ringdecision.mlw 13.7 KB
 Raphael Rieu-Helft committed Feb 26, 2018 1 2 ``````module UnitaryCommutativeRingDecision `````` Andrei Paskevich committed Jun 14, 2018 3 ``````clone algebra.UnitaryCommutativeRing as C with axiom . `````` Raphael Rieu-Helft committed Feb 26, 2018 4 `````` `````` Andrei Paskevich committed Jun 14, 2018 5 ``````clone algebra.UnitaryCommutativeRing as Z with axiom . `````` Raphael Rieu-Helft committed Feb 26, 2018 6 7 8 9 10 11 `````` function morph Z.t : C.t axiom morph_zero: morph Z.zero = C.zero axiom morph_one: morph Z.one = C.one axiom morph_add: forall z1 z2:Z.t. morph (Z.(+) z1 z2) = C.(+) (morph z1) (morph z2) `````` Andrei Paskevich committed Jun 14, 2018 12 13 ``````axiom morph_mul: forall z1 z2:Z.t. morph (Z.(*) z1 z2) = C.(*) (morph z1) (morph z2) `````` Raphael Rieu-Helft committed Feb 26, 2018 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ``````axiom morph_inv: forall z:Z.t. morph (Z.(-_) z) = C.(-_) (morph z) val predicate eq0 (x:Z.t) ensures { result <-> x = Z.zero } use import int.Int use import list.List type t = Var int | Add t t | Mul t t | Cst Z.t type vars = int -> C.t function interp (x:t) (y:vars) : C.t = match x with | Var n -> y n | Add x1 x2 -> C.(+) (interp x1 y) (interp x2 y) `````` Andrei Paskevich committed Jun 14, 2018 29 `````` | Mul x1 x2 -> C.(*) (interp x1 y) (interp x2 y) `````` Raphael Rieu-Helft committed Feb 26, 2018 30 31 32 33 34 35 36 37 38 39 40 41 `````` | Cst c -> morph c end predicate eq (x1 x2:t) = forall y: vars. interp x1 y = interp x2 y (** Conversion *) type m = M Z.t (list int) type t' = list m (* sum of monomials *) function mon (x:list int) (y:vars) : C.t = match x with | Nil -> C.one `````` Andrei Paskevich committed Jun 14, 2018 42 `````` | Cons x r -> C.(*) (y x) (mon r y) end `````` Raphael Rieu-Helft committed Feb 26, 2018 43 44 45 46 `````` function interp' (x:t') (y:vars) : C.t = match x with | Nil -> C.zero `````` Andrei Paskevich committed Jun 14, 2018 47 `````` | Cons (M a m) r -> C.(+) (C.(*) (morph a) (mon m y)) (interp' r y) end `````` Raphael Rieu-Helft committed Feb 26, 2018 48 49 50 51 52 53 54 55 `````` predicate eq_mon (m1 m2: list int) = forall y: vars. mon m1 y = mon m2 y predicate eq' (x1 x2: t') = forall y: vars. interp' x1 y = interp' x2 y use import list.Append use import list.Length let rec lemma mon_append (x1 x2: list int) (y:vars) `````` Andrei Paskevich committed Jun 14, 2018 56 `````` ensures { mon (x1 ++ x2) y = C.(*) (mon x1 y) (mon x2 y) } `````` Raphael Rieu-Helft committed Feb 26, 2018 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 `````` variant { x1 } = match x1 with Nil -> () | Cons _ x -> mon_append x x2 y end lemma interp_nil : forall y:vars. interp' Nil y = C.zero lemma interp_cons : forall m:m, x:t', y:vars. interp' (Cons m x) y = C.(+) (interp' (Cons m Nil) y) (interp' x y) let rec lemma interp_sum (x1 x2:t') (y:vars) ensures { interp' (x1 ++ x2) y = C.(+) (interp' x1 y) (interp' x2 y) } variant { x1 } = match x1 with | Nil -> () | Cons _ x -> interp_sum x x2 y end let ghost function append_mon (m1 m2:m) ensures { forall y. interp' (Cons result Nil) y `````` Andrei Paskevich committed Jun 14, 2018 75 76 `````` = C.(*) (interp' (Cons m1 Nil) y) (interp' (Cons m2 Nil) y) } = match m1,m2 with M a1 l1, M a2 l2 -> M (Z.(*) a1 a2) (l1 ++ l2) end `````` Raphael Rieu-Helft committed Feb 26, 2018 77 78 79 `````` let rec ghost function mul_mon (x:t') (mon:m) : t' ensures { forall y. `````` Andrei Paskevich committed Jun 14, 2018 80 `````` interp' result y = C.(*) (interp' x y) (interp' (Cons mon Nil) y) } `````` Raphael Rieu-Helft committed Feb 26, 2018 81 82 83 84 85 86 87 88 89 ``````= match x with | Nil -> Nil | Cons m r -> let mr = append_mon m mon in let lr = mul_mon r mon in Cons mr lr end let rec ghost function mul_devel (x1 x2:t') : t' `````` Andrei Paskevich committed Jun 14, 2018 90 `````` ensures { forall y. interp' result y = C.(*) (interp' x1 y) (interp' x2 y) } = `````` Raphael Rieu-Helft committed Feb 26, 2018 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 `````` match x1 with | Nil -> Nil | Cons (M a m) r -> (mul_mon x2 (M a m)) ++ (mul_devel r x2) end let rec ghost function conv (x:t) : t' ensures { forall y. interp x y = interp' result y } = match x with | Var v -> Cons (M Z.one (Cons v Nil)) Nil | Add x1 x2 -> (conv x1) ++ (conv x2) | Mul x1 x2 -> mul_devel (conv x1) (conv x2) | Cst n -> Cons (M n Nil) Nil end (** Normalisation *) let rec function insert (x: int) (l: list int) : list int ensures { eq_mon (Cons x l) result } variant { l } = match l with | Nil -> Cons x Nil | Cons y r -> if x <= y then Cons x l else Cons y (insert x r) end (*no need to prove that this actually sorts the list*) let rec function insertion_sort (l: list int) : list int ensures { eq_mon l result } variant { l } = match l with | Nil -> Nil | Cons x r -> insert x (insertion_sort r) end let function sort_mon (x:m) : m ensures { eq' (Cons x Nil) (Cons result Nil) } = match x with M a m -> M a (insertion_sort m) end let rec function sort_mons (x:t') : t' ensures { eq' result x } variant { x } = match x with Nil -> Nil | Cons m r -> Cons (sort_mon m) (sort_mons r) end (*lexicographic order on monomials with variables sorted using sort_mons*) let rec function le_mon (x1 x2: list int) : bool = (length x1 < length x2) || ((length x1 = length x2) && match x1, x2 with | Nil, _ -> true | _, Nil -> false | Cons v1 r1, Cons v2 r2 -> v1 <= v2 && le_mon r1 r2 end) let rec function same (l1 l2: list int) : bool ensures { result -> eq_mon l1 l2 } = match l1, l2 with | Nil, Nil -> true | Nil, _ | _, Nil -> false | Cons x1 l1, Cons x2 l2 -> x1 = x2 && same l1 l2 end lemma squash_sum: forall a1 a2:Z.t, l1 l2: list int. same l1 l2 -> eq' (Cons (M a1 l1) (Cons (M a2 l2) Nil)) (Cons (M (Z.(+) a1 a2) l1) Nil) let lemma squash_append (a1 a2:Z.t) (l1 l2: list int) (r:t') requires { same l1 l2 } ensures { eq' (Cons (M a1 l1) (Cons (M a2 l2) r)) (Cons (M (Z.(+) a1 a2) l1) r) } = () let rec ghost function insert_mon (m: m) (x: t') : t' ensures { eq' result (Cons m x) } variant { length x } = match m,x with | _,Nil -> Cons m Nil | M a1 l1, Cons (M a2 l2) r -> if same l1 l2 then let s = Z.(+) a1 a2 in if eq0 s then (assert { eq' r (Cons m x) by eq' r (Cons (M s l1) r) so eq' (Cons (M s l1) r) (Cons m x)}; r) else Cons (M s l1) r else if le_mon l1 l2 then Cons m x else Cons (M a2 l2) (insert_mon m r) end let rec ghost function insertion_sort_mon (x: t') : t' ensures { eq' result x } variant { x } = match x with | Nil -> Nil | Cons m r -> insert_mon m (insertion_sort_mon r) end let ghost function normalize (x: t') : t' ensures { eq' result x } = (* sort inside each monomial *) let x = sort_mons x in (* sort monomials lexicographically *) insertion_sort_mon x lemma norm': forall x1 x2:t', y:vars. normalize x1 = normalize x2 -> interp' x1 y = interp' x2 y lemma norm: forall x1 x2:t, y:vars. normalize (conv x1) = normalize (conv x2) -> interp x1 y = interp x2 y end (** Tests *) module Tests use import int.Int function id (x:int) : int = x let predicate eq0_int (x:int) = x = 0 `````` Andrei Paskevich committed Jun 14, 2018 214 215 ``````clone export UnitaryCommutativeRingDecision with type C.t = int, constant C.zero = zero, constant C.one = one, function C.(-_) = (-_), function C.(+) = (+), function C.(*) = (*), type Z.t = int, constant Z.zero = zero, constant Z.one = one, function Z.(-_) = (-_), function Z.(+) = (+), function Z.(*) = (*), function morph = id, goal morph_zero, goal morph_one, goal morph_add, goal morph_mul, goal morph_inv, val eq0 = eq0_int, axiom . (* FIXME: replace with "goal" and prove *) `````` Raphael Rieu-Helft committed Feb 26, 2018 216 217 218 219 220 221 222 223 224 225 226 227 228 229 `````` meta "compute_max_steps" 0x10000000 goal g: forall x. (x+3)*(x+2) = x*x + 5*x + 6 (* introduce_premises -> reflection_l norm *) end theory AssocAlgebra type r type a function (+) a a : a `````` Andrei Paskevich committed Jun 14, 2018 230 ``````function (*) a a : a `````` Raphael Rieu-Helft committed Feb 26, 2018 231 `````` `````` Andrei Paskevich committed Jun 14, 2018 232 233 ``````clone algebra.UnitaryCommutativeRing as R with type t = r, axiom . clone algebra.Ring as A with type t = a, function (+) = (+), function (*) = (*), axiom . `````` Raphael Rieu-Helft committed Feb 26, 2018 234 235 236 237 238 239 240 241 242 243 244 245 ``````constant one : a constant zero : a = A.zero axiom AUnitary : forall x:a. one * x = x * one = x axiom ANonTrivial : A.zero <> one (* A is an associative algebra over R *) val function (\$) r a : a axiom ExtDistSumA : forall r: r, x y: a. r \$ (x + y) = r \$ x + r \$ y axiom ExtDistSumR : forall r s: r, x: a. (R.(+) r s)\$x = r\$x + s\$x `````` Andrei Paskevich committed Jun 14, 2018 246 ``````axiom AssocMulExt : forall r s: r, x: a. (R.(*) r s)\$x = r\$(s\$x) `````` Raphael Rieu-Helft committed Feb 26, 2018 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 ``````axiom UnitExt : forall x: a. R.one \$ x = x axiom CommMulExt : forall r: r, x y: a. r\$(x*y) = (r\$x)*y = x*(r\$y) val predicate eq0 (r: r) ensures { result <-> r = R.zero } end module AssocAlgebraDecision use import int.Int use import list.List type r type a val constant rzero : r val constant rone : r val constant aone : a val ghost constant azero : a val function rplus r r : r val function rtimes r r : r val function ropp r : r val function aplus a a : a val function atimes a a : a val function aopp a : a `````` Andrei Paskevich committed Jun 14, 2018 276 ``````clone export AssocAlgebra with type r = r, type a = a, constant one = aone, constant A.zero = azero, constant R.zero = rzero, constant R.one = rone, function R.(+) = rplus, function R.(*) = rtimes, function R.(-_) = ropp, function (+) = aplus, function (*) = atimes, function A.(-_) = aopp, axiom . `````` Raphael Rieu-Helft committed Feb 26, 2018 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 `````` (*axiom azero_def: azero = A.zero*) (* FIXME *) type t = Var int | Add t t | Mul t t | Ext r t | Sub t t type vars = int -> a val function asub (x y:a) : a axiom asub_def: forall x y: a. asub x y = aplus x (aopp y) lemma ext_minone: forall a: a. (\$) (ropp rone) a = aopp a let rec function interp (x: t) (y: vars) : a = match x with | Var n -> y n | Add x1 x2 -> aplus (interp x1 y) (interp x2 y) | Mul x1 x2 -> atimes (interp x1 y) (interp x2 y) | Sub x1 x2 -> asub (interp x1 y) (interp x2 y) | Ext r x -> (\$) r (interp x y) end predicate eq (x1 x2:t) = forall y: vars. interp x1 y = interp x2 y (** Conversion to sum of monomials *) type m = M r (list int) type t' = list m let rec function mon (x: list int) (y: vars) : a = match x with | Nil -> aone | Cons x l -> atimes (y x) (mon l y) end let rec ghost function interp' (x: t') (y: vars) : a = match x with | Nil -> azero | Cons (M r m) l -> aplus ((\$) r (mon m y)) (interp' l y) end predicate eq_mon (m1 m2: list int) = forall y: vars. mon m1 y = mon m2 y predicate eq' (x1 x2: t') = forall y: vars. interp' x1 y = interp' x2 y use import list.Append use import list.Length let rec lemma mon_append (x1 x2: list int) (y: vars) ensures { mon (x1 ++ x2) y = atimes (mon x1 y) (mon x2 y) } variant { x1 } = match x1 with Nil -> () | Cons _ x -> mon_append x x2 y end lemma interp_cons : forall m:m, x:t', y:vars. interp' (Cons m x) y = aplus (interp' x y) (interp' (Cons m Nil) y) let rec lemma interp_sum (x1 x2: t') ensures { forall y: vars. interp' (x1++x2) y = aplus (interp' x1 y) (interp' x2 y) } variant { x1 } = match x1 with | Nil -> () | Cons _ x -> interp_sum x x2 end let function append_mon (m1 m2:m) ensures { forall y. interp' (Cons result Nil) y = atimes (interp' (Cons m1 Nil) y) (interp' (Cons m2 Nil) y) } = match m1,m2 with M r1 l1, M r2 l2 -> M (rtimes r1 r2) (l1 ++ l2) end let rec function mul_mon (mon: m) (x:t'): t' (* mon*x *) ensures { forall y. interp' result y = atimes (interp' (Cons mon Nil) y) (interp' x y) } = match x with | Nil -> Nil | Cons m l -> let mr = append_mon mon m in let lr = mul_mon mon l in Cons mr lr end let rec function mul_devel (x1 x2:t') : t' ensures { forall y. interp' result y = atimes (interp' x1 y) (interp' x2 y) } = match x1 with | Nil -> Nil | Cons (M r m) l -> mul_mon (M r m) x2 ++ mul_devel l x2 end let rec function ext (c:r) (x:t') : t' ensures { forall y. interp' result y = (\$) c (interp' x y) } = match x with | Nil -> Nil | Cons (M r m) l -> Cons (M (rtimes c r) m) (ext c l) end lemma ext_sub: forall x:t', y:vars. interp' (ext (ropp rone) x) y = aopp (interp' x y) let rec function conv (x:t) : t' ensures { forall y. interp x y = interp' result y } = match x with | Var v -> Cons (M rone (Cons v Nil)) Nil | Add x1 x2 -> (conv x1) ++ (conv x2) | Mul x1 x2 -> mul_devel (conv x1) (conv x2) | Ext r x -> ext r (conv x) | Sub x1 x2 -> (conv x1) ++ (ext (ropp rone) (conv x2)) end (** Normalisation *) (*lexicographic order on monomials with variables sorted using sort_mons*) let rec function le_mon (x1 x2: list int) : bool = (length x1 < length x2) || ((length x1 = length x2) && match x1, x2 with | Nil, _ -> true | _, Nil -> false | Cons v1 r1, Cons v2 r2 -> v1 <= v2 && le_mon r1 r2 end) let rec function same (l1 l2: list int) : bool ensures { result -> eq_mon l1 l2 } = match l1, l2 with | Nil, Nil -> true | Nil, _ | _, Nil -> false | Cons x1 l1, Cons x2 l2 -> x1 = x2 && same l1 l2 end lemma squash_append: forall r1 r2: r, l1 l2: list int, l: t'. same l1 l2 -> eq' (Cons (M r1 l1) (Cons (M r2 l2) l)) (Cons (M (rplus r1 r2) l1) l) let rec function insert_mon (m: m) (x: t') : t' ensures { eq' result (Cons m x) } variant { length x } = match m,x with | _,Nil -> Cons m Nil | M r1 l1, Cons (M r2 l2) l -> if same l1 l2 then let s = rplus r1 r2 in if eq0 s then (assert { eq' l (Cons m x) by eq' l (Cons (M s l1) l) so eq' (Cons (M s l1) l) (Cons m x)}; l) else Cons (M s l1) l else if le_mon l1 l2 then Cons m x else Cons (M r2 l2) (insert_mon m l) end let rec function insertion_sort_mon (x:t') : t' ensures { eq' result x } variant { x } = match x with | Nil -> Nil | Cons m l -> insert_mon m (insertion_sort_mon l) end let function normalize' (x:t') : t' ensures { eq' result x } = insertion_sort_mon x let function normalize (x:t) : t' ensures { eq' result (conv x) } = normalize' (conv x) let lemma norm (x1 x2: t) (y:vars) requires { normalize x1 = normalize x2 } ensures { interp x1 y = interp x2 y } = () let lemma norm' (x1 x2: t') requires { normalize' x1 = normalize' x2 } ensures { eq' x1 x2 } = () let norm_f (x1 x2: t) : bool ensures { forall y: vars. result = true -> interp x1 y = interp x2 y } = match normalize' (conv (Sub x1 x2)) with | Nil -> true | _ -> false end meta rewrite_def function interp end module ReifyTests use import int.Int let predicate eq0_int (x:int) = x=0 let function (\$\$) (x y:int) :int = x * y `````` Andrei Paskevich committed Jun 14, 2018 468 469 ``````clone export AssocAlgebraDecision with type r = int, type a = int, val rzero = Int.zero, val rone = Int.one, val rplus = (+), val ropp = (-_), val rtimes = (*), val azero = Int.zero, val aone = Int.one, val aplus = (+), val aopp = (-_), val atimes = (*), val (\$) = (\$\$), goal AUnitary, goal ANonTrivial, goal ExtDistSumA, goal ExtDistSumR, goal AssocMulExt, goal UnitExt, goal CommMulExt, val eq0 = eq0_int, goal A.MulAssoc.Assoc, goal A.Unit_def_l, goal A.Unit_def_r, goal A.Comm, goal A.Assoc, axiom . (* FIXME: replace with "goal" and prove *) `````` Raphael Rieu-Helft committed Feb 26, 2018 470 471 472 473 474 475 `````` goal g: forall x y. x + y = y + x goal h: forall x y z. x \$\$ (y * z) = (x \$\$ y) * z end ``````