(** Random Access Lists. (Okasaki, "Purely Functional Data Structures", 10.1.2.) The code below uses polymorphic recursion (both in the logic and in the programs). Author: Jean-Christophe FilliĆ¢tre (CNRS) *) module RandomAccessList use int.Int use int.ComputerDivision use list.List use list.Length use list.Nth use option.Option type ral 'a = | Empty | Zero (ral ('a, 'a)) | One 'a (ral ('a, 'a)) function flatten (l: list ('a, 'a)) : list 'a = match l with | Nil -> Nil | Cons (x, y) l1 -> Cons x (Cons y (flatten l1)) end let rec lemma length_flatten (l:list ('a, 'a)) ensures { length (flatten l) = 2 * length l } variant { l } = match l with | Cons (_,_) q -> length_flatten q | Nil -> () end function elements (l: ral 'a) : list 'a = match l with | Empty -> Nil | Zero l1 -> flatten (elements l1) | One x l1 -> Cons x (flatten (elements l1)) end let rec size (l: ral 'a) : int variant { l } ensures { result = length (elements l) } = match l with | Empty -> 0 | Zero l1 -> 2 * size l1 | One _ l1 -> 1 + 2 * size l1 end let rec cons (x: 'a) (l: ral 'a) : ral 'a variant { l } ensures { elements result = Cons x (elements l) } = match l with | Empty -> One x Empty | Zero l1 -> One x l1 | One y l1 -> Zero (cons (x, y) l1) end let rec lemma nth_flatten (i: int) (l: list ('a, 'a)) requires { 0 <= i < length l } variant { l } ensures { match nth i l with | None -> false | Some (x0, x1) -> Some x0 = nth (2 * i) (flatten l) /\ Some x1 = nth (2 * i + 1) (flatten l) end } = match l with | Nil -> () | Cons _ r -> if i > 0 then nth_flatten (i-1) r end let rec lookup (i: int) (l: ral 'a) : 'a requires { 0 <= i < length (elements l) } variant { i, l } ensures { nth i (elements l) = Some result } = match l with | Empty -> absurd | One x l1 -> if i = 0 then x else lookup (i-1) (Zero l1) | Zero l1 -> let (x0, x1) = lookup (div i 2) l1 in if mod i 2 = 0 then x0 else x1 end let rec tail (l: ral 'a) : ral 'a requires { elements l <> Nil } variant { l } ensures { match elements l with | Nil -> false | Cons _ l -> elements result = l end } = match l with | Empty -> absurd | One _ l1 -> Zero l1 | Zero l1 -> let (_, x1) = lookup 0 l1 in One x1 (tail l1) end let rec update (i: int) (y: 'a) (l: ral 'a) : ral 'a requires { 0 <= i < length (elements l) } variant { i, l} ensures { nth i (elements result) = Some y} ensures { forall j. 0 <= j < length (elements l) -> j <> i -> nth j (elements result) = nth j (elements l) } ensures { length (elements result) = length (elements l) } ensures { match result, l with | One _ _, One _ _ | Zero _, Zero _ -> true | _ -> false end } = match l with | Empty -> absurd | One x l1 -> if i = 0 then One y l1 else match update (i-1) y (Zero l1) with | Empty | One _ _ -> absurd | Zero l1 -> One x l1 end | Zero l1 -> let (x0, x1) = lookup (div i 2) l1 in let l1' = update (div i 2) (if mod i 2 = 0 then (y,x1) else (x0,y)) l1 in assert { forall j. 0 <= j < length (elements l) -> j <> i -> match nth (div j 2) (elements l1) with | None -> false | Some (x0,_) -> Some x0 = nth (2 * (div j 2)) (elements l) end && nth j (elements l) = nth j (elements (Zero l1')) }; Zero l1' end end (** A straightforward encapsulation with a list ghost model (in anticipation of module refinement) *) module RAL use int.Int use list.List use list.Length use option.Option use list.Nth use RandomAccessList type t 'a = { r: ral 'a; ghost l: list 'a } invariant { l = elements r } let empty () : t 'a ensures { result.l = Nil } = { r = Empty; l = Nil } let size (t: t 'a) : int ensures { result = length t.l } = size t.r let cons (x: 'a) (s: t 'a) : t 'a ensures { result.l = Cons x s.l } = { r = cons x s.r; l = Cons x s.l } let lookup (i: int) (s: t 'a) : 'a requires { 0 <= i < length s.l } ensures { Some result = nth i s.l } = lookup i s.r end (** Model using sequences instead of lists *) module RandomAccessListWithSeq use int.Int use int.ComputerDivision use seq.Seq type ral 'a = | Empty | Zero (ral ('a, 'a)) | One 'a (ral ('a, 'a)) function flatten (s: seq ('a, 'a)) : seq 'a = create (2 * length s) (fun i -> let (x0, x1) = s[div i 2] in if mod i 2 = 0 then x0 else x1) lemma cons_flatten : forall x y :'a,s:seq ('a,'a). let a = flatten (cons (x,y) s) in let b = cons x (cons y (flatten s)) in a = b by a == b by forall i. 0 <= i < a.length -> a[i] = b[i] by (i <= 1 so (cons (x,y) s)[div i 2] = (x,y)) \/ (i >= 2 so (cons (x,y) s)[div i 2] = s[div (i-2) 2] ) function elements (l: ral 'a) : seq 'a = match l with | Empty -> empty | Zero l1 -> flatten (elements l1) | One x l1 -> cons x (flatten (elements l1)) end let rec size (l: ral 'a) : int variant { l } ensures { result = length (elements l) } = match l with | Empty -> 0 | Zero l1 -> 2 * size l1 | One _ l1 -> 1 + 2 * size l1 end let rec cons (x: 'a) (l: ral 'a) : ral 'a variant { l } ensures { elements result == cons x (elements l) } = match l with | Empty -> One x Empty | Zero l1 -> One x l1 | One y l1 -> Zero (cons (x, y) l1) end let rec lookup (i: int) (l: ral 'a) : 'a requires { 0 <= i < length (elements l) } variant { i, l } ensures { (elements l)[i] = result } = match l with | Empty -> absurd | One x l1 -> if i = 0 then x else lookup (i-1) (Zero l1) | Zero l1 -> let (x0, x1) = lookup (div i 2) l1 in if mod i 2 = 0 then x0 else x1 end let rec tail (l: ral 'a) : ral 'a requires { 0 < length (elements l) } variant { l } ensures { elements result == (elements l)[1..] } = match l with | Empty -> absurd | One _ l1 -> Zero l1 | Zero l1 -> let (_, x1) as p = lookup 0 l1 in let tl = tail l1 in assert { elements l1 == cons p (elements tl) }; One x1 tl end (** update in O(log n) for this, we need to pass a function that will update the element when we find it *) function setf (s: seq 'a) (i: int) (f: 'a -> 'a) : seq 'a = set s i (f s[i]) let function aux (i: int) (f: 'a -> 'a) : ('a, 'a) -> ('a, 'a) = fun z -> let (x,y) = z in if mod i 2 = 0 then (f x, y) else (x, f y) let rec fupdate (f: 'a -> 'a) (i: int) (l: ral 'a) : ral 'a requires { 0 <= i < length (elements l) } variant { i, l } ensures { elements result == setf (elements l) i f } = match l with | Empty -> absurd | One x l1 -> if i = 0 then One (f x) l1 else cons x (fupdate f (i-1) (Zero l1)) | Zero l1 -> let ul1 = fupdate (aux i f) (div i 2) l1 in let res = Zero ul1 in assert { forall j. 0 <= j < length (elements res) -> (elements res)[j] = (setf (elements l) i f)[j] by div j 2 <> div i 2 -> (elements ul1)[div j 2] = (elements l1)[div j 2] }; res end let function f (y: 'a) : 'a -> 'a = fun _ -> y let update (i: int) (y: 'a) (l: ral 'a) : ral 'a requires { 0 <= i < length (elements l) } ensures { elements result == set (elements l) i y} = fupdate (f y) i l end