImp as M with type key = char
(*
------------------------+---+----------------
| C |
-----+-----+---+--------+---+----------------
| ... | C | ..!C.. |
+-----+---+--------+
0 m
<----sht c--->
*)
type bad_shift_table = {
pat: string;
sht: M.t int63;
}
invariant { forall j c. 0 <= j < length pat -> c = pat[j] ->
M.mem c sht }
invariant { forall c. M.mem c sht -> 1 <= sht c <= length pat + 1 }
invariant { forall c. M.mem c sht -> forall j. 1 <= j < sht c ->
pat[length pat - j] <> c }
by { pat = ""; sht = M.create () }
let partial make_table (pat: string) : bad_shift_table
= let m = length pat in
let sht = M.create () in
for i = 0 to m - 1 do
invariant { forall j c. 0 <= j < i -> c = pat[j] -> M.mem c sht }
invariant { forall c. M.mem c sht -> 1 <= sht c <= m + 1 }
invariant { forall c. M.mem c sht -> forall j. m - sht c < j < i ->
pat[j] <> c }
M.add pat[i] (m - i) sht;
done;
{ pat = pat; sht = sht }
let lemma shift (bst: bad_shift_table) (text: string) (i: int63)
requires { 0 <= i <= length text - length bst.pat }
requires { M.mem text[i + length bst.pat] bst.sht }
ensures { forall j. i < j < i + M.find text[i + length bst.pat] bst.sht ->
j <= length text - length bst.pat ->
substring text j (length bst.pat) <> bst.pat }
= let m = String.length bst.pat in
let c = Char.get text (to_int i + m) in
let lemma aux (j: int)
requires { i < j < i + M.find c bst.sht }
requires { j <= length text - m }
ensures { substring text j m <> bst.pat }
= assert { (substring text j m)[i + m - j] = c };
assert { bst.pat[m - (j - i)] <> c } in
()
let lemma no_shift (bst: bad_shift_table) (text: string) (i: int63)
requires { 0 <= i < length text - length bst.pat }
requires { not (M.mem text[i + length bst.pat] bst.sht) }
ensures { forall j. i < j <= i + length bst.pat ->
j <= length text - length bst.pat ->
substring text j (length bst.pat) <> bst.pat }
= let m = String.length bst.pat in
let c = Char.get text (to_int i + m) in
assert { forall j. 0 <= j < m -> bst.pat[j] <> c };
let lemma aux (j: int)
requires { i < j <= i + m }
requires { j <= length text - m }
ensures { substring text j m <> bst.pat }
= assert { (substring text j m)[i + m - j] = c } in
()
let partial search (bst: bad_shift_table) (text: string) : int63
requires { length bst.pat <= length text }
ensures { -1 <= result <= length text - length bst.pat }
ensures { if result = -1 then
forall j. not (matches bst.pat text j)
else
matches bst.pat text result }
= let pat = bst.pat in
let m = length pat in
let n = length text in
let ref i = 0 in
while i <= n - m do
invariant { 0 <= i <= n }
invariant { forall j. 0 <= j < i -> j <= n - m ->
substring text j m <> pat }
variant { n - m - i }
if occurs pat text i then return i;
if i = n - m then break;
let c = text[i + m] in
i <- i + if M.mem c bst.sht then (shift bst text i; M.find c bst.sht)
else (no_shift bst text i; m + 1)
done;
-1
end
|
