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author Adam Chlipala <adamc@hcoop.net>
date Fri, 05 Sep 2008 16:48:13 -0400
parents 91e247c68ee8
children 26ad686e68f2
rev   line source
adamc@2 1 (* Copyright (c) 2008, Adam Chlipala
adamc@2 2 *
adamc@2 3 * This work is licensed under a
adamc@2 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@2 5 * Unported License.
adamc@2 6 * The license text is available at:
adamc@2 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@2 8 *)
adamc@2 9
adamc@3 10 (* begin hide *)
adamc@2 11 Require Import List.
adamc@2 12
adamc@2 13 Require Import Tactics.
adamc@14 14
adamc@14 15 Set Implicit Arguments.
adamc@3 16 (* end hide *)
adamc@2 17
adamc@2 18
adamc@9 19 (** I will start off by jumping right in to a fully-worked set of examples, building certified compilers from increasingly complicated source languages to stack machines. We will meet a few useful tactics and see how they can be used in manual proofs, and we will also see how easily these proofs can be automated instead. I assume that you have installed Coq and Proof General.
adamc@9 20
adamc@11 21 As always, you can step through the source file %\texttt{%#<tt>#StackMachine.v#</tt>#%}% for this chapter interactively in Proof General. Alternatively, to get a feel for the whole lifecycle of creating a Coq development, you can enter the pieces of source code in this chapter in a new %\texttt{%#<tt>#.v#</tt>#%}% file in an Emacs buffer. If you do the latter, include a line [Require Import List Tactics] at the start of the file, to match some code hidden from the chapter source, and be sure to run the Coq binary %\texttt{%#<tt>#coqtop#</tt>#%}% with the command-line argument %\texttt{%#<tt>#-I SRC#</tt>#%}%, where %\texttt{%#<tt>#SRC#</tt>#%}% is the path to a directory containing the source for this book. In either case, if you have installed Proof General properly, it should start automatically when you visit a %\texttt{%#<tt>#.v#</tt>#%}% buffer in Emacs.
adamc@11 22
adamc@11 23 With Proof General, the portion of a buffer that Coq has processed is highlighted in some way, like being given a blue background. You step through Coq source files by positioning the point at the position you want Coq to run to and pressing C-C C-RET. This can be used both for normal step-by-step coding, by placing the point inside some command past the end of the highlighted region; and for undoing, by placing the point inside the highlighted region. *)
adamc@9 24
adamc@9 25
adamc@20 26 (** * Arithmetic Expressions Over Natural Numbers *)
adamc@2 27
adamc@9 28 (** We will begin with that staple of compiler textbooks, arithemtic expressions over a single type of numbers. *)
adamc@9 29
adamc@20 30 (** ** Source Language *)
adamc@9 31
adamc@9 32 (** We begin with the syntax of the source language. *)
adamc@2 33
adamc@4 34 Inductive binop : Set := Plus | Times.
adamc@2 35
adamc@9 36 (** Our first line of Coq code should be unsurprising to ML and Haskell programmers. We define an algebraic datatype [binop] to stand for the binary operators of our source language. There are just two wrinkles compared to ML and Haskell. First, we use the keyword [Inductive], in place of %\texttt{%#<tt>#data#</tt>#%}%, %\texttt{%#<tt>#datatype#</tt>#%}%, or %\texttt{%#<tt>#type#</tt>#%}%. This is not just a trivial surface syntax difference; inductive types in Coq are much more expressive than garden variety algebraic datatypes, essentially enabling us to encode all of mathematics, though we begin humbly in this chapter. Second, there is the [: Set] fragment, which declares that we are defining a datatype that should be thought of as a constituent of programs. Later, we will see other options for defining datatypes in the universe of proofs or in an infinite hierarchy of universes, encompassing both programs and proofs, that is useful in higher-order constructions. *)
adamc@9 37
adamc@4 38 Inductive exp : Set :=
adamc@4 39 | Const : nat -> exp
adamc@4 40 | Binop : binop -> exp -> exp -> exp.
adamc@2 41
adamc@9 42 (** Now we define the type of arithmetic expressions. We write that a constant may be built from one argument, a natural number; and a binary operation may be built from a choice of operator and two operand expressions.
adamc@9 43
adamc@9 44 A note for readers following along in the PDF version: coqdoc supports pretty-printing of tokens in LaTeX or HTML. Where you see a right arrow character, the source contains the ASCII text %\texttt{%#<tt>#->#</tt>#%}%. Other examples of this substitution appearing in this chapter are a double right arrow for %\texttt{%#<tt>#=>#</tt>#%}% and the inverted 'A' symbol for %\texttt{%#<tt>#forall#</tt>#%}%. When in doubt about the ASCII version of a symbol, you can consult the chapter source code.
adamc@9 45
adamc@9 46 %\medskip%
adamc@9 47
adamc@9 48 Now we are ready to say what these programs mean. We will do this by writing an interpreter that can be thought of as a trivial operational or denotational semantics. (If you are not familiar with these semantic techniques, no need to worry; we will stick to "common sense" constructions.) *)
adamc@9 49
adamc@4 50 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 51 match b with
adamc@4 52 | Plus => plus
adamc@4 53 | Times => mult
adamc@4 54 end.
adamc@2 55
adamc@9 56 (** The meaning of a binary operator is a binary function over naturals, defined with pattern-matching notation analogous to the %\texttt{%#<tt>#case#</tt>#%}% and %\texttt{%#<tt>#match#</tt>#%}% of ML and Haskell, and referring to the functions [plus] and [mult] from the Coq standard library. The keyword [Definition] is Coq's all-purpose notation for binding a term of the programming language to a name, with some associated syntactic sugar, like the notation we see here for defining a function. That sugar could be expanded to yield this definition:
adamc@9 57
adamc@9 58 [[
adamc@9 59 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
adamc@9 60 match b with
adamc@9 61 | Plus => plus
adamc@9 62 | Times => mult
adamc@9 63 end.
adamc@9 64
adamc@9 65 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 66
adamc@9 67 [[
adamc@9 68 Definition binopDenote := fun b =>
adamc@9 69 match b with
adamc@9 70 | Plus => plus
adamc@9 71 | Times => mult
adamc@9 72 end.
adamc@9 73
adamc@9 74 Languages like Haskell and ML have a convenient %\textit{%#<i>#principal typing#</i>#%}% property, which gives us strong guarantees about how effective type inference will be. Unfortunately, Coq's type system is so expressive that any kind of "complete" type inference is impossible, and the task even seems to be hard heuristically in practice. Nonetheless, Coq includes some very helpful heuristics, many of them copying the workings of Haskell and ML type-checkers for programs that fall in simple fragments of Coq's language.
adamc@9 75
adamc@9 76 This is as good a time as any to mention the preponderance of different languages associated with Coq. The theoretical foundation of Coq is a formal system called the %\textit{%#<i>#Calculus of Inductive Constructions (CIC)#</i>#%}%, which is an extension of the older %\textit{%#<i>#Calculus of Constructions (CoC)#</i>#%}%. CIC is quite a spartan foundation, which is helpful for proving metatheory but not so helpful for real development. Still, it is nice to know that it has been proved that CIC enjoys properties like %\textit{%#<i>#strong normalization#</i>#%}%, meaning that every program (and, more importantly, every proof term) terminates; and %\textit{%#<i>#relative consistency#</i>#%}% with systems like versions of Zermelo Fraenkel set theory, which roughly means that you can believe that Coq proofs mean that the corresponding propositions are "really true," if you believe in set theory.
adamc@9 77
adamc@9 78 Coq is actually based on an extension of CIC called %\textit{%#<i>#Gallina#</i>#%}%. The text after the [:=] and before the period in the last code example is a term of Gallina. Gallina adds many useful features that are not compiled internalluy to more primitive CIC features. The important metatheorems about CIC have not been extended to the full breadth of these features, but most Coq users do not seem to lose much sleep over this omission.
adamc@9 79
adamc@9 80 Commands like [Inductive] and [Definition] are part of %\textit{%#<i>#the vernacular#</i>#%}%, which includes all sorts of useful queries and requests to the Coq system.
adamc@9 81
adamc@9 82 Finally, there is %\textit{%#<i>#Ltac#</i>#%}%, Coq's domain-specific language for writing proofs and decision procedures. We will see some basic examples of Ltac later in this chapter, and much of this book is devoted to more involved Ltac examples.
adamc@9 83
adamc@9 84 %\medskip%
adamc@9 85
adamc@9 86 We can give a simple definition of the meaning of an expression: *)
adamc@9 87
adamc@4 88 Fixpoint expDenote (e : exp) : nat :=
adamc@4 89 match e with
adamc@4 90 | Const n => n
adamc@4 91 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 92 end.
adamc@2 93
adamc@9 94 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 95
adamc@16 96 (** It is convenient to be able to test definitions before starting to prove things about them. We can verify that our semantics is sensible by evaluating some sample uses. *)
adamc@16 97
adamc@16 98 Eval simpl in expDenote (Const 42).
adamc@18 99 (** [[
adamc@18 100 = 42 : nat
adamc@18 101 ]] *)
adamc@16 102 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
adamc@18 103 (** [[
adamc@18 104 = 4 : nat
adamc@18 105 ]] *)
adamc@16 106 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@18 107 (** [[
adamc@18 108 = 28 : nat
adamc@18 109 ]] *)
adamc@9 110
adamc@20 111 (** ** Target Language *)
adamc@4 112
adamc@10 113 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 114
adamc@4 115 Inductive instr : Set :=
adamc@4 116 | IConst : nat -> instr
adamc@4 117 | IBinop : binop -> instr.
adamc@2 118
adamc@4 119 Definition prog := list instr.
adamc@4 120 Definition stack := list nat.
adamc@2 121
adamc@10 122 (** An instruction either pushes a constant onto the stack or pops two arguments, applies a binary operator to them, and pushes the result onto the stack. A program is a list of instructions, and a stack is a list of natural numbers.
adamc@10 123
adamc@10 124 We can give instructions meanings as functions from stacks to optional stacks, where running an instruction results in [None] in case of a stack underflow and results in [Some s'] when the result of execution is the new stack [s']. [::] is the "list cons" operator from the Coq standard library. *)
adamc@10 125
adamc@4 126 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 127 match i with
adamc@4 128 | IConst n => Some (n :: s)
adamc@4 129 | IBinop b =>
adamc@4 130 match s with
adamc@4 131 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 132 | _ => None
adamc@4 133 end
adamc@4 134 end.
adamc@2 135
adamc@10 136 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@10 137
adamc@4 138 Fixpoint progDenote (p : prog) (s : stack) {struct p} : option stack :=
adamc@4 139 match p with
adamc@4 140 | nil => Some s
adamc@4 141 | i :: p' =>
adamc@4 142 match instrDenote i s with
adamc@4 143 | None => None
adamc@4 144 | Some s' => progDenote p' s'
adamc@4 145 end
adamc@4 146 end.
adamc@2 147
adamc@10 148 (** There is one interesting difference compared to our previous example of a [Fixpoint]. This recursive function takes two arguments, [p] and [s]. It is critical for the soundness of Coq that every program terminate, so a shallow syntactic termination check is imposed on every recursive function definition. One of the function parameters must be designated to decrease monotonically across recursive calls. That is, every recursive call must use a version of that argument that has been pulled out of the current argument by some number of [match] expressions. [expDenote] has only one argument, so we did not need to specify which of its arguments decreases. For [progDenote], we resolve the ambiguity by writing [{struct p}] to indicate that argument [p] decreases structurally. *)
adamc@2 149
adamc@4 150
adamc@9 151 (** ** Translation *)
adamc@4 152
adamc@10 153 (** Our compiler itself is now unsurprising. [++] is the list concatenation operator from the Coq standard library. *)
adamc@2 154
adamc@4 155 Fixpoint compile (e : exp) : prog :=
adamc@4 156 match e with
adamc@4 157 | Const n => IConst n :: nil
adamc@4 158 | Binop b e1 e2 => compile e2 ++ compile e1 ++ IBinop b :: nil
adamc@4 159 end.
adamc@2 160
adamc@2 161
adamc@16 162 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 163
adamc@16 164 Eval simpl in compile (Const 42).
adamc@18 165 (** [[
adamc@18 166 = IConst 42 :: nil : prog
adamc@18 167 ]] *)
adamc@16 168 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adamc@18 169 (** [[
adamc@18 170 = IConst 2 :: IConst 2 :: IBinop Plus :: nil : prog
adamc@18 171 ]] *)
adamc@16 172 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@18 173 (** [[
adamc@18 174 = IConst 7 :: IConst 2 :: IConst 2 :: IBinop Plus :: IBinop Times :: nil : prog
adamc@18 175 ]] *)
adamc@16 176
adamc@16 177 (** We can also run our compiled programs and chedk that they give the right results. *)
adamc@16 178
adamc@16 179 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@18 180 (** [[
adamc@18 181 = Some (42 :: nil) : option stack
adamc@18 182 ]] *)
adamc@16 183 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@18 184 (** [[
adamc@18 185 = Some (4 :: nil) : option stack
adamc@18 186 ]] *)
adamc@16 187 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7))) nil.
adamc@18 188 (** [[
adamc@18 189 = Some (28 :: nil) : option stack
adamc@18 190 ]] *)
adamc@16 191
adamc@16 192
adamc@20 193 (** ** Translation Correctness *)
adamc@4 194
adamc@11 195 (** We are ready to prove that our compiler is implemented correctly. We can use a new vernacular command [Theorem] to start a correctness proof, in terms of the semantics we defined earlier: *)
adamc@11 196
adamc@11 197 Theorem compileCorrect : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 198 (* begin hide *)
adamc@11 199 Abort.
adamc@11 200 (* end hide *)
adamc@22 201 (* begin thide *)
adamc@11 202
adamc@11 203 (** Though a pencil-and-paper proof might clock out at this point, writing "by a routine induction on [e]," it turns out not to make sense to attack this proof directly. We need to use the standard trick of %\textit{%#<i>#strengthening the induction hypothesis#</i>#%}%. We do that by proving an auxiliary lemma:
adamc@11 204 *)
adamc@2 205
adamc@15 206 Lemma compileCorrect' : forall e p s, progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 207
adamc@11 208 (** After the period in the [Lemma] command, we are in %\textit{%#<i>#the interactive proof-editing mode#</i>#%}%. We find ourselves staring at this ominous screen of text:
adamc@11 209
adamc@11 210 [[
adamc@11 211 1 subgoal
adamc@11 212
adamc@11 213 ============================
adamc@15 214 forall (e : exp) (p : list instr) (s : stack),
adamc@15 215 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@11 216 ]]
adamc@11 217
adamc@11 218 Coq seems to be restating the lemma for us. What we are seeing is a limited case of a more general protocol for describing where we are in a proof. We are told that we have a single subgoal. In general, during a proof, we can have many pending subgoals, each of which is a logical proposition to prove. Subgoals can be proved in any order, but it usually works best to prove them in the order that Coq chooses.
adamc@11 219
adamc@11 220 Next in the output, we see our single subgoal described in full detail. There is a double-dashed line, above which would be our free variables and hypotheses, if we had any. Below the line is the conclusion, which, in general, is to be proved from the hypotheses.
adamc@11 221
adamc@11 222 We manipulate the proof state by running commands called %\textit{%#<i>#tactics#</i>#%}%. Let us start out by running one of the most important tactics:
adamc@11 223 *)
adamc@11 224
adamc@4 225 induction e.
adamc@2 226
adamc@11 227 (** We declare that this proof will proceed by induction on the structure of the expression [e]. This swaps out our initial subgoal for two new subgoals, one for each case of the inductive proof:
adamc@11 228
adamc@11 229 [[
adamc@11 230 2 subgoals
adamc@11 231
adamc@11 232 n : nat
adamc@11 233 ============================
adamc@11 234 forall (s : stack) (p : list instr),
adamc@11 235 progDenote (compile (Const n) ++ p) s =
adamc@11 236 progDenote p (expDenote (Const n) :: s)
adamc@11 237 ]]
adamc@11 238 [[
adamc@11 239 subgoal 2 is:
adamc@11 240 forall (s : stack) (p : list instr),
adamc@11 241 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 242 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@11 243 ]]
adamc@11 244
adamc@11 245 The first and current subgoal is displayed with the double-dashed line below free variables and hypotheses, while later subgoals are only summarized with their conclusions. We see an example of a free variable in the first subgoal; [n] is a free variable of type [nat]. The conclusion is the original theorem statement where [e] has been replaced by [Const n]. In a similar manner, the second case has [e] replaced by a generalized invocation of the [Binop] expression constructor. We can see that proving both cases corresponds to a standard proof by structural induction.
adamc@11 246
adamc@11 247 We begin the first case with another very common tactic.
adamc@11 248 *)
adamc@11 249
adamc@4 250 intros.
adamc@11 251
adamc@11 252 (** The current subgoal changes to:
adamc@11 253 [[
adamc@11 254
adamc@11 255 n : nat
adamc@11 256 s : stack
adamc@11 257 p : list instr
adamc@11 258 ============================
adamc@11 259 progDenote (compile (Const n) ++ p) s =
adamc@11 260 progDenote p (expDenote (Const n) :: s)
adamc@11 261 ]]
adamc@11 262
adamc@11 263 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 264
adamc@11 265 To progress further, we need to use the definitions of some of the functions appearing in the goal. The [unfold] tactic replaces an identifier with its definition.
adamc@11 266 *)
adamc@11 267
adamc@4 268 unfold compile.
adamc@11 269
adamc@11 270 (** [[
adamc@11 271
adamc@11 272 n : nat
adamc@11 273 s : stack
adamc@11 274 p : list instr
adamc@11 275 ============================
adamc@11 276 progDenote ((IConst n :: nil) ++ p) s =
adamc@11 277 progDenote p (expDenote (Const n) :: s)
adamc@11 278 ]] *)
adamc@11 279
adamc@4 280 unfold expDenote.
adamc@11 281
adamc@11 282 (** [[
adamc@11 283
adamc@11 284 n : nat
adamc@11 285 s : stack
adamc@11 286 p : list instr
adamc@11 287 ============================
adamc@11 288 progDenote ((IConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@11 289 ]]
adamc@11 290
adamc@11 291 We only need to unfold the first occurrence of [progDenote] to prove the goal: *)
adamc@11 292
adamc@11 293 unfold progDenote at 1.
adamc@11 294
adamc@11 295 (** [[
adamc@11 296
adamc@11 297 n : nat
adamc@11 298 s : stack
adamc@11 299 p : list instr
adamc@11 300 ============================
adamc@11 301 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 302 option stack :=
adamc@11 303 match p0 with
adamc@11 304 | nil => Some s0
adamc@11 305 | i :: p' =>
adamc@11 306 match instrDenote i s0 with
adamc@11 307 | Some s' => progDenote p' s'
adamc@11 308 | None => None (A:=stack)
adamc@11 309 end
adamc@11 310 end) ((IConst n :: nil) ++ p) s =
adamc@11 311 progDenote p (n :: s)
adamc@11 312 ]]
adamc@11 313
adamc@11 314 This last [unfold] has left us with an anonymous fixpoint version of [progDenote], which will generally happen when unfolding recursive definitions. Fortunately, in this case, we can eliminate such complications right away, since the structure of the argument [(IConst n :: nil) ++ p] is known, allowing us to simplify the internal pattern match with the [simpl] tactic:
adamc@11 315 *)
adamc@11 316
adamc@4 317 simpl.
adamc@11 318
adamc@11 319 (** [[
adamc@11 320
adamc@11 321 n : nat
adamc@11 322 s : stack
adamc@11 323 p : list instr
adamc@11 324 ============================
adamc@11 325 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 326 option stack :=
adamc@11 327 match p0 with
adamc@11 328 | nil => Some s0
adamc@11 329 | i :: p' =>
adamc@11 330 match instrDenote i s0 with
adamc@11 331 | Some s' => progDenote p' s'
adamc@11 332 | None => None (A:=stack)
adamc@11 333 end
adamc@11 334 end) p (n :: s) = progDenote p (n :: s)
adamc@11 335 ]]
adamc@11 336
adamc@11 337 Now we can unexpand the definition of [progDenote]:
adamc@11 338 *)
adamc@11 339
adamc@11 340 fold progDenote.
adamc@11 341
adamc@11 342 (** [[
adamc@11 343
adamc@11 344 n : nat
adamc@11 345 s : stack
adamc@11 346 p : list instr
adamc@11 347 ============================
adamc@11 348 progDenote p (n :: s) = progDenote p (n :: s)
adamc@11 349 ]]
adamc@11 350
adamc@11 351 It looks like we are at the end of this case, since we have a trivial equality. Indeed, a single tactic finishes us off:
adamc@11 352 *)
adamc@11 353
adamc@4 354 reflexivity.
adamc@2 355
adamc@11 356 (** On to the second inductive case:
adamc@11 357
adamc@11 358 [[
adamc@11 359
adamc@11 360 b : binop
adamc@11 361 e1 : exp
adamc@11 362 IHe1 : forall (s : stack) (p : list instr),
adamc@11 363 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 364 e2 : exp
adamc@11 365 IHe2 : forall (s : stack) (p : list instr),
adamc@11 366 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 367 ============================
adamc@11 368 forall (s : stack) (p : list instr),
adamc@11 369 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 370 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@11 371 ]]
adamc@11 372
adamc@11 373 We see our first example of hypotheses above the double-dashed line. They are the inductive hypotheses [IHe1] and [IHe2] corresponding to the subterms [e1] and [e2], respectively.
adamc@11 374
adamc@11 375 We start out the same way as before, introducing new free variables and unfolding and folding the appropriate definitions. The seemingly frivolous [unfold]/[fold] pairs are actually accomplishing useful work, because [unfold] will sometimes perform easy simplifications. *)
adamc@11 376
adamc@4 377 intros.
adamc@4 378 unfold compile.
adamc@4 379 fold compile.
adamc@4 380 unfold expDenote.
adamc@4 381 fold expDenote.
adamc@11 382
adamc@11 383 (** Now we arrive at a point where the tactics we have seen so far are insufficient:
adamc@11 384
adamc@11 385 [[
adamc@11 386
adamc@11 387 b : binop
adamc@11 388 e1 : exp
adamc@11 389 IHe1 : forall (s : stack) (p : list instr),
adamc@11 390 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 391 e2 : exp
adamc@11 392 IHe2 : forall (s : stack) (p : list instr),
adamc@11 393 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 394 s : stack
adamc@11 395 p : list instr
adamc@11 396 ============================
adamc@11 397 progDenote ((compile e2 ++ compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 398 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 399 ]]
adamc@11 400
adamc@11 401 What we need is the associative law of list concatenation, available as a theorem [app_ass] in the standard library. *)
adamc@11 402
adamc@11 403 Check app_ass.
adamc@11 404
adamc@11 405 (** [[
adamc@11 406
adamc@11 407 app_ass
adamc@11 408 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@11 409 ]]
adamc@11 410
adamc@11 411 We use it to perform a rewrite: *)
adamc@11 412
adamc@4 413 rewrite app_ass.
adamc@11 414
adamc@11 415 (** changing the conclusion to: [[
adamc@11 416
adamc@11 417 progDenote (compile e2 ++ (compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 418 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 419 ]]
adamc@11 420
adamc@11 421 Now we can notice that the lefthand side of the equality matches the lefthand side of the second inductive hypothesis, so we can rewrite with that hypothesis, too: *)
adamc@11 422
adamc@4 423 rewrite IHe2.
adamc@11 424
adamc@11 425 (** [[
adamc@11 426
adamc@11 427 progDenote ((compile e1 ++ IBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 428 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 429 ]]
adamc@11 430
adamc@11 431 The same process lets us apply the remaining hypothesis. *)
adamc@11 432
adamc@4 433 rewrite app_ass.
adamc@4 434 rewrite IHe1.
adamc@11 435
adamc@11 436 (** [[
adamc@11 437
adamc@11 438 progDenote ((IBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 439 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 440 ]]
adamc@11 441
adamc@11 442 Now we can apply a similar sequence of tactics to that that ended the proof of the first case.
adamc@11 443 *)
adamc@11 444
adamc@11 445 unfold progDenote at 1.
adamc@4 446 simpl.
adamc@11 447 fold progDenote.
adamc@4 448 reflexivity.
adamc@11 449
adamc@11 450 (** And the proof is completed, as indicated by the message:
adamc@11 451
adamc@11 452 [[
adamc@11 453 Proof completed.
adamc@11 454
adamc@11 455 And there lies our first proof. Already, even for simple theorems like this, the final proof script is unstructured and not very enlightening to readers. If we extend this approach to more serious theorems, we arrive at the unreadable proof scripts that are the favorite complaints of opponents of tactic-based proving. Fortunately, Coq has rich support for scripted automation, and we can take advantage of such a scripted tactic (defined elsewhere) to make short work of this lemma. We abort the old proof attempt and start again.
adamc@11 456 *)
adamc@11 457
adamc@4 458 Abort.
adamc@2 459
adamc@4 460 Lemma compileCorrect' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 461 progDenote p (expDenote e :: s).
adamc@4 462 induction e; crush.
adamc@4 463 Qed.
adamc@2 464
adamc@11 465 (** We need only to state the basic inductive proof scheme and call a tactic that automates the tedious reasoning in between. In contrast to the period tactic terminator from our last proof, the semicolon tactic separator supports structured, compositional proofs. The tactic [t1; t2] has the effect of running [t1] and then running [t2] on each remaining subgoal. The semicolon is one of the most fundamental building blocks of effective proof automation. The period terminator is very useful for exploratory proving, where you need to see intermediate proof states, but final proofs of any serious complexity should have just one period, terminating a single compound tactic that probably uses semicolons.
adamc@11 466
adamc@11 467 The proof of our main theorem is now easy. We prove it with four period-terminated tactics, though separating them with semicolons would work as well; the version here is easier to step through. *)
adamc@11 468
adamc@4 469 Theorem compileCorrect : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 470 intros.
adamc@11 471
adamc@11 472 (** [[
adamc@11 473
adamc@11 474 e : exp
adamc@11 475 ============================
adamc@11 476 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@11 477 ]]
adamc@11 478
adamc@11 479 At this point, we want to massage the lefthand side to match the statement of [compileCorrect']. A theorem from the standard library is useful: *)
adamc@11 480
adamc@11 481 Check app_nil_end.
adamc@11 482
adamc@11 483 (** [[
adamc@11 484
adamc@11 485 app_nil_end
adamc@11 486 : forall (A : Type) (l : list A), l = l ++ nil
adamc@11 487 ]] *)
adamc@11 488
adamc@4 489 rewrite (app_nil_end (compile e)).
adamc@11 490
adamc@11 491 (** This time, we explicitly specify the value of the variable [l] from the theorem statement, since multiple expressions of list type appear in the conclusion. [rewrite] might choose the wrong place to rewrite if we did not specify which we want.
adamc@11 492
adamc@11 493 [[
adamc@11 494
adamc@11 495 e : exp
adamc@11 496 ============================
adamc@11 497 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@11 498 ]]
adamc@11 499
adamc@11 500 Now we can apply the lemma. *)
adamc@11 501
adamc@4 502 rewrite compileCorrect'.
adamc@11 503
adamc@11 504 (** [[
adamc@11 505
adamc@11 506 e : exp
adamc@11 507 ============================
adamc@11 508 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@11 509 ]]
adamc@11 510
adamc@11 511 We are almost done. The lefthand and righthand sides can be seen to match by simple symbolic evaluation. That means we are in luck, because Coq identifies any pair of terms as equal whenever they normalize to the same result by symbolic evaluation. By the definition of [progDenote], that is the case here, but we do not need to worry about such details. A simple invocation of [reflexivity] does the normalization and checks that the two results are syntactically equal. *)
adamc@11 512
adamc@4 513 reflexivity.
adamc@4 514 Qed.
adamc@22 515 (* end thide *)
adamc@14 516
adamc@14 517
adamc@20 518 (** * Typed Expressions *)
adamc@14 519
adamc@14 520 (** In this section, we will build on the initial example by adding additional expression forms that depend on static typing of terms for safety. *)
adamc@14 521
adamc@20 522 (** ** Source Language *)
adamc@14 523
adamc@15 524 (** We define a trivial language of types to classify our expressions: *)
adamc@15 525
adamc@14 526 Inductive type : Set := Nat | Bool.
adamc@14 527
adamc@15 528 (** Now we define an expanded set of binary operators. *)
adamc@15 529
adamc@14 530 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 531 | TPlus : tbinop Nat Nat Nat
adamc@14 532 | TTimes : tbinop Nat Nat Nat
adamc@14 533 | TEq : forall t, tbinop t t Bool
adamc@14 534 | TLt : tbinop Nat Nat Bool.
adamc@14 535
adamc@15 536 (** The definition of [tbinop] is different from [binop] in an important way. Where we declared that [binop] has type [Set], here we declare that [tbinop] has type [type -> type -> type -> Set]. We define [tbinop] as an %\textit{%#<i>#indexed type family#</i>#%}%. Indexed inductive types are at the heart of Coq's expressive power; almost everything else of interest is defined in terms of them.
adamc@15 537
adamc@15 538 ML and Haskell have indexed algebraic datatypes. For instance, their list types are indexed by the type of data that the list carries. However, compared to Coq, ML and Haskell 98 place two important restrictions on datatype definitions.
adamc@15 539
adamc@15 540 First, the indices of the range of each data constructor must be type variables bound at the top level of the datatype definition. There is no way to do what we did here, where we, for instance, say that [TPlus] is a constructor building a [tbinop] whose indices are all fixed at [Nat]. %\textit{%#<i>#Generalized algebraic datatypes (GADTs)#</i>#%}% are a popular feature in GHC Haskell and other languages that removes this first restriction.
adamc@15 541
adamc@15 542 The second restriction is not lifted by GADTs. In ML and Haskell, indices of types must be types and may not be %\textit{%#<i>#expressions#</i>#%}%. In Coq, types may be indiced by arbitrary Gallina terms. Type indices can live in the same universe as programs, and we can compute with them just like regular programs. Haskell supports a hobbled form of computation in type indices based on multi-parameter type classes, and recent extensions like type functions bring Haskell programming even closer to "real" functional programming with types, but, without dependent typing, there must always be a gap between how one programs with types and how one programs normally.
adamc@15 543 *)
adamc@15 544
adamc@15 545 (** We can define a similar type family for typed expressions. *)
adamc@15 546
adamc@14 547 Inductive texp : type -> Set :=
adamc@14 548 | TNConst : nat -> texp Nat
adamc@14 549 | TBConst : bool -> texp Bool
adamc@14 550 | TBinop : forall arg1 arg2 res, tbinop arg1 arg2 res -> texp arg1 -> texp arg2 -> texp res.
adamc@14 551
adamc@15 552 (** Thanks to our use of dependent types, every well-typed [texp] represents a well-typed source expression, by construction. This turns out to be very convenient for many things we might want to do with expressions. For instance, it is easy to adapt our interpreter approach to defining semantics. We start by defining a function mapping the types of our languages into Coq types: *)
adamc@15 553
adamc@14 554 Definition typeDenote (t : type) : Set :=
adamc@14 555 match t with
adamc@14 556 | Nat => nat
adamc@14 557 | Bool => bool
adamc@14 558 end.
adamc@14 559
adamc@15 560 (** It can take a few moments to come to terms with the fact that [Set], the type of types of programs, is itself a first-class type, and that we can write functions that return [Set]s. Past that wrinkle, the definition of [typeDenote] is trivial, relying on the [nat] and [bool] types from the Coq standard library.
adamc@15 561
adamc@15 562 We need to define a few auxiliary functions, implementing our boolean binary operators that do not appear with the right types in the standard library. They are entirely standard and ML-like, with the one caveat being that the Coq [nat] type uses a unary representation, where [O] is zero and [S n] is the successor of [n].
adamc@15 563 *)
adamc@15 564
adamc@14 565 Definition eq_bool (b1 b2 : bool) : bool :=
adamc@14 566 match b1, b2 with
adamc@14 567 | true, true => true
adamc@14 568 | false, false => true
adamc@14 569 | _, _ => false
adamc@14 570 end.
adamc@14 571
adamc@14 572 Fixpoint eq_nat (n1 n2 : nat) {struct n1} : bool :=
adamc@14 573 match n1, n2 with
adamc@14 574 | O, O => true
adamc@14 575 | S n1', S n2' => eq_nat n1' n2'
adamc@14 576 | _, _ => false
adamc@14 577 end.
adamc@14 578
adamc@14 579 Fixpoint lt (n1 n2 : nat) {struct n1} : bool :=
adamc@14 580 match n1, n2 with
adamc@14 581 | O, S _ => true
adamc@14 582 | S n1', S n2' => lt n1' n2'
adamc@14 583 | _, _ => false
adamc@14 584 end.
adamc@14 585
adamc@15 586 (** Now we can interpret binary operators: *)
adamc@15 587
adamc@14 588 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@14 589 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@14 590 match b in (tbinop arg1 arg2 res) return (typeDenote arg1 -> typeDenote arg2 -> typeDenote res) with
adamc@14 591 | TPlus => plus
adamc@14 592 | TTimes => mult
adamc@14 593 | TEq Nat => eq_nat
adamc@14 594 | TEq Bool => eq_bool
adamc@14 595 | TLt => lt
adamc@14 596 end.
adamc@14 597
adamc@15 598 (** This function has just a few differences from the denotation functions we saw earlier. First, [tbinop] is an indexed type, so its indices become additional arguments to [tbinopDenote]. Second, we need to perform a genuine %\textit{%#<i>#dependent pattern match#</i>#%}% to come up with a definition of this function that type-checks. In each branch of the [match], we need to use branch-specific information about the indices to [tbinop]. General type inference that takes such information into account is undecidable, and Coq avoids pursuing special heuristics for the problem, instead asking users to write annotations, like we see above on the line with [match].
adamc@15 599
adamc@15 600 The [in] annotation restates the type of the term being case-analyzed. Though we use the same names for the indices as we use in the type of the original argument binder, these are actually fresh variables, and they are %\textit{%#<i>#binding occcurrences#</i>#%}%. Their scope is the [return] clause. That is, [arg1], [arg2], and [arg3] are new bound variables bound only within the return clause [typeDenote arg1 -> typeDenote arg2 -> typeDenote res]. By being explicit about the functional relationship between the type indices and the match result, we regain decidable type inference.
adamc@15 601
adamc@15 602 The same tricks suffice to define an expression denotation function in an unsurprising way:
adamc@15 603 *)
adamc@15 604
adamc@14 605 Fixpoint texpDenote t (e : texp t) {struct e} : typeDenote t :=
adamc@14 606 match e in (texp t) return (typeDenote t) with
adamc@14 607 | TNConst n => n
adamc@14 608 | TBConst b => b
adamc@14 609 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 610 end.
adamc@14 611
adamc@17 612 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 613
adamc@17 614 Eval simpl in texpDenote (TNConst 42).
adamc@18 615 (** [[
adamc@18 616 = 42 : typeDenote Nat
adamc@18 617 ]] *)
adamc@17 618 Eval simpl in texpDenote (TBConst true).
adamc@18 619 (** [[
adamc@18 620 = true : typeDenote Bool
adamc@18 621 ]] *)
adamc@17 622 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@18 623 (** [[
adamc@18 624 = 28 : typeDenote Nat
adamc@18 625 ]] *)
adamc@17 626 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@18 627 (** [[
adamc@18 628 = false : typeDenote Bool
adamc@18 629 ]] *)
adamc@17 630 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@18 631 (** [[
adamc@18 632 = true : typeDenote Bool
adamc@18 633 ]] *)
adamc@17 634
adamc@14 635
adamc@20 636 (** ** Target Language *)
adamc@14 637
adamc@18 638 (** Now we want to define a suitable stack machine target for compilation. In the example of the untyped language, stack machine programs could encounter stack underflows and "get stuck." This was unfortunate, since we had to deal with this complication even though we proved that our compiler never produced underflowing programs. We could have used dependent types to force all stack machine programs to be underflow-free.
adamc@18 639
adamc@18 640 For our new languages, besides underflow, we also have the problem of stack slots with naturals instead of bools or vice versa. This time, we will use indexed typed families to avoid the need to reason about potential failures.
adamc@18 641
adamc@18 642 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 643
adamc@14 644 Definition tstack := list type.
adamc@14 645
adamc@18 646 (** Any stack classified by a [tstack] must have exactly as many elements, and each stack element must have the type found in the same position of the stack type.
adamc@18 647
adamc@18 648 We can define instructions in terms of stack types, where every instruction's type tells us what initial stack type it expects and what final stack type it will produce. *)
adamc@18 649
adamc@14 650 Inductive tinstr : tstack -> tstack -> Set :=
adamc@14 651 | TINConst : forall s, nat -> tinstr s (Nat :: s)
adamc@14 652 | TIBConst : forall s, bool -> tinstr s (Bool :: s)
adamc@14 653 | TIBinop : forall arg1 arg2 res s,
adamc@14 654 tbinop arg1 arg2 res
adamc@14 655 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 656
adamc@18 657 (** Stack machine programs must be a similar inductive family, since, if we again used the [list] type family, we would not be able to guarantee that intermediate stack types match within a program. *)
adamc@18 658
adamc@14 659 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 660 | TNil : forall s, tprog s s
adamc@14 661 | TCons : forall s1 s2 s3,
adamc@14 662 tinstr s1 s2
adamc@14 663 -> tprog s2 s3
adamc@14 664 -> tprog s1 s3.
adamc@14 665
adamc@18 666 (** Now, to define the semantics of our new target language, we need a representation for stacks at runtime. We will again take advantage of type information to define types of value stacks that, by construction, contain the right number and types of elements. *)
adamc@18 667
adamc@14 668 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 669 match ts with
adamc@14 670 | nil => unit
adamc@14 671 | t :: ts' => typeDenote t * vstack ts'
adamc@14 672 end%type.
adamc@14 673
adamc@18 674 (** This is another [Set]-valued function. This time it is recursive, which is perfectly valid, since [Set] is not treated specially in determining which functions may be written. We say that the value stack of an empty stack type is any value of type [unit], which has just a single value, [tt]. A nonempty stack type leads to a value stack that is a pair, whose first element has the proper type and whose second element follows the representation for the remainder of the stack type.
adamc@18 675
adamc@18 676 This idea of programming with types can take a while to internalize, but it enables a very simple definition of instruction denotation. We have the same kind of type annotations for the dependent [match], but everything else is like what you might expect from a Lisp-like version of ML that ignored type information. Nonetheless, the fact that [tinstrDenote] passes the type-checker guarantees that our stack machine programs can never go wrong. *)
adamc@18 677
adamc@14 678 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@14 679 match i in (tinstr ts ts') return (vstack ts -> vstack ts') with
adamc@14 680 | TINConst _ n => fun s => (n, s)
adamc@14 681 | TIBConst _ b => fun s => (b, s)
adamc@14 682 | TIBinop _ _ _ _ b => fun s =>
adamc@14 683 match s with
adamc@14 684 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@14 685 end
adamc@14 686 end.
adamc@14 687
adamc@18 688 (** Why do we choose to use an anonymous function to bind the initial stack in every case of the [match]? Consider this well-intentioned but invalid alternative version:
adamc@18 689
adamc@18 690 [[
adamc@18 691 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@18 692 match i in (tinstr ts ts') return (vstack ts') with
adamc@18 693 | TINConst _ n => (n, s)
adamc@18 694 | TIBConst _ b => (b, s)
adamc@18 695 | TIBinop _ _ _ _ b =>
adamc@18 696 match s with
adamc@18 697 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@18 698 end
adamc@18 699 end.
adamc@18 700
adamc@18 701 The Coq type-checker complains that:
adamc@18 702
adamc@18 703 [[
adamc@18 704 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@18 705 while it is expected to have type "vstack (Nat :: t)"
adamc@18 706 ]]
adamc@18 707
adamc@18 708 Recall from our earlier discussion of [match] annotations that we write the annotations to express to the type-checker the relationship between the type indices of the case object and the result type of the [match]. Coq chooses to assign to the wildcard [_] after [TINConst] the name [t], and the type error is telling us that the type checker cannot prove that [t] is the same as [ts]. By moving [s] out of the [match], we lose the ability to express, with [in] and [return] clauses, the relationship between the shared index [ts] of [s] and [i].
adamc@18 709
adamc@18 710 There %\textit{%#<i>#are#</i>#%}% reasonably general ways of getting around this problem without pushing binders inside [match]es. However, the alternatives are significantly more involved, and the technique we use here is almost certainly the best choice, whenever it applies.
adamc@18 711
adamc@18 712 *)
adamc@18 713
adamc@18 714 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 715
adamc@14 716 Fixpoint tprogDenote ts ts' (p : tprog ts ts') {struct p} : vstack ts -> vstack ts' :=
adamc@14 717 match p in (tprog ts ts') return (vstack ts -> vstack ts') with
adamc@14 718 | TNil _ => fun s => s
adamc@14 719 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 720 end.
adamc@14 721
adamc@14 722
adamc@14 723 (** ** Translation *)
adamc@14 724
adamc@19 725 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 726
adamc@14 727 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') {struct p} : tprog ts' ts'' -> tprog ts ts'' :=
adamc@14 728 match p in (tprog ts ts') return (tprog ts' ts'' -> tprog ts ts'') with
adamc@14 729 | TNil _ => fun p' => p'
adamc@14 730 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 731 end.
adamc@14 732
adamc@19 733 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 734
adamc@14 735 Fixpoint tcompile t (e : texp t) (ts : tstack) {struct e} : tprog ts (t :: ts) :=
adamc@14 736 match e in (texp t) return (tprog ts (t :: ts)) with
adamc@14 737 | TNConst n => TCons (TINConst _ n) (TNil _)
adamc@14 738 | TBConst b => TCons (TIBConst _ b) (TNil _)
adamc@14 739 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adamc@14 740 (tconcat (tcompile e1 _) (TCons (TIBinop _ b) (TNil _)))
adamc@14 741 end.
adamc@14 742
adamc@19 743 (** One interesting feature of the definition is the underscores appearing to the right of [=>] arrows. Haskell and ML programmers are quite familiar with compilers that infer type parameters to polymorphic values. In Coq, it is possible to go even further and ask the system to infer arbitary terms, by writing underscores in place of specific values. You may have noticed that we have been calling functions without specifying all of their arguments. For instance, the recursive calls here to [tcompile] omit the [t] argument. Coq's %\textit{%#<i>#implicit argument#</i>#%}% mechanism automatically inserts underscores for arguments that it will probably be able to infer. Inference of such values is far from complete, though; generally, it only works in cases similar to those encountered with polymorphic type instantiation in Haskell and ML.
adamc@19 744
adamc@19 745 The underscores here are being filled in with stack types. That is, the Coq type inferencer is, in a sense, inferring something about the flow of control in the translated programs. We can take a look at exactly which values are filled in: *)
adamc@19 746
adamc@14 747 Print tcompile.
adamc@14 748
adamc@19 749 (** [[
adamc@19 750
adamc@19 751 tcompile =
adamc@19 752 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 753 tprog ts (t :: ts) :=
adamc@19 754 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adamc@19 755 | TNConst n => TCons (TINConst ts n) (TNil (Nat :: ts))
adamc@19 756 | TBConst b => TCons (TIBConst ts b) (TNil (Bool :: ts))
adamc@19 757 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 758 tconcat (tcompile arg2 e2 ts)
adamc@19 759 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adamc@19 760 (TCons (TIBinop ts b) (TNil (res :: ts))))
adamc@19 761 end
adamc@19 762 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adamc@19 763 ]] *)
adamc@19 764
adamc@19 765
adamc@19 766 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 767
adamc@19 768 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adamc@19 769 (** [[
adamc@19 770 = (42, tt) : vstack (Nat :: nil)
adamc@19 771 ]] *)
adamc@19 772 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adamc@19 773 (** [[
adamc@19 774 = (true, tt) : vstack (Bool :: nil)
adamc@19 775 ]] *)
adamc@19 776 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@19 777 (** [[
adamc@19 778 = (28, tt) : vstack (Nat :: nil)
adamc@19 779 ]] *)
adamc@19 780 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@19 781 (** [[
adamc@19 782 = (false, tt) : vstack (Bool :: nil)
adamc@19 783 ]] *)
adamc@19 784 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@19 785 (** [[
adamc@19 786 = (true, tt) : vstack (Bool :: nil)
adamc@19 787 ]] *)
adamc@19 788
adamc@14 789
adamc@20 790 (** ** Translation Correctness *)
adamc@20 791
adamc@20 792 (** We can state a correctness theorem similar to the last one. *)
adamc@20 793
adamc@20 794 Theorem tcompileCorrect : forall t (e : texp t), tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 795 (* begin hide *)
adamc@20 796 Abort.
adamc@20 797 (* end hide *)
adamc@22 798 (* begin thide *)
adamc@20 799
adamc@20 800 (** Again, we need to strengthen the theorem statement so that the induction will go through. This time, I will develop an alternative approach to this kind of proof, stating the key lemma as: *)
adamc@14 801
adamc@14 802 Lemma tcompileCorrect' : forall t (e : texp t)
adamc@14 803 ts (s : vstack ts),
adamc@14 804 tprogDenote (tcompile e ts) s
adamc@14 805 = (texpDenote e, s).
adamc@20 806
adamc@20 807 (** While lemma [compileCorrect'] quantified over a program that is the "continuation" for the expression we are considering, here we avoid drawing in any extra syntactic elements. In addition to the source expression and its type, we also quantify over an initial stack type and a stack compatible with it. Running the compilation of the program starting from that stack, we should arrive at a stack that differs only in having the program's denotation pushed onto it.
adamc@20 808
adamc@20 809 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 810
adamc@14 811 induction e; crush.
adamc@20 812
adamc@20 813 (** We are left with this unproved conclusion:
adamc@20 814
adamc@20 815 [[
adamc@20 816
adamc@20 817 tprogDenote
adamc@20 818 (tconcat (tcompile e2 ts)
adamc@20 819 (tconcat (tcompile e1 (arg2 :: ts))
adamc@20 820 (TCons (TIBinop ts t) (TNil (res :: ts))))) s =
adamc@20 821 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@20 822 ]]
adamc@20 823
adamc@20 824 We need an analogue to the [app_ass] theorem that we used to rewrite the goal in the last section. We can abort this proof and prove such a lemma about [tconcat].
adamc@20 825 *)
adamc@14 826 Abort.
adamc@14 827
adamc@14 828 Lemma tconcatCorrect : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 829 (s : vstack ts),
adamc@14 830 tprogDenote (tconcat p p') s
adamc@14 831 = tprogDenote p' (tprogDenote p s).
adamc@14 832 induction p; crush.
adamc@14 833 Qed.
adamc@14 834
adamc@20 835 (** This one goes through completely automatically.
adamc@20 836
adamc@20 837 Some code behind the scenes registers [app_ass] for use by [crush]. We must register [tconcatCorrect] similarly to get the same effect: *)
adamc@20 838
adamc@14 839 Hint Rewrite tconcatCorrect : cpdt.
adamc@14 840
adamc@20 841 (** We ask that the lemma be used for left-to-right rewriting, and we ask for the hint to be added to the hint database called [cpdt], which is the database used by [crush]. Now we are ready to return to [tcompileCorrect'], proving it automatically this time. *)
adamc@20 842
adamc@14 843 Lemma tcompileCorrect' : forall t (e : texp t)
adamc@14 844 ts (s : vstack ts),
adamc@14 845 tprogDenote (tcompile e ts) s
adamc@14 846 = (texpDenote e, s).
adamc@14 847 induction e; crush.
adamc@14 848 Qed.
adamc@14 849
adamc@20 850 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 851
adamc@14 852 Hint Rewrite tcompileCorrect' : cpdt.
adamc@14 853
adamc@14 854 Theorem tcompileCorrect : forall t (e : texp t), tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 855 crush.
adamc@14 856 Qed.
adamc@22 857 (* end thide *)