(** {1 Sorting lists using mergesort} Author: Jean-Christophe FilliĆ¢tre (CNRS) *) module Elt use export int.Int use export list.List use export list.Length use export list.Append use export list.Permut type elt val predicate le elt elt clone relations.TotalPreOrder with type t = elt, predicate rel = le, axiom . clone export list.Sorted with type t = elt, predicate le = le, goal Transitive.Trans end (** recursive (and naive) merging of two sorted lists *) module Merge (* : MergeSpec *) clone export Elt with axiom . let rec merge (l1 l2: list elt) : list elt requires { sorted l1 /\ sorted l2 } ensures { sorted result } ensures { 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 (** tail recursive implementation *) module EfficientMerge (* : MergeSpec *) clone export Elt with axiom . use list.Mem use list.Reverse use list.RevAppend lemma sorted_reverse_cons: forall acc x1. sorted (reverse acc) -> (forall x. mem x acc -> le x x1) -> sorted (reverse (Cons x1 acc)) 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 } ensures { 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 (** Mergesort. This implementation splits the input list in two according to even- and odd-order elements (see function `split` below). Thus it is not stable. For a stable implementation, see below module `OCamlMergesort`. *) module Mergesort clone Merge (* or EfficientMerge *) with axiom . 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 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 end in split_aux Nil Nil l0 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) end end (** {2 OCaml's List.sort} There are several ideas here: - instead of splitting the list in two, sort takes the length of the prefix to be sorted; hence there is nothing to do to get the first half and the second half is obtained with chop (which does not allocate at all) - all functions are tail recursive. To avoid the extra cost of List.rev, sort is duplicated in two versions that respectively sort in order and in reverse order (`sort` and `sort_rev`) and merge is duplicated as well (`rev_merge` and `rev_merge_rev`). Note: this is a stable sort, but stability is not proved here. *) module OCamlMergesort clone export Elt with axiom . use list.Mem use list.Reverse use list.RevAppend lemma sorted_reverse_cons: forall acc x1. sorted (reverse acc) -> (forall x. mem x acc -> le x x1) -> sorted (reverse (Cons x1 acc)) lemma sorted_rev_append: forall acc l: list elt. sorted (reverse acc) -> sorted l -> (forall x y. mem x acc -> mem y l -> le x y) -> sorted (reverse (rev_append l acc)) let rec rev_merge (l1 l2 accu: list elt) : list elt requires { sorted (reverse accu) /\ sorted l1 /\ sorted l2 } requires { forall x y: elt. mem x accu -> mem y l1 -> le x y } requires { forall x y: elt. mem x accu -> mem y l2 -> le x y } ensures { sorted (reverse result) } ensures { permut result (accu ++ l1 ++ l2) } variant { length l1 + length l2 } = match l1, l2 with | Nil, _ -> rev_append l2 accu | _, Nil -> rev_append l1 accu | Cons h1 t1, Cons h2 t2 -> if le h1 h2 then rev_merge t1 l2 (Cons h1 accu) else rev_merge l1 t2 (Cons h2 accu) end lemma sorted_reverse_mem: forall x l. sorted (reverse (Cons x l)) -> forall y. mem y l -> le y x lemma sorted_reverse_cons2: forall x l. sorted (reverse (Cons x l)) -> sorted (reverse l) let rec rev_merge_rev (l1 l2 accu: list elt) : list elt requires { sorted accu /\ sorted (reverse l1) /\ sorted (reverse l2) } requires { forall x y: elt. mem x accu -> mem y l1 -> le y x } requires { forall x y: elt. mem x accu -> mem y l2 -> le y x } ensures { sorted result } ensures { permut result (accu ++ l1 ++ l2) } variant { length l1 + length l2 } = match l1, l2 with | Nil, _ -> rev_append l2 accu | _, Nil -> rev_append l1 accu | Cons h1 t1, Cons h2 t2 -> if not (le h1 h2) then rev_merge_rev t1 l2 (Cons h1 accu) else rev_merge_rev l1 t2 (Cons h2 accu) end function prefix int (list 'a) : list 'a axiom prefix_def1: forall l: list 'a. prefix 0 l = Nil axiom prefix_def2: forall n: int, x: 'a, l: list 'a. n > 0 -> prefix n (Cons x l) = Cons x (prefix (n-1) l) let rec lemma prefix_length (n: int) (l: list 'a) requires { 0 <= n <= length l } ensures { length (prefix n l) = n } variant { n } = if n > 0 then match l with Nil -> () | Cons _ r -> prefix_length (n-1) r end let rec lemma prefix_append (n: int) (l1 l2: list 'a) requires { length l1 <= n <= length l1 + length l2 } ensures { prefix n (l1 ++ l2) = prefix (length l1) l1 ++ prefix (n - length l1) l2 } variant { l1 } = match l1 with Nil -> () | Cons _ t -> prefix_append (n-1) t l2 end let rec chop (n: int) (l: list 'a) : list 'a requires { 0 <= n <= length l } ensures { l = prefix n l ++ result } variant { n } = if n = 0 then l else match l with | Cons _ t -> chop (n-1) t | Nil -> absurd end (** `sort n l` sorts `prefix n l` and `rev_sort n l` sorts `prefix n l` in reverse order. *) use mach.int.Int let rec sort (n: int) (l: list elt) : list elt requires { 2 <= n <= length l } ensures { sorted result } ensures { permut result (prefix n l) } variant { n } = if n = 2 then match l with | Cons x1 (Cons x2 _) -> if le x1 x2 then Cons x1 (Cons x2 Nil) else Cons x2 (Cons x1 Nil) | _ -> absurd end else if n = 3 then match l with | Cons x1 (Cons x2 (Cons x3 _)) -> if le x1 x2 then if le x2 x3 then Cons x1 (Cons x2 (Cons x3 Nil)) else if le x1 x3 then Cons x1 (Cons x3 (Cons x2 Nil)) else Cons x3 (Cons x1 (Cons x2 Nil)) else if le x1 x3 then Cons x2 (Cons x1 (Cons x3 Nil)) else if le x2 x3 then Cons x2 (Cons x3 (Cons x1 Nil)) else Cons x3 (Cons x2 (Cons x1 Nil)) | _ -> absurd end else begin let n1 = n / 2 in let n2 = n - n1 in let l2 = chop n1 l in assert { prefix n1 l ++ prefix n2 l2 = prefix n l }; let s1 = rev_sort n1 l in let s2 = rev_sort n2 l2 in rev_merge_rev s1 s2 Nil end with rev_sort (n: int) (l: list elt) : list elt requires { 2 <= n <= length l } ensures { sorted (reverse result) } ensures { permut result (prefix n l) } variant { n } = if n = 2 then match l with | Cons x1 (Cons x2 _) -> if not (le x1 x2) then Cons x1 (Cons x2 Nil) else Cons x2 (Cons x1 Nil) | _ -> absurd end else if n = 3 then match l with | Cons x1 (Cons x2 (Cons x3 _)) -> if not (le x1 x2) then if not (le x2 x3) then Cons x1 (Cons x2 (Cons x3 Nil)) else if not (le x1 x3) then Cons x1 (Cons x3 (Cons x2 Nil)) else Cons x3 (Cons x1 (Cons x2 Nil)) else if not (le x1 x3) then Cons x2 (Cons x1 (Cons x3 Nil)) else if not (le x2 x3) then Cons x2 (Cons x3 (Cons x1 Nil)) else Cons x3 (Cons x2 (Cons x1 Nil)) | _ -> absurd end else begin let n1 = n / 2 in let n2 = n - n1 in let l2 = chop n1 l in assert { prefix n1 l ++ prefix n2 l2 = prefix n l }; let s1 = sort n1 l in let s2 = sort n2 l2 in rev_merge s1 s2 Nil end lemma permut_prefix: forall l: list elt. permut (prefix (length l) l) l let mergesort (l: list elt) : list elt ensures { sorted result /\ permut result l } = let n = length l in if n < 2 then begin assert { sorted l by match l with Nil | Cons _ Nil -> sorted l | _ -> false end }; l end else sort n l end