random_access_list.mlw 6.95 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 `````` 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 add (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 (add (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 `````` Léon Gondelman committed Jun 23, 2015 77 `````` `````` Jean-Christophe Filliatre committed May 02, 2015 78 79 80 81 82 83 84 85 86 87 88 `````` let rec get (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 get (i-1) (Zero l1) | Zero l1 -> let (x0, x1) = get (div i 2) l1 in if mod i 2 = 0 then x0 else x1 end `````` Léon Gondelman committed Jun 23, 2015 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 `````` let rec tail (l: ral 'a) : ral 'a requires { elements l <> Nil } variant { l } ensures {let m = elements l in match nth 0 m with | None -> false | Some x -> m = Cons x (elements result) end } = match l with | Empty -> absurd | One _ l1 -> Zero l1 | Zero l1 -> let (_, x1) = get 0 l1 in One x1 (tail l1) end let rec set (y: 'a) (i: int) (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 set y (i-1) (Zero l1) with | Empty | One _ _ -> absurd | Zero l1 -> One x l1 end | Zero l1 -> let (x0, x1) = get (div i 2) l1 in let l1' = set (if mod i 2 = 0 then (y,x1) else (x0,y)) (div i 2) 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 `````` Jean-Christophe Filliatre committed May 02, 2015 134 ``````end `````` Jean-Christophe Filliatre committed May 02, 2015 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 `````` (** 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 } = { r = add x s.r; l = Cons x s.l } let get (i: int) (s: t 'a) : 'a requires { 0 <= i < length s.l } ensures { Some result = nth i s.l } = get i s.r end `````` Jean-Christophe Filliatre committed Jun 22, 2015 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 ``````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 let rec add (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 (add (x, y) l1) end `````` Léon Gondelman committed Jun 23, 2015 216 `````` let rec get (i: int) (l: ral 'a) : 'a `````` Jean-Christophe Filliatre committed Jun 22, 2015 217 218 219 220 221 222 223 224 225 226 `````` 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 get (i-1) (Zero l1) | Zero l1 -> let (x0, x1) = get (div i 2) l1 in if mod i 2 = 0 then x0 else x1 end `````` Léon Gondelman committed Jun 23, 2015 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 `````` let rec tail (l: ral 'a) : ral 'a requires { not ((elements l) == empty) } variant { l } ensures { (elements l) == cons (elements l)[0] (elements result) } = match l with | Empty -> absurd | One _ l1 -> Zero l1 | Zero l1 -> let (_, x1) = get 0 l1 in One x1 (tail l1) end let rec set (y: 'a) (i: int) (l: ral 'a) : ral 'a requires { 0 <= i < length (elements l) } variant { i, l} ensures { elements result == set (elements l) i y} = match l with | Empty -> absurd | One x l1 -> if i = 0 then One y l1 else match set y (i-1) (Zero l1) with | Empty | One _ _ -> absurd | Zero l1 -> One x l1 end | Zero l1 -> let (x0, x1) = get (div i 2) l1 in Zero (set (if mod i 2 = 0 then (y,x1) else (x0,y)) (div i 2) l1) end `````` Jean-Christophe Filliatre committed Jun 22, 2015 253 ``end``