isa_planner.why 10.6 KB
 Jean-Christophe Filliâtre committed Sep 17, 2015 1 2 3 4 5 6 7 8 9 10 11 12 13 14 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 `````` (** Benchmarks goals for proof by induction From http://dream.inf.ed.ac.uk/projects/isaplanner/ *) theory Nat type nat = Zero | Suc nat function (+) (x y: nat) : nat = match x with | Zero -> y | Suc xs -> Suc (xs + y) end function ( * ) (x y: nat) : nat = match y with | Zero -> Zero | Suc ys -> x + x * ys end function (-) (x y: nat) : nat = match x with | Zero -> Zero | Suc xs -> match y with | Zero -> Suc xs | Suc ys -> xs - ys end end predicate ( < ) (x y: nat) = match y with | Zero -> false | Suc ys -> match x with | Zero -> true | Suc xs -> xs < ys end end predicate ( <= ) (x y: nat) = match x with | Zero -> true | Suc xs -> match y with | Zero -> false | Suc ys -> xs <= ys end end function max (x y: nat) : nat = match x with | Zero -> y | Suc xs -> match y with | Zero -> Suc xs | Suc ys -> Suc (max xs ys) end end function min (x y: nat) : nat = match x with | Zero -> Zero | Suc xs -> match y with | Zero -> y | Suc ys -> Suc (min xs ys) end end end theory List `````` Andrei Paskevich committed Jun 15, 2018 58 `````` use Nat `````` Jean-Christophe Filliâtre committed Sep 17, 2015 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 `````` type list 'a = Nil | Cons 'a (list 'a) function len (l : list 'a) : nat = match l with | Nil -> Zero | Cons _ s -> Suc (len s) end predicate mem (x: 'a) (l: list 'a) = match l with | Nil -> false | Cons y ys -> if x = y then true else mem x ys end function (++) (l r : list 'a) : list 'a = match l with | Nil -> r | Cons x ls -> Cons x (ls ++ r) end function rev (l: list 'a) : list 'a = match l with | Nil -> Nil | Cons x xs -> rev xs ++ Cons x Nil end function insert (x : 'a) (l : list 'a) : list 'a = match l with | Nil -> Cons x Nil | Cons h t -> if x = h then Cons x t else Cons h (insert x t) end function delete (x: 'a) (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons h t -> if x = h then delete x t else Cons h (delete x t) end function take (n: nat) (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons h t -> match n with | Zero -> Nil | Suc m -> Cons h (take m t) end end function drop (n: nat) (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons h t -> match n with | Zero -> Cons h t | Suc m -> drop m t end end function last (l : list 'a) : 'a axiom last_single : forall x: 'a. last (Cons x Nil) = x axiom last_cons : forall x: 'a, l : list 'a. l <> Nil -> last (Cons x l) = last l function butlast (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons _ Nil -> Nil | Cons x xs -> Cons x (butlast xs) end function zip (l r : list 'a) : list ('a, 'a) = match r with | Nil -> Nil | Cons y rs -> match l with | Nil -> Nil | Cons x ls -> Cons (x,y) (zip ls rs) end end function count (x: 'a) (l: list 'a) : nat = match l with | Nil -> Zero | Cons y ys -> if x = y then Suc (count x ys) else count x ys end function map (f: 'a -> 'b) (l: list 'a) : list 'b = match l with | Nil -> Nil | Cons x xs -> Cons (f x) (map f xs) end function filter (p: 'a -> bool) (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons x xs -> if p x then Cons x (filter p xs) else filter p xs end function takeWhile (p: 'a -> bool) (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons x xs -> if p x then Cons x (takeWhile p xs) else Nil end function dropWhile (p: 'a -> bool) (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons x xs -> if p x then (dropWhile p xs) else (Cons x xs) end `````` DAILLER Sylvain committed Aug 20, 2019 153 `````` predicate pfalse (_x: 'a) = false `````` Jean-Christophe Filliâtre committed Sep 17, 2015 154 155 156 157 158 159 `````` function dropWhile_False (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons x xs -> if pfalse x then (dropWhile_False xs) else (Cons x xs) end `````` DAILLER Sylvain committed Aug 20, 2019 160 `````` predicate ptrue (_x: 'a) = true `````` Jean-Christophe Filliâtre committed Sep 17, 2015 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 `````` function takeWhile_True (l : list 'a) : list 'a = match l with | Nil -> Nil | Cons x xs -> if ptrue x then Cons x (takeWhile_True xs) else Nil end (******************** INSERTION SORT WITH NAT LIST ***************) predicate sorted (l : list nat) = match l with | Nil -> true | Cons x xs -> match xs with | Nil -> true | Cons y _ -> x <= y && sorted xs end end function insert_nat (n : nat) (l : list nat) : list nat = match l with | Nil -> Cons n Nil | Cons h t -> if n < h then Cons n (Cons h t) else Cons h (insert_nat n t) end function insertion_sort_aux (x : nat) (l : list nat) : list nat = match l with | Nil -> Cons x Nil | Cons y ys -> if x <= y then Cons x (Cons y ys) else Cons y (insertion_sort_aux x ys) end function insertion_sort (l : list nat) : list nat = match l with | Nil -> Nil | Cons x xs -> insertion_sort_aux x (insertion_sort xs) end end theory Tree `````` Andrei Paskevich committed Jun 15, 2018 198 `````` use Nat `````` Jean-Christophe Filliâtre committed Sep 17, 2015 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 `````` type tree 'a = | Leaf | Node (tree 'a) 'a (tree 'a) function mirror (t: tree 'a) : tree 'a = match t with | Leaf -> Leaf | Node l x r -> Node (mirror r) x (mirror l) end function nodes (t: tree 'a) : nat = match t with | Leaf -> Zero | Node l _ r-> (Suc Zero) + nodes l + nodes r end function height (t: tree 'a) : nat = match t with | Leaf -> Zero | Node l _ r -> Suc (max (height l) (height r)) end end (******************************************************************************) (************************** ISAPLANNER THEOREMS *******************************) (******************************************************************************) (*Pas d'induction(13): 4, 5, 11, 13, 16, 17, 35, 39, 40, 42, 44, 45, 46 *) (*Induction sur une variable(22): 2, 3, 6, 7, 8, 10, 12, 14, 15, 18, 19, 21, 26, 27, 28, 29, 30, 32, 36, 37, 38, 43 *) (*Induction sur plusieurs variables à cause de raisonnement par cas (9): 1, 9, 22, 23, 24, 25, 31, 33, 34, *) (*Problème résolu avec un lemme supplémentaire (2): 20(15), 47(23) *) (******************************************************************************) theory IsaPlanner_all `````` Andrei Paskevich committed Jun 15, 2018 234 235 236 `````` use Nat use List use Tree `````` Jean-Christophe Filliâtre committed Sep 17, 2015 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 `````` goal P1: forall l: list 'a, n : nat. take n l ++ drop n l = l goal P2: forall l m: list 'a, x: 'a. count x l + count x m = count x (l ++ m) goal P3: forall l m: list 'a, x: 'a. count x l <= count x (l ++ m) goal P4: forall l: list 'a, x: 'a. Suc Zero + count x l = count x (Cons x l) goal P5: forall l: list 'a, x y : 'a. x = y -> Suc Zero + count x l = count x (Cons y l) goal P6: forall n m: nat. n - (n + m) = Zero goal P7: forall n m: nat. (n + m) - n = m `````` Guillaume Melquiond committed Jan 12, 2018 259 `````` goal P8: forall k [@induction] n m: nat. `````` Jean-Christophe Filliâtre committed Sep 17, 2015 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 `````` (k + m) - (k + n) = (m - n) goal P9: forall i j k: nat. (i - j) - k = i - (j + k) goal P10: forall m: nat. m - m = Zero goal P11: forall l: list 'a. drop Zero l = l goal P12: forall f: 'a -> 'b, l, n. drop n (map f l) = map f (drop n l) goal P13: forall n: nat, x: 'a, ls: list 'a. drop (Suc n) (Cons x ls) = drop n ls goal P14: forall p, xs ys: list 'a. filter p (xs ++ ys) = filter p xs ++ filter p ys goal P15: forall l: list nat, n: nat. len (insertion_sort_aux n l) = Suc (len l) goal P16: forall l: list 'a, x: 'a. l = Nil -> last (Cons x l) = x goal P17: forall n: nat. (n <= Zero) <-> (n = Zero) goal P18: forall i m: nat. i < (Suc (i + m)) goal P19: forall l: list 'a, n: nat. len (drop n l) = (len l) - n (* requires the lemma forall l: list nat, n: nat. len (insertion_sort_aux n l) = Suc (len l) *) goal P20: forall l: list nat. len (insertion_sort l) = len l goal P21: forall n m: nat. n <= (n + m) `````` Guillaume Melquiond committed Jan 12, 2018 303 `````` goal P22: forall a [@induction] b [@induction] c [@induction]: nat . `````` Jean-Christophe Filliâtre committed Sep 17, 2015 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 `````` max (max a b) c = max a (max b c) goal P23: forall a b: nat. max a b = max b a goal P24: forall a b: nat. (max a b = a) <-> b <= a goal P25: forall a b: nat. (max a b = b) <-> a <= b goal P26: forall l t: list 'a, x: 'a. mem x l -> mem x (l ++ t) goal P27: forall l t: list 'a, x: 'a. mem x t -> mem x (l ++ t) goal P28: forall l: list 'a, x: 'a. mem x (l ++ Cons x Nil) goal P29: forall l : list nat, x : nat. mem x (insert_nat x l) goal P30: forall l: list 'a, x: 'a. mem x (insert x l) `````` Guillaume Melquiond committed Jan 12, 2018 330 `````` goal P31: forall a [@induction] b [@induction] c [@induction]: nat. `````` Jean-Christophe Filliâtre committed Sep 17, 2015 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 `````` min (min a b) c = min a (min b c) goal P32: forall a b: nat. min a b = min b a goal P33: forall a b: nat. (min a b = a) <-> a <= b goal P34: forall a b: nat. (min a b = b) <-> b <= a goal P35: forall xs : list 'a. dropWhile_False xs = xs goal P36: forall xs: list 'a. takeWhile_True xs = xs goal P37: forall l: list 'a, x: 'a. not mem x (delete x l) goal P38: forall l: list 'a, n: 'a. count n (l ++ Cons n Nil) = Suc (count n l) goal P39: forall t: list 'a, n h: 'a. count n (Cons h Nil) + count n t = count n (Cons h t) goal P40: forall xs: list 'a. take Zero xs = Nil goal P41: forall f, xs : list 'a, n: nat. take n (map f xs : list 'b) = map f (take n xs) goal P42: forall xs: list 'a, n: nat, x: 'a. take (Suc n) (Cons x xs) = Cons x (take n xs) goal P43: forall p, xs : list 'a. takeWhile p xs ++ dropWhile p xs = xs goal P44: forall xs ys: list 'a, x: 'a. zip (Cons x xs) ys = match ys with | Nil -> Nil | Cons y ys -> Cons (x,y) (zip xs ys) end goal P45: forall xs ys: list 'a, x y: 'a . zip (Cons x xs) (Cons y ys) = Cons (x, y) (zip xs ys) goal P46: forall ys: list 'a. zip Nil ys = Nil (* requires P23: forall a b: nat. max a b = max b a *) goal P47: forall a : tree 'a. height (mirror a) = height a end theory IsaPlanner_beyond `````` Andrei Paskevich committed Jun 15, 2018 390 391 392 `````` use Nat use List use Tree `````` Jean-Christophe Filliâtre committed Sep 17, 2015 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 `````` goal P48: forall xs : list 'a. xs <> Nil -> butlast xs ++ (Cons (last xs) Nil) = xs goal P49: forall xs ys: list 'a . butlast (xs ++ ys) = (if ys = Nil then butlast xs else xs ++ butlast ys) goal P50: forall xs : list 'a. butlast xs = take ((len xs) - (Suc Zero)) xs goal P51: forall xs : list 'a, x: 'a. butlast (xs ++ Cons x Nil) = xs goal P52: forall l : list 'a, n: 'a. count n l = count n (rev l) goal P53: forall l : list nat, x : nat. count x l = count x (insertion_sort l) goal P54: forall m n: nat. (m + n) - n = m `````` Guillaume Melquiond committed Jan 12, 2018 415 `````` goal P55: forall i [@induction] k [@induction] j [@induction] : nat. `````` Jean-Christophe Filliâtre committed Sep 17, 2015 416 417 418 419 420 `````` (i - j) - k = i - (k - j) goal P56: forall xs ys: list 'a, n: nat. drop n (xs ++ ys) = drop n xs ++ drop (n - (len xs)) ys `````` Guillaume Melquiond committed Jan 12, 2018 421 422 `````` goal P57: forall n [@induction] m [@induction]: nat, xs [@induction]: list nat. `````` Jean-Christophe Filliâtre committed Sep 17, 2015 423 424 `````` drop n (drop m xs) = drop (n + m) xs `````` Guillaume Melquiond committed Jan 12, 2018 425 426 `````` goal P58: forall xs [@induction]: list 'a, m [@induction] n [@induction]: nat. `````` Jean-Christophe Filliâtre committed Sep 17, 2015 427 428 429 `````` drop n (take m xs) = take (m - n) (drop n xs) end``````