lineardecision.mlw 19.1 KB
 Raphael Rieu-Helft committed Dec 01, 2017 1 2 ``````module LinearEquationsCoeffs `````` Raphael Rieu-Helft committed Dec 14, 2017 3 4 5 6 ``````type a function (+) a a : a function ( *) a a : a function (-_) a : a `````` Raphael Rieu-Helft committed Dec 18, 2017 7 8 9 ``````function azero: a function aone: a predicate ale a a `````` Raphael Rieu-Helft committed Dec 14, 2017 10 `````` `````` Raphael Rieu-Helft committed Dec 18, 2017 11 ``````clone algebra.OrderedUnitaryCommutativeRing as A with type t = a, function (+) = (+), function ( *) = ( *), function (-_) = (-_), constant zero = azero, constant one=aone, predicate (<=) = ale `````` Raphael Rieu-Helft committed Dec 01, 2017 12 13 `````` type t `````` Raphael Rieu-Helft committed Dec 14, 2017 14 ``````type vars = int -> a `````` Raphael Rieu-Helft committed Dec 01, 2017 15 16 17 `````` exception Unknown `````` Raphael Rieu-Helft committed Dec 14, 2017 18 ``````function interp t vars : a `````` Raphael Rieu-Helft committed Dec 01, 2017 19 `````` `````` 20 21 ``````val constant czero : t val constant cone : t `````` Raphael Rieu-Helft committed Dec 01, 2017 22 `````` `````` Raphael Rieu-Helft committed Dec 18, 2017 23 24 ``````axiom zero_def: forall y. interp czero y = azero axiom one_def: forall y. interp cone y = aone `````` Raphael Rieu-Helft committed Dec 14, 2017 25 `````` `````` Raphael Rieu-Helft committed Dec 01, 2017 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 `````` val add (a b: t) : t ensures { forall v: vars. interp result v = interp a v + interp b v } raises { Unknown -> true } val mul (a b: t) : t ensures { forall v: vars. interp result v = interp a v * interp b v } val opp (a:t) : t ensures { forall v: vars. interp result v = - (interp a v) } val predicate eq (a b:t) ensures { result <-> forall y:vars. interp a y = interp b y } axiom eq_def: forall a b: t. a=b -> eq a b val inv (a:t) : t `````` 43 `````` requires { not (eq a czero) } `````` Raphael Rieu-Helft committed Dec 18, 2017 44 `````` ensures { forall v: vars. interp result v * interp a v = aone } `````` 45 46 `````` ensures { not (eq result czero) } raises { Unknown -> true } `````` Raphael Rieu-Helft committed Dec 01, 2017 47 48 `````` val le (a b:t) : bool `````` Raphael Rieu-Helft committed Dec 18, 2017 49 `````` ensures { result -> forall y:vars. ale (interp a y) (interp b y) } `````` Raphael Rieu-Helft committed Dec 01, 2017 50 51 52 `````` raises { Unknown -> true } `````` 53 ``````(*FIXME equality test, extensionality, specs for le and eq ? *) `````` Raphael Rieu-Helft committed Dec 01, 2017 54 55 56 57 58 59 60 61 62 63 `````` end module LinearEquationsDecision use import int.Int type coeff clone LinearEquationsCoeffs as C with type t = coeff `````` Raphael Rieu-Helft committed Dec 14, 2017 64 ``````type vars = C.vars `````` Raphael Rieu-Helft committed Dec 01, 2017 65 `````` `````` 66 ``````type expr = Term coeff int | Add expr expr | Cst coeff | UTerm int `````` Raphael Rieu-Helft committed Dec 01, 2017 67 68 69 70 `````` let rec predicate valid_expr (e:expr) variant { e } = match e with `````` 71 `````` | Term _ i | UTerm i -> 0 <= i `````` Raphael Rieu-Helft committed Dec 01, 2017 72 73 74 75 76 77 78 `````` | Cst _ -> true | Add e1 e2 -> valid_expr e1 && valid_expr e2 end let rec predicate expr_bound (e:expr) (b:int) variant { e } = match e with `````` 79 `````` | Term _ i | UTerm i -> 0 <= i <= b `````` Raphael Rieu-Helft committed Dec 01, 2017 80 81 82 83 `````` | Cst _ -> true | Add e1 e2 -> expr_bound e1 b && expr_bound e2 b end `````` Raphael Rieu-Helft committed Dec 14, 2017 84 ``````function interp (e:expr) (y:vars) (z:vars) : C.a `````` Raphael Rieu-Helft committed Dec 01, 2017 85 ``````= match e with `````` 86 `````` | UTerm v -> y v `````` Raphael Rieu-Helft committed Dec 14, 2017 87 88 `````` | Term c v -> C.( *) (C.interp c z) (y v) | Add e1 e2 -> C.(+) (interp e1 y z) (interp e2 y z) `````` Raphael Rieu-Helft committed Dec 01, 2017 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 `````` | Cst c -> C.interp c z end use import bool.Bool use import list.List type equality = (expr, expr) type context = list equality let predicate valid_eq (eq:equality) = match eq with (e1,e2) -> valid_expr e1 && valid_expr e2 end let predicate eq_bound (eq:equality) (b:int) = match eq with (e1,e2) -> expr_bound e1 b && expr_bound e2 b end let rec predicate valid_ctx (ctx:context) = match ctx with Nil -> true | Cons eq t -> valid_eq eq && valid_ctx t end let rec predicate ctx_bound (ctx:context) (b:int) = match ctx with Nil -> true | Cons eq t -> eq_bound eq b && ctx_bound t b end let rec lemma expr_bound_w (e:expr) (b1 b2:int) requires { b1 <= b2 } requires { expr_bound e b1 } ensures { expr_bound e b2 } variant { e } = match e with | Add e1 e2 -> expr_bound_w e1 b1 b2; expr_bound_w e2 b1 b2 | Cst _ -> () | Term _ _ -> () `````` 119 `````` | UTerm _ -> () `````` Raphael Rieu-Helft committed Dec 01, 2017 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 `````` end lemma eq_bound_w: forall e:equality, b1 b2:int. eq_bound e b1 -> b1 <= b2 -> eq_bound e b2 let rec lemma ctx_bound_w (l:context) (b1 b2:int) requires { ctx_bound l b1 } requires { b1 <= b2 } ensures { ctx_bound l b2 } variant { l } = match l with Nil -> () | Cons _ t -> ctx_bound_w t b1 b2 end function interp_eq (g:equality) (y:vars) (z:C.vars) : bool = match g with (g1, g2) -> interp g1 y z = interp g2 y z end function interp_ctx (l: context) (g: equality) (y: vars) (z:C.vars) : bool = match l with | Nil -> interp_eq g y z `````` Raphael Rieu-Helft committed Dec 14, 2017 137 `````` | Cons h t -> (interp_eq h y z) -> (interp_ctx t g y z) `````` Raphael Rieu-Helft committed Dec 01, 2017 138 139 140 141 142 143 144 145 146 `````` end use import array.Array use import matrix.Matrix let apply_r (m: matrix coeff) (v: array coeff) : array coeff requires { v.length = m.columns } ensures { result.length = m.rows } raises { C.Unknown -> true } `````` 147 ``````= let r = Array.make m.rows C.czero in `````` Raphael Rieu-Helft committed Dec 01, 2017 148 149 150 151 152 153 154 155 156 157 158 `````` for i = 0 to m.rows - 1 do for j = 0 to m.columns - 1 do r[i] <- C.add r[i] (C.mul (get m i j) v[j]); done done; r let apply_l (v: array coeff) (m: matrix coeff) : array coeff requires { v.length = m.rows } ensures { result.length = m.columns } raises { C.Unknown -> true } `````` 159 ``````= let r = Array.make m.columns C.czero in `````` Raphael Rieu-Helft committed Dec 01, 2017 160 161 162 163 164 165 166 167 168 169 170 171 `````` for j = 0 to m.columns - 1 do for i = 0 to m.rows - 1 do r[j] <- C.add r[j] (C.mul (get m i j) v[i]); done done; r use import ref.Ref let sprod (a b: array coeff) : coeff requires { a.length = b.length } raises { C.Unknown -> true } `````` 172 ``````= let r = ref C.czero in `````` Raphael Rieu-Helft committed Dec 01, 2017 173 174 175 176 177 178 179 180 181 182 183 184 `````` for i = 0 to a.length - 1 do r := C.add !r (C.mul a[i] b[i]); done; !r let m_append (m: matrix coeff) (v:array coeff) : matrix coeff requires { m.rows = v.length } ensures { result.rows = m.rows } ensures { result.columns = m.columns + 1 } ensures { forall i j. 0 <= i < m.rows -> 0 <= j < m.columns -> result.elts i j = m.elts i j } ensures { forall i. 0 <= i < m.rows -> result.elts i m.columns = v[i] } `````` 185 ``````= let r = Matrix.make m.rows (m.columns + 1) C.czero in `````` Raphael Rieu-Helft committed Dec 01, 2017 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 `````` for i = 0 to m.rows - 1 do invariant { forall k j. 0 <= k < i -> 0 <= j < m.columns -> r.elts k j = m.elts k j } invariant { forall k. 0 <= k < i -> r.elts k m.columns = v[k] } for j = 0 to m.columns - 1 do invariant { forall k j. 0 <= k < i -> 0 <= j < m.columns -> r.elts k j = m.elts k j } invariant { forall k. 0 <= k < i -> r.elts k m.columns = v[k] } invariant { forall l. 0 <= l < j -> r.elts i l = m.elts i l } set r i j (get m i j) done; set r i m.columns v[i] done; r let v_append (v: array coeff) (c: coeff) : array coeff ensures { length result = length v + 1 } ensures { forall k. 0 <= k < v.length -> result[k] = v[k] } ensures { result[v.length] = c } = let r = Array.make (v.length + 1) c in for i = 0 to v.length - 1 do invariant { forall k. 0 <= k < i -> r[k] = v[k] } invariant { r[v.length] = c } r[i] <- v[i] done; r `````` Raphael Rieu-Helft committed Dec 14, 2017 213 ``````let predicate (==) (a b: array coeff) `````` Raphael Rieu-Helft committed Dec 01, 2017 214 215 `````` ensures { result = true -> length a = length b /\ forall i. 0 <= i < length a -> C.eq a[i] b[i] } `````` Raphael Rieu-Helft committed Dec 14, 2017 216 217 218 219 220 221 222 223 224 ``````= if length a <> length b then false else let r = ref true in for i = 0 to length a - 1 do invariant { !r = true -> forall j. 0 <= j < i -> C.eq a[j] b[j] } if not (C.eq a[i] b[i]) then r := false; done; !r `````` Raphael Rieu-Helft committed Dec 01, 2017 225 226 227 228 229 230 231 232 233 234 `````` use import int.MinMax use import list.Length let rec function max_var (e:expr) : int variant { e } requires { valid_expr e } ensures { 0 <= result } ensures { expr_bound e result } = match e with `````` 235 `````` | Term _ i | UTerm i -> i `````` Raphael Rieu-Helft committed Dec 01, 2017 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 `````` | Cst _ -> 0 | Add e1 e2 -> max (max_var e1) (max_var e2) end let function max_var_e (e:equality) : int requires { valid_eq e } ensures { 0 <= result } ensures { eq_bound e result } = match e with (e1,e2) -> max (max_var e1) (max_var e2) end let rec function max_var_ctx (l:context) : int variant { l } requires { valid_ctx l } ensures { 0 <= result } ensures { ctx_bound l result } = match l with | Nil -> 0 | Cons e t -> max (max_var_e e) (max_var_ctx t) end let rec function opp_expr (e:expr) : expr `````` Raphael Rieu-Helft committed Dec 14, 2017 257 `````` (* ensures { forall y z. interp result y z = C.(-_) (interp e y z) }*) `````` Raphael Rieu-Helft committed Dec 01, 2017 258 259 260 261 262 `````` ensures { forall b. expr_bound e b -> expr_bound result b } variant { e } = match e with | Cst c -> Cst (C.opp c) | Term c j -> Term (C.opp c) j `````` 263 `````` | UTerm j -> Term (C.opp C.cone) j `````` Raphael Rieu-Helft committed Dec 01, 2017 264 265 266 267 268 `````` | Add e1 e2 -> Add (opp_expr e1) (opp_expr e2) end predicate no_cst (e:expr) = match e with `````` 269 270 `````` | Cst c -> C.eq c C.czero | Term _ _ | UTerm _ -> true `````` Raphael Rieu-Helft committed Dec 01, 2017 271 272 273 `````` | Add e1 e2 -> no_cst e1 && no_cst e2 end `````` 274 275 276 277 ``````(*TODO put this back in norm_eq*) let rec norm_eq_aux (ex acc_e:expr) (acc_c:coeff) : (expr, coeff) requires { no_cst acc_e } returns { (rex, rc) -> forall y z. `````` Raphael Rieu-Helft committed Dec 14, 2017 278 279 280 `````` C.(+) (interp rex y z) (interp (Cst rc) y z) = C.(+) (interp ex y z) (C.(+) (interp acc_e y z) (interp (Cst acc_c) y z)) } `````` 281 282 283 284 285 286 287 288 289 290 291 292 `````` returns { (rex, _) -> no_cst rex } returns { (rex, _) -> forall b:int. expr_bound ex b /\ expr_bound acc_e b -> expr_bound rex b } raises { C.Unknown -> true } variant { ex } = match ex with | Cst c -> acc_e, (C.add c acc_c) | Term _ _ | UTerm _ -> (Add acc_e ex, acc_c) | Add e1 e2 -> let ae, ac = norm_eq_aux e1 acc_e acc_c in norm_eq_aux e2 ae ac end `````` Raphael Rieu-Helft committed Dec 01, 2017 293 ``````let norm_eq (e:equality) : (expr, coeff) `````` Raphael Rieu-Helft committed Dec 14, 2017 294 295 `````` (* returns { (ex, c) -> forall y z. interp_eq e y z -> interp_eq (ex, Cst c) y z }*) `````` Raphael Rieu-Helft committed Dec 01, 2017 296 297 298 299 300 301 302 `````` returns { (ex, _) -> no_cst ex } returns { (ex, _) -> forall b:int. eq_bound e b -> expr_bound ex b } raises { C.Unknown -> true } = match e with | (e1, e2) -> let s = Add e1 (opp_expr e2) in assert { forall b. eq_bound e b -> expr_bound s b }; `````` 303 `````` match norm_eq_aux s (Cst C.czero) C.czero with `````` Raphael Rieu-Helft committed Dec 01, 2017 304 305 306 307 308 309 310 311 312 `````` (e, c) -> e, C.opp c end end (*TODO: predicate that says that matrices a, b represent the context*) let transpose (m:matrix coeff) : matrix coeff ensures { result.rows = m.columns /\ result.columns = m.rows } = `````` 313 `````` let r = Matrix.make m.columns m.rows C.czero in `````` Raphael Rieu-Helft committed Dec 01, 2017 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 `````` for i = 0 to m.rows - 1 do for j = 0 to m.columns - 1 do set r j i (get m i j) done done; r let swap_rows (m:matrix coeff) (i1 i2: int) : unit requires { 0 <= i1 < m.rows /\ 0 <= i2 < m.rows } = for j = 0 to m.columns - 1 do let c = get m i1 j in set m i1 j (get m i2 j); set m i2 j c done let mul_row (m:matrix coeff) (i: int) (c: coeff) : unit requires { 0 <= i < m.rows } `````` 331 `````` requires { not (C.eq c C.czero) } `````` Raphael Rieu-Helft committed Dec 01, 2017 332 333 334 335 336 337 338 339 340 341 342 343 344 345 ``````= for j = 0 to m.columns - 1 do set m i j (C.mul c (get m i j)) done let addmul_row (m:matrix coeff) (src dst: int) (c: coeff) : unit requires { 0 <= src < m.rows /\ 0 <= dst < m.rows } raises { C.Unknown -> true } = for j = 0 to m.columns - 1 do set m dst j (C.add (get m dst j) (C.mul c (get m src j))) done use import ref.Refint use import option.Option `````` 346 ``````(*TODO this goes inside gauss_jordan*) `````` Raphael Rieu-Helft committed Dec 14, 2017 347 348 349 350 ``````val breakpoint (a: matrix coeff) : unit writes { a } let a_breakpoint (v:array coeff) : unit = any unit `````` 351 `````` `````` Raphael Rieu-Helft committed Dec 01, 2017 352 353 354 355 356 357 358 359 ``````let gauss_jordan (a: matrix coeff) : option (array coeff) (*AX=B, a=(A|B), result=X*) returns { Some r -> Array.length r = a.columns - 1 | None -> true } requires { 1 <= a.rows /\ 1 <= a.columns } raises { C.Unknown -> true } = let n = a.rows in let m = a.columns in `````` Raphael Rieu-Helft committed Dec 14, 2017 360 361 362 363 364 365 366 367 368 369 370 `````` let rec find_nonz (i j:int) requires { 0 <= i <= n } requires { 0 <= j < m } variant { n-i } ensures { i <= result <= n } ensures { result < n -> not (C.eq (a.elts result j) C.czero) } = if i >= n then n else if C.eq (get a i j) C.czero then find_nonz (i+1) j else i in `````` Raphael Rieu-Helft committed Dec 01, 2017 371 372 373 374 375 376 377 378 `````` let pivots = Array.make n 0 in let r = ref (-1) in for j = 0 to m-1 do invariant { -1 <= !r < n } invariant { forall i. 0 <= i <= !r -> 0 <= pivots[i] } invariant { forall i1 i2: int. 0 <= i1 < i2 <= !r -> pivots[i1] < pivots[i2] } invariant { !r >= 0 -> pivots[!r] < j } label Start in `````` Raphael Rieu-Helft committed Dec 14, 2017 379 `````` let k = find_nonz (!r+1) j in `````` Raphael Rieu-Helft committed Dec 01, 2017 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 `````` if k < n then begin incr r; pivots[!r] <- j; mul_row a k (C.inv(get a k j)); if k <> !r then swap_rows a k !r; for i = 0 to n-1 do if i <> !r then addmul_row a !r i (C.opp(get a i j)) done; end done; if !r < 0 then None (* matrix is all zeroes *) else if pivots[!r] >= m-1 then None (*pivot on last column, no solution*) else begin `````` 395 `````` let v = Array.make (m-1) C.czero in `````` Raphael Rieu-Helft committed Dec 01, 2017 396 397 398 `````` for i = 0 to !r do v[pivots[i]] <- get a i (m-1) done; `````` Raphael Rieu-Helft committed Dec 14, 2017 399 `````` (* a_breakpoint v;*) `````` Raphael Rieu-Helft committed Dec 01, 2017 400 401 402 `````` Some v end `````` 403 `````` `````` Raphael Rieu-Helft committed Dec 01, 2017 404 405 406 ``````let linear_decision (l: context) (g: equality) : bool requires { valid_ctx l } requires { valid_eq g } `````` 407 `````` ensures { forall y z. result -> interp_ctx l g y z } `````` Raphael Rieu-Helft committed Dec 01, 2017 408 409 410 `````` raises { C.Unknown -> true } = let nv = max (max_var_e g) (max_var_ctx l) in `````` 411 412 413 `````` let a = Matrix.make (length l) (nv+1) C.czero in let b = Array.make (length l) C.czero in (* ax = b *) let v = Array.make (nv+1) C.czero in (* goal *) `````` Raphael Rieu-Helft committed Dec 18, 2017 414 `````` (*v[0] <- any coeff;*) `````` Raphael Rieu-Helft committed Dec 14, 2017 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 `````` let rec fill_expr (ex: expr) (i:int): unit variant { ex } requires { no_cst ex } raises { C.Unknown -> true } requires { 0 <= i < length l } requires { expr_bound ex nv } = match ex with | Cst c -> if C.eq c C.czero then () else absurd | Term c j -> set a i j (C.add (get a i j) c) | UTerm j -> set a i j (C.add (get a i j) C.cone) | Add e1 e2 -> fill_expr e1 i; fill_expr e2 i end in let rec fill_ctx (ctx:context) (i:int) : unit requires { ctx_bound ctx nv } variant { length l - i } requires { length l - i = length ctx } requires { 0 <= i <= length l } raises { C.Unknown -> true } = match ctx with | Nil -> () | Cons e t -> assert { i < length l }; let ex, c = norm_eq e in if (not (C.eq c C.czero)) then b[i] <- C.add b[i] c; fill_expr ex i; fill_ctx t (i+1) end in let rec fill_goal (ex:expr) : unit requires { expr_bound ex nv } variant { ex } requires { no_cst ex } raises { C.Unknown -> true } = match ex with | Cst c -> if C.eq c C.czero then () else absurd | Term c j -> v[j] <- C.add v[j] c | UTerm j -> v[j] <- C.add v[j] C.cone | Add e1 e2 -> fill_goal e1; fill_goal e2 `````` Raphael Rieu-Helft committed Dec 18, 2017 452 `````` end in `````` Raphael Rieu-Helft committed Dec 14, 2017 453 `````` fill_ctx l 0; `````` Raphael Rieu-Helft committed Dec 01, 2017 454 `````` let (ex, d) = norm_eq g in `````` Raphael Rieu-Helft committed Dec 14, 2017 455 456 457 `````` fill_goal ex; (*let show (a: matrix coeff) (b v: array coeff) (d: coeff) = breakpoint a in show a b v d;*) `````` Raphael Rieu-Helft committed Dec 01, 2017 458 459 460 461 462 463 464 465 `````` let ab = m_append a b in let cd = v_append v d in let ab' = transpose ab in match gauss_jordan (m_append ab' cd) with | Some r -> apply_l r a == v && C.eq (sprod r b) d | None -> false end `````` 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 ``````(* forall eq in list interp_eq est vraie -> interp_eq (toute combinaison linéaire) est vraie *) end module RationalCoeffs use import int.Int use import real.RealInfix use import real.FromInt use import int.Abs meta coercion function from_int type t = (int, int) type rvars = int -> real function of_int (n:int) : t = (n,1) (*meta coercion function of_int*) let constant rzero = (0,1) let constant rone = (1,1) function rinterp (t:t) (v:rvars) : real = match t with `````` Raphael Rieu-Helft committed Dec 14, 2017 490 `````` | (n,d) -> from_int n /. from_int d (*todo if d = 1 then n...*) `````` 491 492 493 494 495 496 497 498 499 `````` end use import int.ComputerDivision use import ref.Ref use import number.Gcd let gcd (x:int) (y:int) requires { x >= 0 /\ y >= 0 } ensures { result = gcd x y } `````` Raphael Rieu-Helft committed Dec 14, 2017 500 `````` ensures { x > 0 -> result > 0 } `````` 501 `````` = `````` Raphael Rieu-Helft committed Dec 14, 2017 502 `````` let ghost ox = x in `````` 503 504 505 506 507 508 `````` let x = ref x in let y = ref y in label Pre in while (!y > 0) do invariant { !x >= 0 /\ !y >= 0 } invariant { gcd !x !y = gcd (!x at Pre) (!y at Pre) } variant { !y } `````` Raphael Rieu-Helft committed Dec 14, 2017 509 `````` invariant { ox > 0 -> !x > 0 } `````` 510 511 512 513 514 `````` let r = mod !x !y in let ghost q = div !x !y in assert { r = !x - q * !y }; x := !y; y := r; done; !x `````` Raphael Rieu-Helft committed Dec 14, 2017 515 ``````(* `````` 516 517 518 519 ``````let simp (t:t) : t ensures { forall v:rvars. rinterp result v = rinterp t v } = match t with | (n,d) -> `````` Raphael Rieu-Helft committed Dec 14, 2017 520 521 522 523 524 525 526 527 `````` let g = gcd (abs n) (abs d) in if g > 1 then let n', d' = (div n g, div d g) in assert { n = g * n' /\ d = g * d' }; assert { n /. d = n' /. d' }; (n', d') else (n, d) `````` 528 `````` end `````` Raphael Rieu-Helft committed Dec 14, 2017 529 ``````*) `````` 530 531 ``````let radd (a b:t) = match (a,b) with `````` Raphael Rieu-Helft committed Dec 14, 2017 532 `````` | (n1,d1), (n2,d2) -> (*simp*) ((n1*d2 + n2*d1),(d1*d2)) `````` 533 534 535 536 `````` end let rmul (a b:t) = match (a,b) with `````` Raphael Rieu-Helft committed Dec 14, 2017 537 `````` | (n1,d1), (n2, d2) -> (*simp*) (n1*n2, d1*d2) `````` 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 `````` end let ropp (a:t) = match a with | (n,d) -> (-n, d) end let predicate req (a b:t) = match (a,b) with | (n1,d1), (n2,d2) -> n1 * d2 = n2 * d1 end let rinv a = match a with | (n,d) -> (d,n) end let function rle (a b:t) = match (a,b) with | (n1,d1), (n2,d2) -> n1 * d2 <= n2 * d1 end predicate (<=) (a b:t) = rle a b end module LinearDecisionRational use import RationalCoeffs `````` Raphael Rieu-Helft committed Dec 14, 2017 567 568 569 ``````use import real.RealInfix use import real.FromInt `````` Raphael Rieu-Helft committed Dec 18, 2017 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 ``````clone export LinearEquationsDecision with type C.a = real, function C.(+) = (+.), function C.( *) = ( *.), type coeff = t, function C.interp=rinterp, constant C.azero = Real.zero, constant C.aone = Real.one, predicate C.ale = (<=.), val C.czero=rzero, val C.cone=rone, lemma C.zero_def, lemma C.one_def, val C.add=radd, val C.mul=rmul, val C.opp=ropp, val C.eq=req, val C.inv=rinv, lemma C.eq_def, val C.le=rle(*, goal ZeroLessOne, goal C.CompatOrderAdd, goal C.CompatOrderMult, goal C.Unitary, goal C.NonTrivialRing, goal C.Mul_distr_l, goal C.Mul_distr_r, goal C.Inv_def_l, goal C.Inv_def_r, goal C.MulAssoc.Assoc, goal C.Assoc, goal C.MulComm.Comm, goal C.Comm, goal C.Unit_def_l, goal C.Unit_def_r*) end module LinearDecisionInt use import int.Int function id (t:int) (v:int -> int) : int = t let predicate eq (a b:int) = a=b exception Unknown let inv (t:int) : int ensures { forall v: int -> int. id result v * id t v = one } ensures { not (eq result zero) } raises { Unknown -> true } = raise Unknown clone export LinearEquationsDecision with type C.a = int, function C.(+)=(+), function C.(*) = (*), type coeff = int, function C.interp = id, constant C.azero = zero, constant C.aone = one, predicate C.ale= (<=), val C.czero = zero, val C.cone = one, lemma C.zero_def, lemma C.one_def, val C.add = (+), val C.mul = ( *), val C.opp = (-_), val C.eq = eq, val C.inv = inv, lemma C.eq_def, val C.le = (<=)(*, goal ZeroLessOne, goal C.CompatOrderAdd, goal C.CompatOrderMult, goal C.Unitary, goal C.NonTrivialRing, goal C.Mul_distr_l, goal C.Mul_distr_r, goal C.Inv_def_l, goal C.Inv_def_r, goal C.MulAssoc.Assoc, goal C.Assoc, goal C.MulComm.Comm, goal C.Comm, goal C.Unit_def_l, goal C.Unit_def_r*) use import real.FromInt axiom from_int_ext: forall x y: int. from_int x = from_int y -> x = y (*FIXME put this in real.why ?*) use import RationalCoeffs use LinearDecisionRational as R use import list.List let function m (x:int) : (int, int) ensures { forall z. rinterp result z = from_int x } = (x,1) val function m_y (y:int->int): (int -> real) ensures { forall i. result i = from_int (y i) } `````` 605 `````` `````` Raphael Rieu-Helft committed Dec 18, 2017 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 ``````let rec function m_expr (e:expr) : R.expr ensures { forall y z. R.interp result (m_y y) (m_y z) = from_int (interp e y z) } ensures { valid_expr e -> R.valid_expr result } = match e with | Cst c -> R.Cst (m c) | Add e1 e2 -> R.Add (m_expr e1) (m_expr e2) | Term c n -> R.Term (m c) n | UTerm n -> R.UTerm n end let function m_eq (eq:equality) : R.equality ensures { forall y z. R.interp_eq result (m_y y) (m_y z) <-> interp_eq eq y z } ensures { valid_eq eq -> R.valid_eq result } = match eq with (e1,e2) -> (m_expr e1, m_expr e2) end let rec function m_ctx (ctx:context) : R.context ensures { forall y z g. R.interp_ctx result (m_eq g) (m_y y) (m_y z) <-> interp_ctx ctx g y z } ensures { valid_ctx ctx -> R.valid_ctx result } variant { ctx } = match ctx with | Nil -> Nil | Cons h t -> let r = Cons (m_eq h) (m_ctx t) in r end let int_decision (l: context) (g: equality) : bool requires { valid_ctx l } requires { valid_eq g } ensures { forall y z. result -> interp_ctx l g y z } (*raises { R.Unknown -> true } ??? *) = R.linear_decision (m_ctx l) (m_eq g) `````` 640 641 642 `````` end `````` Raphael Rieu-Helft committed Dec 18, 2017 643 644 `````` module Test `````` 645 646 647 648 649 650 651 652 653 654 655 656 657 `````` use import RationalCoeffs use import LinearDecisionRational use import int.Int use import real.RealInfix use import real.FromInt meta coercion function from_int goal g: forall x y: real. (from_int 3 /. from_int 1) *. x +. (from_int 2/. from_int 1) *. y = (from_int 21/. from_int 1) -> (from_int 7 /. from_int 1) *. x +. (from_int 4/. from_int 1) *. y = (from_int 47/. from_int 1) -> x = (from_int 5 /. from_int 1) `````` Raphael Rieu-Helft committed Dec 18, 2017 658 659 660 661 662 663 664 665 666 667 668 669 ``````end module TestInt use import LinearDecisionInt use import int.Int goal g: forall x y:int. 3 * x + 2 * y = 21 -> 7 * x + 4 * y = 47 -> x = 5 `````` Raphael Rieu-Helft committed Dec 01, 2017 670 ``end``