random_access_list.mlw 7.24 KB
 Jean-Christophe Filliatre committed May 02, 2015 1 2 `````` (** Random Access Lists. `````` Jean-Christophe Filliatre committed May 20, 2015 3 `````` (Okasaki, "Purely Functional Data Structures", 10.1.2.) `````` Jean-Christophe Filliatre committed May 02, 2015 4 5 6 7 `````` The code below uses polymorphic recursion (both in the logic and in the programs). `````` Jean-Christophe Filliatre committed May 20, 2015 8 `````` Author: Jean-Christophe Filliâtre (CNRS) `````` Jean-Christophe Filliatre committed May 02, 2015 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 `````` *) module RandomAccessList use import int.Int use import int.ComputerDivision use import list.List use import list.Length use import list.Nth use import option.Option type ral 'a = | Empty | Zero (ral ('a, 'a)) | One 'a (ral ('a, 'a)) `````` Jean-Christophe Filliatre committed May 20, 2015 25 `````` function flatten (l: list ('a, 'a)) : list 'a `````` Jean-Christophe Filliatre committed May 02, 2015 26 27 28 29 30 `````` = match l with | Nil -> Nil | Cons (x, y) l1 -> Cons x (Cons y (flatten l1)) end `````` Jean-Christophe Filliatre committed May 20, 2015 31 `````` let rec lemma length_flatten (l:list ('a, 'a)) `````` Martin Clochard committed May 04, 2015 32 33 34 35 36 37 38 `````` ensures { length (flatten l) = 2 * length l } variant { l } = match l with | Cons (_,_) q -> length_flatten q | Nil -> () end `````` Jean-Christophe Filliatre committed May 02, 2015 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 `````` 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 `````` Léon Gondelman committed Jul 03, 2015 56 `````` let rec cons (x: 'a) (l: ral 'a) : ral 'a `````` Jean-Christophe Filliatre committed May 02, 2015 57 58 59 60 61 `````` variant { l } ensures { elements result = Cons x (elements l) } = match l with | Empty -> One x Empty | Zero l1 -> One x l1 `````` Léon Gondelman committed Jul 03, 2015 62 `````` | One y l1 -> Zero (cons (x, y) l1) `````` Jean-Christophe Filliatre committed May 02, 2015 63 64 65 66 67 68 69 70 71 72 73 74 75 76 `````` 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 `````` Léon Gondelman committed Jul 03, 2015 77 `````` let rec lookup (i: int) (l: ral 'a) : 'a `````` Jean-Christophe Filliatre committed May 02, 2015 78 79 80 81 82 `````` requires { 0 <= i < length (elements l) } variant { i, l } ensures { nth i (elements l) = Some result } = match l with | Empty -> absurd `````` Léon Gondelman committed Jul 03, 2015 83 84 `````` | 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 `````` Jean-Christophe Filliatre committed May 02, 2015 85 86 87 `````` if mod i 2 = 0 then x0 else x1 end `````` Jean-Christophe Filliatre committed Jun 24, 2015 88 `````` let rec tail (l: ral 'a) : ral 'a `````` Léon Gondelman committed Jun 23, 2015 89 `````` requires { elements l <> Nil } `````` Jean-Christophe Filliatre committed Jun 24, 2015 90 91 92 93 94 `````` variant { l } ensures { match elements l with | Nil -> false | Cons _ l -> elements result = l end } `````` Léon Gondelman committed Jun 23, 2015 95 96 97 `````` = match l with | Empty -> absurd | One _ l1 -> Zero l1 `````` Léon Gondelman committed Jul 03, 2015 98 `````` | Zero l1 -> let (_, x1) = lookup 0 l1 in One x1 (tail l1) `````` Léon Gondelman committed Jun 23, 2015 99 100 `````` end `````` Léon Gondelman committed Jul 03, 2015 101 `````` let rec update (i: int) (y: 'a) (l: ral 'a) : ral 'a `````` Léon Gondelman committed Jun 23, 2015 102 103 104 105 106 107 108 109 110 111 112 113 114 `````` 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 `````` Léon Gondelman committed Jul 03, 2015 115 `````` match update (i-1) y (Zero l1) with `````` Léon Gondelman committed Jun 23, 2015 116 117 118 119 `````` | Empty | One _ _ -> absurd | Zero l1 -> One x l1 end | Zero l1 -> `````` Léon Gondelman committed Jul 03, 2015 120 121 `````` 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 `````` Léon Gondelman committed Jun 23, 2015 122 123 124 125 126 127 128 129 130 `````` 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 `````` Jean-Christophe Filliatre committed May 02, 2015 131 ``````end `````` Jean-Christophe Filliatre committed May 02, 2015 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 `````` (** A straightforward encapsulation with a list ghost model (in anticipation of module refinement) *) module RAL use import int.Int use import RandomAccessList as R use import list.List use import list.Length use import option.Option use import list.Nth type t 'a = { r: ral 'a; ghost l: list 'a } invariant { self.l = elements self.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 } = `````` Léon Gondelman committed Jul 03, 2015 161 `````` { r = cons x s.r; l = Cons x s.l } `````` Jean-Christophe Filliatre committed May 02, 2015 162 `````` `````` Léon Gondelman committed Jul 03, 2015 163 `````` let lookup (i: int) (s: t 'a) : 'a `````` Jean-Christophe Filliatre committed May 02, 2015 164 165 166 `````` requires { 0 <= i < length s.l } ensures { Some result = nth i s.l } = `````` Léon Gondelman committed Jul 03, 2015 167 `````` lookup i s.r `````` Jean-Christophe Filliatre committed May 02, 2015 168 169 170 `````` end `````` Jean-Christophe Filliatre committed Jun 24, 2015 171 172 ``````(** Model using sequences instead of lists *) `````` Jean-Christophe Filliatre committed Jun 22, 2015 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 ``````module RandomAccessListWithSeq use import int.Int use import int.ComputerDivision use import 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) (\ i: int. let (x0, x1) = s[div i 2] in if mod i 2 = 0 then x0 else x1) 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 `````` Léon Gondelman committed Jul 03, 2015 206 `````` let rec cons (x: 'a) (l: ral 'a) : ral 'a `````` Jean-Christophe Filliatre committed Jun 22, 2015 207 208 209 210 211 `````` variant { l } ensures { elements result == cons x (elements l) } = match l with | Empty -> One x Empty | Zero l1 -> One x l1 `````` Léon Gondelman committed Jul 03, 2015 212 `````` | One y l1 -> Zero (cons (x, y) l1) `````` Jean-Christophe Filliatre committed Jun 22, 2015 213 214 `````` end `````` Léon Gondelman committed Jul 03, 2015 215 `````` let rec lookup (i: int) (l: ral 'a) : 'a `````` Jean-Christophe Filliatre committed Jun 22, 2015 216 217 218 219 220 `````` requires { 0 <= i < length (elements l) } variant { i, l } ensures { (elements l)[i] = result } = match l with | Empty -> absurd `````` Léon Gondelman committed Jul 03, 2015 221 222 `````` | 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 `````` Jean-Christophe Filliatre committed Jun 22, 2015 223 224 225 `````` if mod i 2 = 0 then x0 else x1 end `````` Léon Gondelman committed Jun 23, 2015 226 `````` let rec tail (l: ral 'a) : ral 'a `````` Jean-Christophe Filliatre committed Jun 24, 2015 227 228 229 `````` requires { 0 < length (elements l) } variant { l } ensures { elements result == (elements l)[1 .. ] } `````` Léon Gondelman committed Jun 23, 2015 230 231 232 `````` = match l with | Empty -> absurd | One _ l1 -> Zero l1 `````` Léon Gondelman committed Jul 03, 2015 233 `````` | Zero l1 -> let (_, x1) = lookup 0 l1 in One x1 (tail l1) `````` Léon Gondelman committed Jun 23, 2015 234 235 `````` end `````` Léon Gondelman committed Jul 03, 2015 236 237 238 239 240 241 242 243 244 `````` function aux (i: int) (f: 'a -> 'a) : ('a,'a) -> ('a, 'a) = \ z. let (x,y) = z in if mod i 2 = 0 then (f x, y) else (x, f y) function setf (s: seq 'a) (i:int) (f: 'a -> 'a) : seq 'a = set s i (f s[i]) let rec fupdate_aux (f: 'a -> 'a) (i: int) (l: ral 'a) : ral 'a `````` Léon Gondelman committed Jun 23, 2015 245 246 `````` requires { 0 <= i < length (elements l) } variant { i, l} `````` Léon Gondelman committed Jul 03, 2015 247 `````` ensures { elements result == setf (elements l) i f} `````` Léon Gondelman committed Jun 23, 2015 248 249 `````` = match l with | Empty -> absurd `````` Léon Gondelman committed Jul 03, 2015 250 251 252 `````` | One x l1 -> if i = 0 then One (f x) l1 else cons x (fupdate_aux f (i-1) (Zero l1)) | Zero l1 -> Zero (fupdate_aux (aux i f) (div i 2) l1) `````` Léon Gondelman committed Jun 23, 2015 253 254 `````` end `````` Léon Gondelman committed Jul 03, 2015 255 256 257 258 259 260 261 `````` function f (y: 'a) : 'a -> 'a = \ _. y let fupdate (i:int) (y: 'a) (l: ral 'a) : ral 'a requires { 0 <= i < length (elements l) } ensures { elements result == set (elements l) i y} = fupdate_aux (f y) i l `````` Jean-Christophe Filliatre committed Jun 22, 2015 262 ``end``