mergesort_list: generic code, a bit of cleaning up

parent 981d11a6
(* Sorting a list of integers using mergesort *)
module M
module Elt
use import int.Int
use import list.Length
use import list.SortedInt
use import list.Append
use import list.Permut
use export int.Int
use export list.Length
use export list.Append
use export list.Permut
let split (l0 : list 'a)
type elt
predicate le elt elt
clone relations.TotalPreOrder with type t = elt, predicate rel = le
clone export list.Sorted with type t = elt, predicate le = le
end
module Merge
use export Elt
let rec merge (l1 l2: list elt) : list elt
requires { sorted l1 /\ sorted l2 }
ensures { sorted result /\ permut result (l1 ++ l2) }
variant { length l1 + length l2 }
= match l1, l2 with
| Nil, _ -> l2
| _, Nil -> l1
| Cons x1 r1, Cons x2 r2 ->
if le x1 x2 then Cons x1 (merge r1 l2) else Cons x2 (merge l1 r2)
end
end
(* TODO: proof to be completed
module EfficientMerge
use export Elt
use import list.Mem
use import list.Reverse
use import list.RevAppend
let rec merge_aux (acc l1 l2: list elt) : list elt
requires { sorted (reverse acc) /\ sorted l1 /\ sorted l2 }
requires { forall x y: elt. mem x acc -> mem y l1 -> le x y }
requires { forall x y: elt. mem x acc -> mem y l2 -> le x y }
ensures { sorted result /\ permut result (acc ++ l1 ++ l2) }
variant { length l1 + length l2 }
= match l1, l2 with
| Nil, _ -> rev_append acc l2
| _, Nil -> rev_append acc l1
| Cons x1 r1, Cons x2 r2 ->
if le x1 x2 then merge_aux (Cons x1 acc) r1 l2
else merge_aux (Cons x2 acc) l1 r2
end
let merge (l1 l2: list elt) : list elt
requires { sorted l1 /\ sorted l2 }
ensures { sorted result /\ permut result (l1 ++ l2) }
=
merge_aux Nil l1 l2
end
*)
module Mergesort
use import Merge
let split (l0: list 'a) : (list 'a, list 'a)
requires { length l0 >= 2 }
ensures { let (l1, l2) = result in
1 <= length l1 /\ 1 <= length l2 /\ permut l0 (l1 ++ l2) }
= let rec split_aux (l1 : list 'a) l2 l variant { length l }
= let rec split_aux (l1 l2 l: list 'a) : (list 'a, list 'a)
requires { length l2 = length l1 \/ length l2 = length l1 + 1 }
ensures { let r1, r2 = result in
(length r2 = length r1 \/ length r2 = length r1 + 1) /\
permut (r1 ++ r2) (l1 ++ (l2 ++ l)) }
variant { length l }
= match l with
| Nil -> (l1, l2)
| Cons x r -> split_aux l2 (Cons x l1) r
......@@ -25,18 +84,9 @@ module M
in
split_aux Nil Nil l0
let rec merge l1 l2 variant { length l1 + length l2 }
requires { sorted l1 /\ sorted l2 }
ensures { sorted result /\ permut result (l1 ++ l2) }
= match l1, l2 with
| Nil, _ -> l2
| _, Nil -> l1
| Cons x1 r1, Cons x2 r2 ->
if x1 <= x2 then Cons x1 (merge r1 l2) else Cons x2 (merge l1 r2)
end
let rec mergesort l variant { length l }
let rec mergesort (l: list elt) : list elt
ensures { sorted result /\ permut result l }
variant { length l }
= match l with
| Nil | Cons _ Nil -> l
| _ -> let l1, l2 = split l in merge (mergesort l1) (mergesort l2)
......
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