Mercurial > cpdt > repo

annotate src/Firstorder.v @ 157:2022e3f2aa26

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Simplify Concrete
author Adam Chlipala <adamc@hcoop.net>
date Sun, 02 Nov 2008 14:41:01 -0500
parents be15ed0a8bae
children 6087a32b1926
rev   line source
adamc@152 1 (* Copyright (c) 2008, Adam Chlipala
adamc@152 2 *
adamc@152 3 * This work is licensed under a
adamc@152 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@152 5 * Unported License.
adamc@152 6 * The license text is available at:
adamc@152 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@152 8 *)
adamc@152 9
adamc@152 10 (* begin hide *)
adamc@156 11 Require Import Arith String List.
adamc@152 12
adamc@152 13 Require Import Tactics.
adamc@152 14
adamc@152 15 Set Implicit Arguments.
adamc@152 16 (* end hide *)
adamc@152 17
adamc@152 18
adamc@153 19 (** %\part{Formalizing Programming Languages and Compilers}
adamc@152 20
adamc@153 21 \chapter{First-Order Abstract Syntax}% *)
adamc@152 22
adamc@152 23 (** TODO: Prose for this chapter *)
adamc@152 24
adamc@152 25
adamc@152 26 (** * Concrete Binding *)
adamc@152 27
adamc@152 28 Module Concrete.
adamc@152 29
adamc@152 30 Definition var := string.
adamc@152 31 Definition var_eq := string_dec.
adamc@152 32
adamc@152 33 Inductive exp : Set :=
adamc@152 34 | Const : bool -> exp
adamc@152 35 | Var : var -> exp
adamc@152 36 | App : exp -> exp -> exp
adamc@152 37 | Abs : var -> exp -> exp.
adamc@152 38
adamc@152 39 Inductive type : Set :=
adamc@152 40 | Bool : type
adamc@152 41 | Arrow : type -> type -> type.
adamc@152 42
adamc@152 43 Infix "-->" := Arrow (right associativity, at level 60).
adamc@152 44
adamc@152 45 Definition ctx := list (var * type).
adamc@152 46
adamc@152 47 Reserved Notation "G |-v x : t" (no associativity, at level 90, x at next level).
adamc@152 48
adamc@152 49 Inductive lookup : ctx -> var -> type -> Prop :=
adamc@152 50 | First : forall x t G,
adamc@152 51 (x, t) :: G |-v x : t
adamc@152 52 | Next : forall x t x' t' G,
adamc@152 53 x <> x'
adamc@152 54 -> G |-v x : t
adamc@152 55 -> (x', t') :: G |-v x : t
adamc@152 56
adamc@152 57 where "G |-v x : t" := (lookup G x t).
adamc@152 58
adamc@152 59 Hint Constructors lookup.
adamc@152 60
adamc@152 61 Reserved Notation "G |-e e : t" (no associativity, at level 90, e at next level).
adamc@152 62
adamc@152 63 Inductive hasType : ctx -> exp -> type -> Prop :=
adamc@152 64 | TConst : forall G b,
adamc@152 65 G |-e Const b : Bool
adamc@152 66 | TVar : forall G v t,
adamc@152 67 G |-v v : t
adamc@152 68 -> G |-e Var v : t
adamc@152 69 | TApp : forall G e1 e2 dom ran,
adamc@152 70 G |-e e1 : dom --> ran
adamc@152 71 -> G |-e e2 : dom
adamc@152 72 -> G |-e App e1 e2 : ran
adamc@152 73 | TAbs : forall G x e' dom ran,
adamc@152 74 (x, dom) :: G |-e e' : ran
adamc@152 75 -> G |-e Abs x e' : dom --> ran
adamc@152 76
adamc@152 77 where "G |-e e : t" := (hasType G e t).
adamc@152 78
adamc@152 79 Hint Constructors hasType.
adamc@152 80
adamc@155 81 Lemma weaken_lookup : forall x t G' G1,
adamc@155 82 G1 |-v x : t
adamc@155 83 -> G1 ++ G' |-v x : t.
adamc@152 84 induction G1 as [ | [x' t'] tl ]; crush;
adamc@152 85 match goal with
adamc@152 86 | [ H : _ |-v _ : _ |- _ ] => inversion H; crush
adamc@152 87 end.
adamc@152 88 Qed.
adamc@152 89
adamc@152 90 Hint Resolve weaken_lookup.
adamc@152 91
adamc@152 92 Theorem weaken_hasType' : forall G' G e t,
adamc@152 93 G |-e e : t
adamc@155 94 -> G ++ G' |-e e : t.
adamc@152 95 induction 1; crush; eauto.
adamc@152 96 Qed.
adamc@152 97
adamc@155 98 Theorem weaken_hasType : forall e t,
adamc@155 99 nil |-e e : t
adamc@155 100 -> forall G', G' |-e e : t.
adamc@155 101 intros; change G' with (nil ++ G');
adamc@152 102 eapply weaken_hasType'; eauto.
adamc@152 103 Qed.
adamc@152 104
adamc@152 105 Theorem weaken_hasType_closed : forall e t,
adamc@152 106 nil |-e e : t
adamc@152 107 -> forall G, G |-e e : t.
adamc@152 108 intros; rewrite (app_nil_end G); apply weaken_hasType; auto.
adamc@152 109 Qed.
adamc@152 110
adamc@157 111 Hint Resolve weaken_hasType_closed.
adamc@152 112
adamc@152 113 Section subst.
adamc@152 114 Variable x : var.
adamc@152 115 Variable e1 : exp.
adamc@152 116
adamc@152 117 Fixpoint subst (e2 : exp) : exp :=
adamc@152 118 match e2 with
adamc@152 119 | Const b => Const b
adamc@152 120 | Var x' =>
adamc@152 121 if var_eq x' x
adamc@152 122 then e1
adamc@152 123 else Var x'
adamc@152 124 | App e1 e2 => App (subst e1) (subst e2)
adamc@152 125 | Abs x' e' =>
adamc@152 126 Abs x' (if var_eq x' x
adamc@152 127 then e'
adamc@152 128 else subst e')
adamc@152 129 end.
adamc@152 130
adamc@152 131 Variable xt : type.
adamc@152 132 Hypothesis Ht' : nil |-e e1 : xt.
adamc@152 133
adamc@157 134 Notation "x # G" := (forall t' : type, In (x, t') G -> False) (no associativity, at level 90).
adamc@152 135
adamc@157 136 Lemma subst_lookup' : forall x' t,
adamc@152 137 x <> x'
adamc@155 138 -> forall G1, G1 ++ (x, xt) :: nil |-v x' : t
adamc@155 139 -> G1 |-v x' : t.
adamc@152 140 induction G1 as [ | [x'' t'] tl ]; crush;
adamc@152 141 match goal with
adamc@152 142 | [ H : _ |-v _ : _ |- _ ] => inversion H
adamc@152 143 end; crush.
adamc@152 144 Qed.
adamc@152 145
adamc@157 146 Hint Resolve subst_lookup'.
adamc@152 147
adamc@157 148 Lemma subst_lookup : forall x' t G1,
adamc@157 149 x' # G1
adamc@155 150 -> G1 ++ (x, xt) :: nil |-v x' : t
adamc@155 151 -> t = xt.
adamc@152 152 induction G1 as [ | [x'' t'] tl ]; crush; eauto;
adamc@152 153 match goal with
adamc@152 154 | [ H : _ |-v _ : _ |- _ ] => inversion H
adamc@155 155 end; crush; elimtype False; eauto;
adamc@155 156 match goal with
adamc@155 157 | [ H : nil |-v _ : _ |- _ ] => inversion H
adamc@155 158 end.
adamc@152 159 Qed.
adamc@152 160
adamc@157 161 Implicit Arguments subst_lookup [x' t G1].
adamc@152 162
adamc@155 163 Lemma shadow_lookup : forall v t t' G1,
adamc@152 164 G1 |-v x : t'
adamc@155 165 -> G1 ++ (x, xt) :: nil |-v v : t
adamc@155 166 -> G1 |-v v : t.
adamc@152 167 induction G1 as [ | [x'' t''] tl ]; crush;
adamc@152 168 match goal with
adamc@152 169 | [ H : nil |-v _ : _ |- _ ] => inversion H
adamc@152 170 | [ H1 : _ |-v _ : _, H2 : _ |-v _ : _ |- _ ] =>
adamc@152 171 inversion H1; crush; inversion H2; crush
adamc@152 172 end.
adamc@152 173 Qed.
adamc@152 174
adamc@155 175 Lemma shadow_hasType' : forall G e t,
adamc@152 176 G |-e e : t
adamc@155 177 -> forall G1, G = G1 ++ (x, xt) :: nil
adamc@152 178 -> forall t'', G1 |-v x : t''
adamc@155 179 -> G1 |-e e : t.
adamc@152 180 Hint Resolve shadow_lookup.
adamc@152 181
adamc@152 182 induction 1; crush; eauto;
adamc@152 183 match goal with
adamc@152 184 | [ H : (?x0, _) :: _ ++ (x, _) :: _ |-e _ : _ |- _ ] =>
adamc@152 185 destruct (var_eq x0 x); subst; eauto
adamc@152 186 end.
adamc@155 187 Qed.
adamc@152 188
adamc@155 189 Lemma shadow_hasType : forall G1 e t t'',
adamc@155 190 G1 ++ (x, xt) :: nil |-e e : t
adamc@152 191 -> G1 |-v x : t''
adamc@155 192 -> G1 |-e e : t.
adamc@152 193 intros; eapply shadow_hasType'; eauto.
adamc@152 194 Qed.
adamc@152 195
adamc@152 196 Hint Resolve shadow_hasType.
adamc@152 197
adamc@157 198 Lemma disjoint_cons : forall x x' t (G : ctx),
adamc@157 199 x # G
adamc@157 200 -> x' <> x
adamc@157 201 -> x # (x', t) :: G.
adamc@157 202 firstorder;
adamc@157 203 match goal with
adamc@157 204 | [ H : (_, _) = (_, _) |- _ ] => injection H
adamc@157 205 end; crush.
adamc@157 206 Qed.
adamc@157 207
adamc@157 208 Hint Resolve disjoint_cons.
adamc@157 209
adamc@152 210 Theorem subst_hasType : forall G e2 t,
adamc@152 211 G |-e e2 : t
adamc@155 212 -> forall G1, G = G1 ++ (x, xt) :: nil
adamc@157 213 -> x # G1
adamc@155 214 -> G1 |-e subst e2 : t.
adamc@152 215 induction 1; crush;
adamc@152 216 try match goal with
adamc@152 217 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@152 218 end; crush; eauto 6;
adamc@152 219 match goal with
adamc@157 220 | [ H1 : x # _, H2 : _ |-v x : _ |- _ ] =>
adamc@157 221 rewrite (subst_lookup H1 H2)
adamc@152 222 end; crush.
adamc@152 223 Qed.
adamc@152 224
adamc@152 225 Theorem subst_hasType_closed : forall e2 t,
adamc@152 226 (x, xt) :: nil |-e e2 : t
adamc@152 227 -> nil |-e subst e2 : t.
adamc@155 228 intros; eapply subst_hasType; eauto.
adamc@152 229 Qed.
adamc@152 230 End subst.
adamc@152 231
adamc@154 232 Hint Resolve subst_hasType_closed.
adamc@154 233
adamc@154 234 Notation "[ x ~> e1 ] e2" := (subst x e1 e2) (no associativity, at level 80).
adamc@154 235
adamc@154 236 Inductive val : exp -> Prop :=
adamc@154 237 | VConst : forall b, val (Const b)
adamc@154 238 | VAbs : forall x e, val (Abs x e).
adamc@154 239
adamc@154 240 Hint Constructors val.
adamc@154 241
adamc@154 242 Reserved Notation "e1 ==> e2" (no associativity, at level 90).
adamc@154 243
adamc@154 244 Inductive step : exp -> exp -> Prop :=
adamc@154 245 | Beta : forall x e1 e2,
adamc@154 246 App (Abs x e1) e2 ==> [x ~> e2] e1
adamc@154 247 | Cong1 : forall e1 e2 e1',
adamc@154 248 e1 ==> e1'
adamc@154 249 -> App e1 e2 ==> App e1' e2
adamc@154 250 | Cong2 : forall e1 e2 e2',
adamc@154 251 val e1
adamc@154 252 -> e2 ==> e2'
adamc@154 253 -> App e1 e2 ==> App e1 e2'
adamc@154 254
adamc@154 255 where "e1 ==> e2" := (step e1 e2).
adamc@154 256
adamc@154 257 Hint Constructors step.
adamc@154 258
adamc@155 259 Lemma progress' : forall G e t, G |-e e : t
adamc@154 260 -> G = nil
adamc@154 261 -> val e \/ exists e', e ==> e'.
adamc@154 262 induction 1; crush; eauto;
adamc@154 263 try match goal with
adamc@154 264 | [ H : _ |-e _ : _ --> _ |- _ ] => inversion H
adamc@154 265 end;
adamc@154 266 repeat match goal with
adamc@154 267 | [ H : _ |- _ ] => solve [ inversion H; crush; eauto ]
adamc@154 268 end.
adamc@154 269 Qed.
adamc@154 270
adamc@155 271 Theorem progress : forall e t, nil |-e e : t
adamc@155 272 -> val e \/ exists e', e ==> e'.
adamc@155 273 intros; eapply progress'; eauto.
adamc@155 274 Qed.
adamc@155 275
adamc@155 276 Lemma preservation' : forall G e t, G |-e e : t
adamc@154 277 -> G = nil
adamc@154 278 -> forall e', e ==> e'
adamc@155 279 -> nil |-e e' : t.
adamc@154 280 induction 1; inversion 2; crush; eauto;
adamc@154 281 match goal with
adamc@154 282 | [ H : _ |-e Abs _ _ : _ |- _ ] => inversion H
adamc@154 283 end; eauto.
adamc@154 284 Qed.
adamc@154 285
adamc@155 286 Theorem preservation : forall e t, nil |-e e : t
adamc@155 287 -> forall e', e ==> e'
adamc@155 288 -> nil |-e e' : t.
adamc@155 289 intros; eapply preservation'; eauto.
adamc@155 290 Qed.
adamc@155 291
adamc@152 292 End Concrete.
adamc@156 293
adamc@156 294
adamc@156 295 (** * De Bruijn Indices *)
adamc@156 296
adamc@156 297 Module DeBruijn.
adamc@156 298
adamc@156 299 Definition var := nat.
adamc@156 300 Definition var_eq := eq_nat_dec.
adamc@156 301
adamc@156 302 Inductive exp : Set :=
adamc@156 303 | Const : bool -> exp
adamc@156 304 | Var : var -> exp
adamc@156 305 | App : exp -> exp -> exp
adamc@156 306 | Abs : exp -> exp.
adamc@156 307
adamc@156 308 Inductive type : Set :=
adamc@156 309 | Bool : type
adamc@156 310 | Arrow : type -> type -> type.
adamc@156 311
adamc@156 312 Infix "-->" := Arrow (right associativity, at level 60).
adamc@156 313
adamc@156 314 Definition ctx := list type.
adamc@156 315
adamc@156 316 Reserved Notation "G |-v x : t" (no associativity, at level 90, x at next level).
adamc@156 317
adamc@156 318 Inductive lookup : ctx -> var -> type -> Prop :=
adamc@156 319 | First : forall t G,
adamc@156 320 t :: G |-v O : t
adamc@156 321 | Next : forall x t t' G,
adamc@156 322 G |-v x : t
adamc@156 323 -> t' :: G |-v S x : t
adamc@156 324
adamc@156 325 where "G |-v x : t" := (lookup G x t).
adamc@156 326
adamc@156 327 Hint Constructors lookup.
adamc@156 328
adamc@156 329 Reserved Notation "G |-e e : t" (no associativity, at level 90, e at next level).
adamc@156 330
adamc@156 331 Inductive hasType : ctx -> exp -> type -> Prop :=
adamc@156 332 | TConst : forall G b,
adamc@156 333 G |-e Const b : Bool
adamc@156 334 | TVar : forall G v t,
adamc@156 335 G |-v v : t
adamc@156 336 -> G |-e Var v : t
adamc@156 337 | TApp : forall G e1 e2 dom ran,
adamc@156 338 G |-e e1 : dom --> ran
adamc@156 339 -> G |-e e2 : dom
adamc@156 340 -> G |-e App e1 e2 : ran
adamc@156 341 | TAbs : forall G e' dom ran,
adamc@156 342 dom :: G |-e e' : ran
adamc@156 343 -> G |-e Abs e' : dom --> ran
adamc@156 344
adamc@156 345 where "G |-e e : t" := (hasType G e t).
adamc@156 346
adamc@156 347 Hint Constructors hasType.
adamc@156 348
adamc@156 349 Lemma weaken_lookup : forall G' v t G,
adamc@156 350 G |-v v : t
adamc@156 351 -> G ++ G' |-v v : t.
adamc@156 352 induction 1; crush.
adamc@156 353 Qed.
adamc@156 354
adamc@156 355 Hint Resolve weaken_lookup.
adamc@156 356
adamc@156 357 Theorem weaken_hasType' : forall G' G e t,
adamc@156 358 G |-e e : t
adamc@156 359 -> G ++ G' |-e e : t.
adamc@156 360 induction 1; crush; eauto.
adamc@156 361 Qed.
adamc@156 362
adamc@156 363 Theorem weaken_hasType : forall e t,
adamc@156 364 nil |-e e : t
adamc@156 365 -> forall G', G' |-e e : t.
adamc@156 366 intros; change G' with (nil ++ G');
adamc@156 367 eapply weaken_hasType'; eauto.
adamc@156 368 Qed.
adamc@156 369
adamc@156 370 Theorem weaken_hasType_closed : forall e t,
adamc@156 371 nil |-e e : t
adamc@156 372 -> forall G, G |-e e : t.
adamc@156 373 intros; rewrite (app_nil_end G); apply weaken_hasType; auto.
adamc@156 374 Qed.
adamc@156 375
adamc@157 376 Hint Resolve weaken_hasType_closed.
adamc@156 377
adamc@156 378 Section subst.
adamc@156 379 Variable e1 : exp.
adamc@156 380
adamc@156 381 Fixpoint subst (x : var) (e2 : exp) : exp :=
adamc@156 382 match e2 with
adamc@156 383 | Const b => Const b
adamc@156 384 | Var x' =>
adamc@156 385 if var_eq x' x
adamc@156 386 then e1
adamc@156 387 else Var x'
adamc@156 388 | App e1 e2 => App (subst x e1) (subst x e2)
adamc@156 389 | Abs e' => Abs (subst (S x) e')
adamc@156 390 end.
adamc@156 391
adamc@156 392 Variable xt : type.
adamc@156 393
adamc@156 394 Lemma subst_eq : forall t G1,
adamc@156 395 G1 ++ xt :: nil |-v length G1 : t
adamc@156 396 -> t = xt.
adamc@156 397 induction G1; inversion 1; crush.
adamc@156 398 Qed.
adamc@156 399
adamc@156 400 Implicit Arguments subst_eq [t G1].
adamc@156 401
adamc@156 402 Lemma subst_eq' : forall t G1 x,
adamc@156 403 G1 ++ xt :: nil |-v x : t
adamc@156 404 -> x <> length G1
adamc@156 405 -> G1 |-v x : t.
adamc@156 406 induction G1; inversion 1; crush;
adamc@156 407 match goal with
adamc@156 408 | [ H : nil |-v _ : _ |- _ ] => inversion H
adamc@156 409 end.
adamc@156 410 Qed.
adamc@156 411
adamc@156 412 Hint Resolve subst_eq'.
adamc@156 413
adamc@156 414 Lemma subst_neq : forall v t G1,
adamc@156 415 G1 ++ xt :: nil |-v v : t
adamc@156 416 -> v <> length G1
adamc@156 417 -> G1 |-e Var v : t.
adamc@156 418 induction G1; inversion 1; crush.
adamc@156 419 Qed.
adamc@156 420
adamc@156 421 Hint Resolve subst_neq.
adamc@156 422
adamc@156 423 Hypothesis Ht' : nil |-e e1 : xt.
adamc@156 424
adamc@156 425 Lemma hasType_push : forall dom G1 e' ran,
adamc@156 426 dom :: G1 |-e subst (length (dom :: G1)) e' : ran
adamc@156 427 -> dom :: G1 |-e subst (S (length G1)) e' : ran.
adamc@156 428 trivial.
adamc@156 429 Qed.
adamc@156 430
adamc@156 431 Hint Resolve hasType_push.
adamc@156 432
adamc@156 433 Theorem subst_hasType : forall G e2 t,
adamc@156 434 G |-e e2 : t
adamc@156 435 -> forall G1, G = G1 ++ xt :: nil
adamc@156 436 -> G1 |-e subst (length G1) e2 : t.
adamc@156 437 induction 1; crush;
adamc@156 438 try match goal with
adamc@156 439 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@156 440 end; crush; eauto 6;
adamc@156 441 try match goal with
adamc@156 442 | [ H : _ |-v _ : _ |- _ ] =>
adamc@156 443 rewrite (subst_eq H)
adamc@156 444 end; crush.
adamc@156 445 Qed.
adamc@156 446
adamc@156 447 Theorem subst_hasType_closed : forall e2 t,
adamc@156 448 xt :: nil |-e e2 : t
adamc@156 449 -> nil |-e subst O e2 : t.
adamc@156 450 intros; change O with (length (@nil type)); eapply subst_hasType; eauto.
adamc@156 451 Qed.
adamc@156 452 End subst.
adamc@156 453
adamc@156 454 Hint Resolve subst_hasType_closed.
adamc@156 455
adamc@156 456 Notation "[ x ~> e1 ] e2" := (subst e1 x e2) (no associativity, at level 80).
adamc@156 457
adamc@156 458 Inductive val : exp -> Prop :=
adamc@156 459 | VConst : forall b, val (Const b)
adamc@156 460 | VAbs : forall e, val (Abs e).
adamc@156 461
adamc@156 462 Hint Constructors val.
adamc@156 463
adamc@156 464 Reserved Notation "e1 ==> e2" (no associativity, at level 90).
adamc@156 465
adamc@156 466 Inductive step : exp -> exp -> Prop :=
adamc@156 467 | Beta : forall e1 e2,
adamc@156 468 App (Abs e1) e2 ==> [O ~> e2] e1
adamc@156 469 | Cong1 : forall e1 e2 e1',
adamc@156 470 e1 ==> e1'
adamc@156 471 -> App e1 e2 ==> App e1' e2
adamc@156 472 | Cong2 : forall e1 e2 e2',
adamc@156 473 val e1
adamc@156 474 -> e2 ==> e2'
adamc@156 475 -> App e1 e2 ==> App e1 e2'
adamc@156 476
adamc@156 477 where "e1 ==> e2" := (step e1 e2).
adamc@156 478
adamc@156 479 Hint Constructors step.
adamc@156 480
adamc@156 481 Lemma progress' : forall G e t, G |-e e : t
adamc@156 482 -> G = nil
adamc@156 483 -> val e \/ exists e', e ==> e'.
adamc@156 484 induction 1; crush; eauto;
adamc@156 485 try match goal with
adamc@156 486 | [ H : _ |-e _ : _ --> _ |- _ ] => inversion H
adamc@156 487 end;
adamc@156 488 repeat match goal with
adamc@156 489 | [ H : _ |- _ ] => solve [ inversion H; crush; eauto ]
adamc@156 490 end.
adamc@156 491 Qed.
adamc@156 492
adamc@156 493 Theorem progress : forall e t, nil |-e e : t
adamc@156 494 -> val e \/ exists e', e ==> e'.
adamc@156 495 intros; eapply progress'; eauto.
adamc@156 496 Qed.
adamc@156 497
adamc@156 498 Lemma preservation' : forall G e t, G |-e e : t
adamc@156 499 -> G = nil
adamc@156 500 -> forall e', e ==> e'
adamc@156 501 -> nil |-e e' : t.
adamc@156 502 induction 1; inversion 2; crush; eauto;
adamc@156 503 match goal with
adamc@156 504 | [ H : _ |-e Abs _ : _ |- _ ] => inversion H
adamc@156 505 end; eauto.
adamc@156 506 Qed.
adamc@156 507
adamc@156 508 Theorem preservation : forall e t, nil |-e e : t
adamc@156 509 -> forall e', e ==> e'
adamc@156 510 -> nil |-e e' : t.
adamc@156 511 intros; eapply preservation'; eauto.
adamc@156 512 Qed.
adamc@156 513
adamc@156 514 End DeBruijn.