verifythis_2018_le_rouge_et_le_noir_1.mlw 12.4 KB
 Jean-Christophe Filliâtre committed May 02, 2018 1 2 3 4 5 6 7 8 ``````(** {1 VerifyThis @ ETAPS 2018 competition Challenge 2: Le rouge et le noir} Author: Raphaël Rieu-Helft (LRI, Université Paris Sud) *) `````` Raphael Rieu-Helft committed Apr 30, 2018 9 10 ``````module ColoredTiles `````` Andrei Paskevich committed Jun 15, 2018 11 12 13 14 ``````use int.Int use set.Fset use set.FsetComprehension use seq.Seq `````` Raphael Rieu-Helft committed Apr 30, 2018 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 `````` type color = Red | Black type coloring = seq color predicate tworedneighbors (c: coloring) (i:int) = ((c[i-2] = Red /\ c[i-1] = Red /\ 2 <= i) \/ (c[i-1] = Red /\ c[i+1] = Red /\ 1 <= i <= length c - 2) \/ (c[i+1] = Red /\ c[i+2] = Red /\ i <= length c - 3)) predicate valid (c:coloring) = forall i. 0 <= i < length c -> c[i] = Red -> tworedneighbors c i function black (n:int) : color = Black function red (n:int) : color = Red function colorings0 : set coloring = add (create 0 black) Fset.empty function colorings1 : set coloring = add (create 1 black) Fset.empty function colorings2 : set coloring = add (create 2 black) Fset.empty function colorings3: set coloring = add (create 3 red) (add (create 3 black) Fset.empty) lemma valid_contr: forall c i. valid c -> 0 <= i < length c -> not (tworedneighbors c i) -> c[i] = Black lemma colo_0 : forall c: coloring. length c = 0 -> (valid c <-> mem c colorings0 by Seq.(==) c (create 0 black)) lemma colo_1 : forall c: coloring. length c = 1 -> (valid c <-> mem c colorings1 by c[0] = Black so Seq.(==) c (create 1 black)) lemma colo_2 : forall c: coloring. length c = 2 -> (valid c <-> mem c colorings2 by c[0] = Black = c[1] so Seq.(==) c (create 2 black) so c = create 2 black) lemma colo_3 : forall c: coloring. length c = 3 -> (valid c <-> mem c colorings3 by if c[0] = Black then (c[0]=c[1]=c[2]=Black so c == create 3 black so c = create 3 black) else (c[0]=c[1]=c[2]=Red so c == create 3 red so c = create 3 red)) let lemma valid_split_fb (c:coloring) (k: int) requires { 3 <= k < length c } requires { forall i. 0 <= i < k -> c[i] = Red } requires { valid c[k+1 ..] } ensures { valid c } = let c' = c[k+1 ..] in assert { forall i. k+1 <= i < length c -> c[i] = Red -> (tworedneighbors c i by c'[i - (k+1)] = Red so [@case_split] tworedneighbors c' (i - (k+1))) } let lemma valid_restrict (c: coloring) (k: int) requires { valid c } requires { 0 <= k < length c } requires { c[k] = Black } ensures { valid c[k+1 ..] } = () (*1st black tile starting at i *) let rec function first_black_tile (c:coloring) : int ensures { 0 <= result <= length c } ensures { forall j. 0 <= j < result <= length c -> c[j] = Red } ensures { result < length c -> c[result] = Black } ensures { valid c -> result = 0 \/ 3 <= result } variant { length c } = if Seq.length c = 0 then 0 else match c[0] with | Black -> 0 | Red -> assert { valid c -> c[1]=Red /\ c[2] = Red }; let r = first_black_tile c[1 ..] in assert { forall j. 1 <= j < 1+r -> c[j] = Red by c[1 ..][j-1] = Red }; 1+r end let rec function addleft (nr:int) (c:coloring) : coloring variant { nr } ensures { nr >= 0 -> Seq.length result = Seq.length c + nr + 1 } = if nr <= 0 then cons Black c else cons Red (addleft (nr-1) c) (* add nr red tiles and a black tile to the left of each coloring *) function mapaddleft (s:set coloring) (nr:int) : set coloring = map (addleft nr) s lemma addleft_fb: forall c nr. 0 <= nr -> first_black_tile (addleft nr c) = nr lemma mapaddleft_fb: forall s c nr. 0 <= nr -> mem c (mapaddleft s nr) -> first_black_tile c = nr predicate reciprocal (f: 'a -> 'b) (g: 'b -> 'a) = forall y. g (f y) = y let lemma bij_image (u: set 'a) (f: 'a -> 'b) (g: 'b -> 'a) requires { reciprocal f g } ensures { subset u (map g (map f u)) } = assert { forall x. mem x u -> mem (f x) (map f u) -> mem (g (f x)) (map g (map f u)) -> mem x (map g (map f u)) } let lemma bij_cardinal (u: set 'a) (f: 'a -> 'b) (g: 'b -> 'a) requires { reciprocal f g } ensures { cardinal (map f u) = cardinal u } = assert { cardinal (map f u) <= cardinal u }; assert { cardinal (map g (map f u)) <= cardinal (map f u) }; assert { cardinal u <= cardinal (map g (map f u)) } function rmleft (nr:int) (c:coloring) : coloring = c[nr+1 ..] `````` Andrei Paskevich committed Jun 15, 2018 136 ``````use seq.FreeMonoid `````` Raphael Rieu-Helft committed Apr 30, 2018 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 `````` lemma ext: forall c1 c2: coloring. Seq.(==) c1 c2 -> c1 = c2 lemma app_eq: forall c1 c2 c3 c4: coloring. c1 = c2 -> c3 = c4 -> c1 ++ c3 = c2 ++ c4 let rec lemma addleft_result (c:coloring) (nr:int) requires { 0 <= nr } ensures { addleft nr c = (Seq.create nr red) ++ (cons Black c) } variant { nr } = if nr = 0 then assert { addleft nr c = (Seq.create nr red) ++ (cons Black c) } else begin let cnr = create nr red in let cnrm = create (nr - 1) red in addleft_result c (nr-1); assert { addleft (nr-1) c = cnrm ++ cons Black c }; assert { cons Red cnrm = cnr by Seq.(==) (cons Red cnrm) cnr }; assert { addleft nr c = cnr ++ cons Black c by addleft nr c = cons Red (addleft (nr-1) c) = cons Red (cnrm ++ cons Black c) = (cons Red cnrm) ++ cons Black c = cnr ++ cons Black c } end let lemma addleft_bijective (nr:int) requires { 0 <= nr } ensures { reciprocal (addleft nr) (rmleft nr) } = assert { forall c i. 0 <= i < length c -> (rmleft nr (addleft nr c))[i] = c[i] }; assert { forall c. Seq.(==) (rmleft nr (addleft nr c)) c } let lemma mapaddleft_card (s: set coloring) (nr: int) requires { 0 <= nr } ensures { cardinal (mapaddleft s nr) = cardinal s } = addleft_bijective nr; bij_cardinal s (addleft nr) (rmleft nr) let lemma addleft_valid (c:coloring) (nr:int) requires { nr = 0 \/ 3 <= nr } requires { valid c } ensures { valid (addleft nr c) } = addleft_result c nr; if nr = 0 then assert { valid (addleft 0 c) } else valid_split_fb (addleft nr c) nr let lemma mapaddleft_valid (s: set coloring) (nr: int) requires { forall c. mem c s -> valid c } requires { nr = 0 \/ 3 <= nr } ensures { forall c. mem c (mapaddleft s nr) -> valid c } = assert { forall c. mem c (mapaddleft s nr) -> valid c by mem c (map (addleft nr) s) so (exists y. mem y s /\ c = addleft nr y) } let lemma mapaddleft_length (s: set coloring) (nr: int) (l1 l2: int) requires { forall c. mem c s -> Seq.length c = l1 } requires { 0 <= nr } requires { l2 = l1 + nr + 1 } ensures { forall c. mem c (mapaddleft s nr) -> Seq.length c = l2 } = () let rec disjoint_union (s1 s2: set coloring) : set coloring requires { forall x. mem x s1 -> not mem x s2 } ensures { result = union s1 s2 } ensures { cardinal result = cardinal s1 + cardinal s2 } variant { cardinal s1 } = if is_empty s1 then begin assert { union s1 s2 = s2 by (forall x. mem x (union s1 s2) -> mem x s1 \/ mem x s2 -> mem x s2) so subset (union s1 s2) s2 }; s2 end else let x = choose s1 in let s1' = remove x s1 in let s2' = add x s2 in let u = disjoint_union s1' s2' in assert { u = union s1 s2 by u = union s1' s2' so (forall y. (mem y s2' <-> (mem y s2 \/ y = x))) so (forall y. ((mem y s1' \/ y = x) <-> mem y s1)) so (forall y. mem y u <-> mem y s1' \/ mem y s2' <-> mem y s1' \/ mem y s2 \/ y = x <-> mem y s1 \/ mem y s2 <-> mem y (union s1 s2)) so (forall y. mem y u <-> mem y (union s1 s2)) so Fset.(==) u (union s1 s2)}; u `````` Andrei Paskevich committed Jun 15, 2018 227 ``````use array.Array `````` Raphael Rieu-Helft committed Apr 30, 2018 228 `````` `````` Andrei Paskevich committed Jun 07, 2018 229 230 ``````let enum () : (count: array int, ghost sets: array (set coloring)) ensures { Array.length count = 51 = Array.length sets `````` Raphael Rieu-Helft committed Apr 30, 2018 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 `````` /\ (forall i. 0 <= i <= 50 -> (forall c: coloring. Seq.length c = i -> (valid c <-> mem c (sets[i])))) /\ (forall i. 0 <= i < 50 -> count[i] = cardinal (sets[i])) } = let count = Array.make 51 0 in let ghost sets : array (set coloring) = Array.make 51 Fset.empty in count[0] <- 1; sets[0] <- colorings0; assert { forall c. ((Seq.length c = 0 /\ valid c) <-> mem c (sets[0])) }; count[1] <- 1; sets[1] <- colorings1; assert { forall i c. (i=0 \/ i=1) -> ((Seq.length c = i /\ valid c) <-> mem c (sets[i])) }; count[2] <- 1; sets[2] <- colorings2; assert { forall i c. (i=0 \/ i=1 \/ i=2) -> ((Seq.length c = i /\ valid c) <-> mem c (sets[i])) }; count[3] <- 2; sets[3] <- colorings3; assert { sets[3] = colorings3 }; assert { forall i c. (i=0 \/ i=1 \/ i=2 \/ i = 3) -> ((Seq.length c = i /\ valid c) <-> mem c (sets[i])) }; assert { cardinal colorings3 = 2 }; for n = 4 to 50 do invariant { forall c i. 0 <= i < n -> Seq.length c = i -> valid c -> mem c (sets[i]) } invariant { forall c i. 0 <= i < n -> mem c (sets[i]) -> (Seq.length c = i /\ valid c) } invariant { forall i. 0 <= i < n -> count[i] = cardinal (sets[i]) } label StartLoop in (* colorings with first_black_tile = 0 *) count[n] <- count[n-1]; mapaddleft_valid (sets[n-1]) 0; sets[n] <- mapaddleft (sets[n-1]) 0; assert { forall i. 0 <= i < n -> sets[i] = sets[i] at StartLoop }; assert { forall i. 0 <= i < n -> count[i] = count[i] at StartLoop }; assert { forall c. Seq.length c = n -> valid c -> first_black_tile c < 3 -> mem c sets[n] by first_black_tile c = 0 so valid c[1 ..] so mem c[1 ..] (sets[n-1]) so addleft 0 c[1 ..] = c so mem c (mapaddleft sets[n-1] 0) }; for k = 3 to n-1 do invariant { forall c i. 0 <= i < n -> Seq.length c = i -> valid c -> mem c (sets[i]) } invariant { forall c i. 0 <= i < n -> mem c (sets[i]) -> (Seq.length c = i /\ valid c) } invariant { forall i. 0 <= i < n -> count[i] = cardinal (sets[i]) } invariant { forall c. (mem c (sets[n]) <-> (Seq.length c = n /\ valid c /\ first_black_tile c < k)) } invariant { count[n] = cardinal (sets[n]) } label InnerLoop in (* colorings with first_black_tile = k *) count[n] <- count [n] + count [n-k-1]; mapaddleft_length (sets[n-k-1]) k (n-k-1) n; mapaddleft_valid (sets[n-k-1]) k; mapaddleft_card (sets[n-k-1]) k; let ghost ns = mapaddleft sets[n-k-1] k in assert { forall c. mem c ns -> first_black_tile c = k }; assert { forall c. Seq.length c = n -> valid c -> first_black_tile c = k -> mem c ns by valid c[k+1 ..] so mem c[k+1 ..] (sets[n-k-1]) so let c' = addleft k c[k+1 ..] in ((forall i. 0 <= i < n -> Seq.get c i = Seq.get c' i) by c[k+1 ..] = c'[k+1 ..]) so Seq.(==) c' c so c' = c so mem c (mapaddleft sets[n-k-1] k) }; sets[n] <- disjoint_union (sets[n]) ns; assert { forall i. 0 <= i < n -> sets[i] = sets[i] at InnerLoop }; assert { forall i. 0 <= i < n -> count[i] = count[i] at InnerLoop }; done; (* coloring with first_black_tile = n *) label LastAdd in let ghost r = Seq.create n red in let ghost sr = Fset.singleton r in assert { forall c. mem c sets[n] -> first_black_tile c < n }; assert { first_black_tile r = n }; assert { valid r /\ Seq.length r = n }; count[n] <- count[n]+1; sets[n] <- disjoint_union (sets[n]) sr; assert { forall c. mem c sets[n] -> valid c /\ Seq.length c = n by [@case_split] mem c (sets[n] at LastAdd) \/ mem c sr }; assert { forall c. Seq.length c = n -> first_black_tile c = n -> mem c sets[n] by (forall k. 0 <= k < n -> Seq.get c k = Red) so c == r so c = r }; assert { forall i. 0 <= i < n -> sets[i] = sets[i] at LastAdd }; assert { forall i. 0 <= i < n -> count[i] = count[i] at LastAdd }; done; count, sets `````` Andrei Paskevich committed Jun 07, 2018 329 ``end``