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author Adam Chlipala <adamc@hcoop.net>
date Tue, 07 Oct 2008 22:05:37 -0400
parents 9b0d118abbc9
children 8caa3b3f8fc0
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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@25 19 (** %\chapter{Some Quick Examples}% *)
adamc@25 20
adamc@25 21
adamc@30 22 (** 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. The code in this book is tested with Coq version 8.1pl3, though parts may work with other versions.
adamc@9 23
adamc@25 24 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 in this rendering of 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, you will need to run %\texttt{%#<tt>#make#</tt>#%}% in the root directory of the source distribution for the book before getting started. If you have installed Proof General properly, it should start automatically when you visit a %\texttt{%#<tt>#.v#</tt>#%}% buffer in Emacs.
adamc@25 25
adamc@25 26 There are some minor headaches associated with getting Proof General to pass the proper command line arguments to the %\texttt{%#<tt>#coqtop#</tt>#%}% program. The best way to add settings that will be shared by many source files is to add a custom variable setting to your %\texttt{%#<tt>#.emacs#</tt>#%}% file, like this:
adamc@25 27 %\begin{verbatim}%#<pre>#(custom-set-variables
adamc@25 28 ...
adamc@90 29 '(coq-prog-args '("-I" "/path/to/cpdt/src"))
adamc@25 30 ...
adamc@25 31 )#</pre>#%\end{verbatim}%
adamc@25 32 The extra arguments demonstrated here are the proper choices for working with the code for this book. The ellipses stand for other Emacs customization settings you may already have. It can be helpful to save several alternate sets of flags in your %\texttt{%#<tt>#.emacs#</tt>#%}% file, with all but one commented out within the %\texttt{%#<tt>#custom-set-variables#</tt>#%}% block at any given time.
adamc@11 33
adamc@11 34 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 35
adamc@9 36
adamc@20 37 (** * Arithmetic Expressions Over Natural Numbers *)
adamc@2 38
adamc@40 39 (** We will begin with that staple of compiler textbooks, arithmetic expressions over a single type of numbers. *)
adamc@9 40
adamc@20 41 (** ** Source Language *)
adamc@9 42
adamc@9 43 (** We begin with the syntax of the source language. *)
adamc@2 44
adamc@4 45 Inductive binop : Set := Plus | Times.
adamc@2 46
adamc@9 47 (** 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 48
adamc@4 49 Inductive exp : Set :=
adamc@4 50 | Const : nat -> exp
adamc@4 51 | Binop : binop -> exp -> exp -> exp.
adamc@2 52
adamc@9 53 (** 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 54
adamc@9 55 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 56
adamc@9 57 %\medskip%
adamc@9 58
adamc@9 59 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 60
adamc@4 61 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 62 match b with
adamc@4 63 | Plus => plus
adamc@4 64 | Times => mult
adamc@4 65 end.
adamc@2 66
adamc@9 67 (** 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 68
adamc@9 69 [[
adamc@9 70 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
adamc@9 71 match b with
adamc@9 72 | Plus => plus
adamc@9 73 | Times => mult
adamc@9 74 end.
adamc@9 75
adamc@9 76 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 77
adamc@9 78 [[
adamc@9 79 Definition binopDenote := fun b =>
adamc@9 80 match b with
adamc@9 81 | Plus => plus
adamc@9 82 | Times => mult
adamc@9 83 end.
adamc@9 84
adamc@9 85 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 86
adamc@9 87 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 88
adamc@40 89 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 internally 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 90
adamc@9 91 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 92
adamc@9 93 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 94
adamc@9 95 %\medskip%
adamc@9 96
adamc@9 97 We can give a simple definition of the meaning of an expression: *)
adamc@9 98
adamc@4 99 Fixpoint expDenote (e : exp) : nat :=
adamc@4 100 match e with
adamc@4 101 | Const n => n
adamc@4 102 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 103 end.
adamc@2 104
adamc@9 105 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 106
adamc@16 107 (** 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@16 108
adamc@16 109 Eval simpl in expDenote (Const 42).
adamc@18 110 (** [[
adamc@18 111 = 42 : nat
adamc@18 112 ]] *)
adamc@16 113 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
adamc@18 114 (** [[
adamc@18 115 = 4 : nat
adamc@18 116 ]] *)
adamc@16 117 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@18 118 (** [[
adamc@18 119 = 28 : nat
adamc@18 120 ]] *)
adamc@9 121
adamc@20 122 (** ** Target Language *)
adamc@4 123
adamc@10 124 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 125
adamc@4 126 Inductive instr : Set :=
adamc@4 127 | IConst : nat -> instr
adamc@4 128 | IBinop : binop -> instr.
adamc@2 129
adamc@4 130 Definition prog := list instr.
adamc@4 131 Definition stack := list nat.
adamc@2 132
adamc@10 133 (** 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@10 134
adamc@10 135 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@10 136
adamc@4 137 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 138 match i with
adamc@4 139 | IConst n => Some (n :: s)
adamc@4 140 | IBinop b =>
adamc@4 141 match s with
adamc@4 142 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 143 | _ => None
adamc@4 144 end
adamc@4 145 end.
adamc@2 146
adamc@10 147 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@10 148
adamc@4 149 Fixpoint progDenote (p : prog) (s : stack) {struct p} : option stack :=
adamc@4 150 match p with
adamc@4 151 | nil => Some s
adamc@4 152 | i :: p' =>
adamc@4 153 match instrDenote i s with
adamc@4 154 | None => None
adamc@4 155 | Some s' => progDenote p' s'
adamc@4 156 end
adamc@4 157 end.
adamc@2 158
adamc@10 159 (** 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@2 160
adamc@4 161
adamc@9 162 (** ** Translation *)
adamc@4 163
adamc@10 164 (** Our compiler itself is now unsurprising. [++] is the list concatenation operator from the Coq standard library. *)
adamc@2 165
adamc@4 166 Fixpoint compile (e : exp) : prog :=
adamc@4 167 match e with
adamc@4 168 | Const n => IConst n :: nil
adamc@4 169 | Binop b e1 e2 => compile e2 ++ compile e1 ++ IBinop b :: nil
adamc@4 170 end.
adamc@2 171
adamc@2 172
adamc@16 173 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 174
adamc@16 175 Eval simpl in compile (Const 42).
adamc@18 176 (** [[
adamc@18 177 = IConst 42 :: nil : prog
adamc@18 178 ]] *)
adamc@16 179 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adamc@18 180 (** [[
adamc@18 181 = IConst 2 :: IConst 2 :: IBinop Plus :: nil : prog
adamc@18 182 ]] *)
adamc@16 183 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@18 184 (** [[
adamc@18 185 = IConst 7 :: IConst 2 :: IConst 2 :: IBinop Plus :: IBinop Times :: nil : prog
adamc@18 186 ]] *)
adamc@16 187
adamc@40 188 (** We can also run our compiled programs and check that they give the right results. *)
adamc@16 189
adamc@16 190 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@18 191 (** [[
adamc@18 192 = Some (42 :: nil) : option stack
adamc@18 193 ]] *)
adamc@16 194 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@18 195 (** [[
adamc@18 196 = Some (4 :: nil) : option stack
adamc@18 197 ]] *)
adamc@16 198 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7))) nil.
adamc@18 199 (** [[
adamc@18 200 = Some (28 :: nil) : option stack
adamc@18 201 ]] *)
adamc@16 202
adamc@16 203
adamc@20 204 (** ** Translation Correctness *)
adamc@4 205
adamc@11 206 (** 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@11 207
adamc@26 208 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 209 (* begin hide *)
adamc@11 210 Abort.
adamc@11 211 (* end hide *)
adamc@22 212 (* begin thide *)
adamc@11 213
adamc@11 214 (** 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@11 215 *)
adamc@2 216
adamc@26 217 Lemma compile_correct' : forall e p s, progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 218
adamc@11 219 (** 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@11 220
adamc@11 221 [[
adamc@11 222 1 subgoal
adamc@11 223
adamc@11 224 ============================
adamc@15 225 forall (e : exp) (p : list instr) (s : stack),
adamc@15 226 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@11 227 ]]
adamc@11 228
adamc@11 229 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@11 230
adamc@11 231 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@11 232
adamc@11 233 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@11 234 *)
adamc@11 235
adamc@4 236 induction e.
adamc@2 237
adamc@11 238 (** 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@11 239
adamc@11 240 [[
adamc@11 241 2 subgoals
adamc@11 242
adamc@11 243 n : nat
adamc@11 244 ============================
adamc@11 245 forall (s : stack) (p : list instr),
adamc@11 246 progDenote (compile (Const n) ++ p) s =
adamc@11 247 progDenote p (expDenote (Const n) :: s)
adamc@11 248 ]]
adamc@11 249 [[
adamc@11 250 subgoal 2 is:
adamc@11 251 forall (s : stack) (p : list instr),
adamc@11 252 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 253 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@11 254 ]]
adamc@11 255
adamc@11 256 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@11 257
adamc@11 258 We begin the first case with another very common tactic.
adamc@11 259 *)
adamc@11 260
adamc@4 261 intros.
adamc@11 262
adamc@11 263 (** The current subgoal changes to:
adamc@11 264 [[
adamc@11 265
adamc@11 266 n : nat
adamc@11 267 s : stack
adamc@11 268 p : list instr
adamc@11 269 ============================
adamc@11 270 progDenote (compile (Const n) ++ p) s =
adamc@11 271 progDenote p (expDenote (Const n) :: s)
adamc@11 272 ]]
adamc@11 273
adamc@11 274 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 275
adamc@11 276 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@11 277 *)
adamc@11 278
adamc@4 279 unfold compile.
adamc@11 280
adamc@11 281 (** [[
adamc@11 282
adamc@11 283 n : nat
adamc@11 284 s : stack
adamc@11 285 p : list instr
adamc@11 286 ============================
adamc@11 287 progDenote ((IConst n :: nil) ++ p) s =
adamc@11 288 progDenote p (expDenote (Const n) :: s)
adamc@11 289 ]] *)
adamc@11 290
adamc@4 291 unfold expDenote.
adamc@11 292
adamc@11 293 (** [[
adamc@11 294
adamc@11 295 n : nat
adamc@11 296 s : stack
adamc@11 297 p : list instr
adamc@11 298 ============================
adamc@11 299 progDenote ((IConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@11 300 ]]
adamc@11 301
adamc@11 302 We only need to unfold the first occurrence of [progDenote] to prove the goal: *)
adamc@11 303
adamc@11 304 unfold progDenote at 1.
adamc@11 305
adamc@11 306 (** [[
adamc@11 307
adamc@11 308 n : nat
adamc@11 309 s : stack
adamc@11 310 p : list instr
adamc@11 311 ============================
adamc@11 312 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 313 option stack :=
adamc@11 314 match p0 with
adamc@11 315 | nil => Some s0
adamc@11 316 | i :: p' =>
adamc@11 317 match instrDenote i s0 with
adamc@11 318 | Some s' => progDenote p' s'
adamc@11 319 | None => None (A:=stack)
adamc@11 320 end
adamc@11 321 end) ((IConst n :: nil) ++ p) s =
adamc@11 322 progDenote p (n :: s)
adamc@11 323 ]]
adamc@11 324
adamc@11 325 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@11 326 *)
adamc@11 327
adamc@4 328 simpl.
adamc@11 329
adamc@11 330 (** [[
adamc@11 331
adamc@11 332 n : nat
adamc@11 333 s : stack
adamc@11 334 p : list instr
adamc@11 335 ============================
adamc@11 336 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 337 option stack :=
adamc@11 338 match p0 with
adamc@11 339 | nil => Some s0
adamc@11 340 | i :: p' =>
adamc@11 341 match instrDenote i s0 with
adamc@11 342 | Some s' => progDenote p' s'
adamc@11 343 | None => None (A:=stack)
adamc@11 344 end
adamc@11 345 end) p (n :: s) = progDenote p (n :: s)
adamc@11 346 ]]
adamc@11 347
adamc@11 348 Now we can unexpand the definition of [progDenote]:
adamc@11 349 *)
adamc@11 350
adamc@11 351 fold progDenote.
adamc@11 352
adamc@11 353 (** [[
adamc@11 354
adamc@11 355 n : nat
adamc@11 356 s : stack
adamc@11 357 p : list instr
adamc@11 358 ============================
adamc@11 359 progDenote p (n :: s) = progDenote p (n :: s)
adamc@11 360 ]]
adamc@11 361
adamc@11 362 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@11 363 *)
adamc@11 364
adamc@4 365 reflexivity.
adamc@2 366
adamc@11 367 (** On to the second inductive case:
adamc@11 368
adamc@11 369 [[
adamc@11 370
adamc@11 371 b : binop
adamc@11 372 e1 : exp
adamc@11 373 IHe1 : forall (s : stack) (p : list instr),
adamc@11 374 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 375 e2 : exp
adamc@11 376 IHe2 : forall (s : stack) (p : list instr),
adamc@11 377 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 378 ============================
adamc@11 379 forall (s : stack) (p : list instr),
adamc@11 380 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 381 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@11 382 ]]
adamc@11 383
adamc@11 384 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@11 385
adamc@11 386 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@11 387
adamc@4 388 intros.
adamc@4 389 unfold compile.
adamc@4 390 fold compile.
adamc@4 391 unfold expDenote.
adamc@4 392 fold expDenote.
adamc@11 393
adamc@44 394 (** Now we arrive at a point where the tactics we have seen so far are insufficient. No further definition unfoldings get us anywhere, so we will need to try something different.
adamc@11 395
adamc@11 396 [[
adamc@11 397
adamc@11 398 b : binop
adamc@11 399 e1 : exp
adamc@11 400 IHe1 : forall (s : stack) (p : list instr),
adamc@11 401 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 402 e2 : exp
adamc@11 403 IHe2 : forall (s : stack) (p : list instr),
adamc@11 404 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 405 s : stack
adamc@11 406 p : list instr
adamc@11 407 ============================
adamc@11 408 progDenote ((compile e2 ++ compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 409 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 410 ]]
adamc@11 411
adamc@11 412 What we need is the associative law of list concatenation, available as a theorem [app_ass] in the standard library. *)
adamc@11 413
adamc@11 414 Check app_ass.
adamc@11 415
adamc@11 416 (** [[
adamc@11 417
adamc@11 418 app_ass
adamc@11 419 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@11 420 ]]
adamc@11 421
adamc@11 422 We use it to perform a rewrite: *)
adamc@11 423
adamc@4 424 rewrite app_ass.
adamc@11 425
adamc@11 426 (** changing the conclusion to: [[
adamc@11 427
adamc@11 428 progDenote (compile e2 ++ (compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 429 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 430 ]]
adamc@11 431
adamc@11 432 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@11 433
adamc@4 434 rewrite IHe2.
adamc@11 435
adamc@11 436 (** [[
adamc@11 437
adamc@11 438 progDenote ((compile e1 ++ IBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 439 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 440 ]]
adamc@11 441
adamc@11 442 The same process lets us apply the remaining hypothesis. *)
adamc@11 443
adamc@4 444 rewrite app_ass.
adamc@4 445 rewrite IHe1.
adamc@11 446
adamc@11 447 (** [[
adamc@11 448
adamc@11 449 progDenote ((IBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 450 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 451 ]]
adamc@11 452
adamc@11 453 Now we can apply a similar sequence of tactics to that that ended the proof of the first case.
adamc@11 454 *)
adamc@11 455
adamc@11 456 unfold progDenote at 1.
adamc@4 457 simpl.
adamc@11 458 fold progDenote.
adamc@4 459 reflexivity.
adamc@11 460
adamc@11 461 (** And the proof is completed, as indicated by the message:
adamc@11 462
adamc@11 463 [[
adamc@11 464 Proof completed.
adamc@11 465
adamc@11 466 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@11 467 *)
adamc@11 468
adamc@4 469 Abort.
adamc@2 470
adamc@26 471 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 472 progDenote p (expDenote e :: s).
adamc@4 473 induction e; crush.
adamc@4 474 Qed.
adamc@2 475
adamc@11 476 (** 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@11 477
adamc@11 478 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@11 479
adamc@26 480 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 481 intros.
adamc@11 482
adamc@11 483 (** [[
adamc@11 484
adamc@11 485 e : exp
adamc@11 486 ============================
adamc@11 487 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@11 488 ]]
adamc@11 489
adamc@26 490 At this point, we want to massage the lefthand side to match the statement of [compile_correct']. A theorem from the standard library is useful: *)
adamc@11 491
adamc@11 492 Check app_nil_end.
adamc@11 493
adamc@11 494 (** [[
adamc@11 495
adamc@11 496 app_nil_end
adamc@11 497 : forall (A : Type) (l : list A), l = l ++ nil
adamc@11 498 ]] *)
adamc@11 499
adamc@4 500 rewrite (app_nil_end (compile e)).
adamc@11 501
adamc@11 502 (** 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@11 503
adamc@11 504 [[
adamc@11 505
adamc@11 506 e : exp
adamc@11 507 ============================
adamc@11 508 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@11 509 ]]
adamc@11 510
adamc@11 511 Now we can apply the lemma. *)
adamc@11 512
adamc@26 513 rewrite compile_correct'.
adamc@11 514
adamc@11 515 (** [[
adamc@11 516
adamc@11 517 e : exp
adamc@11 518 ============================
adamc@11 519 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@11 520 ]]
adamc@11 521
adamc@11 522 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@11 523
adamc@4 524 reflexivity.
adamc@4 525 Qed.
adamc@22 526 (* end thide *)
adamc@14 527
adamc@14 528
adamc@20 529 (** * Typed Expressions *)
adamc@14 530
adamc@14 531 (** 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@14 532
adamc@20 533 (** ** Source Language *)
adamc@14 534
adamc@15 535 (** We define a trivial language of types to classify our expressions: *)
adamc@15 536
adamc@14 537 Inductive type : Set := Nat | Bool.
adamc@14 538
adamc@15 539 (** Now we define an expanded set of binary operators. *)
adamc@15 540
adamc@14 541 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 542 | TPlus : tbinop Nat Nat Nat
adamc@14 543 | TTimes : tbinop Nat Nat Nat
adamc@14 544 | TEq : forall t, tbinop t t Bool
adamc@14 545 | TLt : tbinop Nat Nat Bool.
adamc@14 546
adamc@15 547 (** 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@15 548
adamc@15 549 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@15 550
adamc@15 551 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@15 552
adamc@40 553 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 indexed 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@15 554 *)
adamc@15 555
adamc@15 556 (** We can define a similar type family for typed expressions. *)
adamc@15 557
adamc@14 558 Inductive texp : type -> Set :=
adamc@14 559 | TNConst : nat -> texp Nat
adamc@14 560 | TBConst : bool -> texp Bool
adamc@14 561 | TBinop : forall arg1 arg2 res, tbinop arg1 arg2 res -> texp arg1 -> texp arg2 -> texp res.
adamc@14 562
adamc@15 563 (** 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@15 564
adamc@14 565 Definition typeDenote (t : type) : Set :=
adamc@14 566 match t with
adamc@14 567 | Nat => nat
adamc@14 568 | Bool => bool
adamc@14 569 end.
adamc@14 570
adamc@15 571 (** 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@15 572
adamc@15 573 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@15 574 *)
adamc@15 575
adamc@14 576 Definition eq_bool (b1 b2 : bool) : bool :=
adamc@14 577 match b1, b2 with
adamc@14 578 | true, true => true
adamc@14 579 | false, false => true
adamc@14 580 | _, _ => false
adamc@14 581 end.
adamc@14 582
adamc@14 583 Fixpoint eq_nat (n1 n2 : nat) {struct n1} : bool :=
adamc@14 584 match n1, n2 with
adamc@14 585 | O, O => true
adamc@14 586 | S n1', S n2' => eq_nat n1' n2'
adamc@14 587 | _, _ => false
adamc@14 588 end.
adamc@14 589
adamc@14 590 Fixpoint lt (n1 n2 : nat) {struct n1} : bool :=
adamc@14 591 match n1, n2 with
adamc@14 592 | O, S _ => true
adamc@14 593 | S n1', S n2' => lt n1' n2'
adamc@14 594 | _, _ => false
adamc@14 595 end.
adamc@14 596
adamc@15 597 (** Now we can interpret binary operators: *)
adamc@15 598
adamc@14 599 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@14 600 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@14 601 match b in (tbinop arg1 arg2 res) return (typeDenote arg1 -> typeDenote arg2 -> typeDenote res) with
adamc@14 602 | TPlus => plus
adamc@14 603 | TTimes => mult
adamc@14 604 | TEq Nat => eq_nat
adamc@14 605 | TEq Bool => eq_bool
adamc@14 606 | TLt => lt
adamc@14 607 end.
adamc@14 608
adamc@15 609 (** 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@15 610
adamc@40 611 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 occurrences#</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@15 612
adamc@15 613 The same tricks suffice to define an expression denotation function in an unsurprising way:
adamc@15 614 *)
adamc@15 615
adamc@14 616 Fixpoint texpDenote t (e : texp t) {struct e} : typeDenote t :=
adamc@14 617 match e in (texp t) return (typeDenote t) with
adamc@14 618 | TNConst n => n
adamc@14 619 | TBConst b => b
adamc@14 620 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 621 end.
adamc@14 622
adamc@17 623 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 624
adamc@17 625 Eval simpl in texpDenote (TNConst 42).
adamc@18 626 (** [[
adamc@18 627 = 42 : typeDenote Nat
adamc@18 628 ]] *)
adamc@17 629 Eval simpl in texpDenote (TBConst true).
adamc@18 630 (** [[
adamc@18 631 = true : typeDenote Bool
adamc@18 632 ]] *)
adamc@17 633 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@18 634 (** [[
adamc@18 635 = 28 : typeDenote Nat
adamc@18 636 ]] *)
adamc@17 637 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@18 638 (** [[
adamc@18 639 = false : typeDenote Bool
adamc@18 640 ]] *)
adamc@17 641 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@18 642 (** [[
adamc@18 643 = true : typeDenote Bool
adamc@18 644 ]] *)
adamc@17 645
adamc@14 646
adamc@20 647 (** ** Target Language *)
adamc@14 648
adamc@18 649 (** 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@18 650
adamc@18 651 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@18 652
adamc@18 653 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 654
adamc@14 655 Definition tstack := list type.
adamc@14 656
adamc@18 657 (** 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@18 658
adamc@18 659 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@18 660
adamc@14 661 Inductive tinstr : tstack -> tstack -> Set :=
adamc@14 662 | TINConst : forall s, nat -> tinstr s (Nat :: s)
adamc@14 663 | TIBConst : forall s, bool -> tinstr s (Bool :: s)
adamc@14 664 | TIBinop : forall arg1 arg2 res s,
adamc@14 665 tbinop arg1 arg2 res
adamc@14 666 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 667
adamc@18 668 (** 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@18 669
adamc@14 670 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 671 | TNil : forall s, tprog s s
adamc@14 672 | TCons : forall s1 s2 s3,
adamc@14 673 tinstr s1 s2
adamc@14 674 -> tprog s2 s3
adamc@14 675 -> tprog s1 s3.
adamc@14 676
adamc@18 677 (** 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@18 678
adamc@14 679 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 680 match ts with
adamc@14 681 | nil => unit
adamc@14 682 | t :: ts' => typeDenote t * vstack ts'
adamc@14 683 end%type.
adamc@14 684
adamc@18 685 (** 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@18 686
adamc@18 687 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@18 688
adamc@14 689 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@14 690 match i in (tinstr ts ts') return (vstack ts -> vstack ts') with
adamc@14 691 | TINConst _ n => fun s => (n, s)
adamc@14 692 | TIBConst _ b => fun s => (b, s)
adamc@14 693 | TIBinop _ _ _ _ b => fun s =>
adamc@14 694 match s with
adamc@14 695 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@14 696 end
adamc@14 697 end.
adamc@14 698
adamc@18 699 (** 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@18 700
adamc@18 701 [[
adamc@18 702 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@18 703 match i in (tinstr ts ts') return (vstack ts') with
adamc@18 704 | TINConst _ n => (n, s)
adamc@18 705 | TIBConst _ b => (b, s)
adamc@18 706 | TIBinop _ _ _ _ b =>
adamc@18 707 match s with
adamc@18 708 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@18 709 end
adamc@18 710 end.
adamc@18 711
adamc@18 712 The Coq type-checker complains that:
adamc@18 713
adamc@18 714 [[
adamc@18 715 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@18 716 while it is expected to have type "vstack (Nat :: t)"
adamc@18 717 ]]
adamc@18 718
adamc@18 719 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@18 720
adamc@18 721 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@18 722
adamc@18 723 *)
adamc@18 724
adamc@18 725 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 726
adamc@14 727 Fixpoint tprogDenote ts ts' (p : tprog ts ts') {struct p} : vstack ts -> vstack ts' :=
adamc@14 728 match p in (tprog ts ts') return (vstack ts -> vstack ts') with
adamc@14 729 | TNil _ => fun s => s
adamc@14 730 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 731 end.
adamc@14 732
adamc@14 733
adamc@14 734 (** ** Translation *)
adamc@14 735
adamc@19 736 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 737
adamc@14 738 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') {struct p} : tprog ts' ts'' -> tprog ts ts'' :=
adamc@14 739 match p in (tprog ts ts') return (tprog ts' ts'' -> tprog ts ts'') with
adamc@14 740 | TNil _ => fun p' => p'
adamc@14 741 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 742 end.
adamc@14 743
adamc@19 744 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 745
adamc@14 746 Fixpoint tcompile t (e : texp t) (ts : tstack) {struct e} : tprog ts (t :: ts) :=
adamc@14 747 match e in (texp t) return (tprog ts (t :: ts)) with
adamc@14 748 | TNConst n => TCons (TINConst _ n) (TNil _)
adamc@14 749 | TBConst b => TCons (TIBConst _ b) (TNil _)
adamc@14 750 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adamc@14 751 (tconcat (tcompile e1 _) (TCons (TIBinop _ b) (TNil _)))
adamc@14 752 end.
adamc@14 753
adamc@40 754 (** 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 arbitrary 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@19 755
adamc@19 756 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@19 757
adamc@14 758 Print tcompile.
adamc@14 759
adamc@19 760 (** [[
adamc@19 761
adamc@19 762 tcompile =
adamc@19 763 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 764 tprog ts (t :: ts) :=
adamc@19 765 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adamc@19 766 | TNConst n => TCons (TINConst ts n) (TNil (Nat :: ts))
adamc@19 767 | TBConst b => TCons (TIBConst ts b) (TNil (Bool :: ts))
adamc@19 768 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 769 tconcat (tcompile arg2 e2 ts)
adamc@19 770 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adamc@19 771 (TCons (TIBinop ts b) (TNil (res :: ts))))
adamc@19 772 end
adamc@19 773 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adamc@19 774 ]] *)
adamc@19 775
adamc@19 776
adamc@19 777 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 778
adamc@19 779 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adamc@19 780 (** [[
adamc@19 781 = (42, tt) : vstack (Nat :: nil)
adamc@19 782 ]] *)
adamc@19 783 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adamc@19 784 (** [[
adamc@19 785 = (true, tt) : vstack (Bool :: nil)
adamc@19 786 ]] *)
adamc@19 787 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@19 788 (** [[
adamc@19 789 = (28, tt) : vstack (Nat :: nil)
adamc@19 790 ]] *)
adamc@19 791 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@19 792 (** [[
adamc@19 793 = (false, tt) : vstack (Bool :: nil)
adamc@19 794 ]] *)
adamc@19 795 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@19 796 (** [[
adamc@19 797 = (true, tt) : vstack (Bool :: nil)
adamc@19 798 ]] *)
adamc@19 799
adamc@14 800
adamc@20 801 (** ** Translation Correctness *)
adamc@20 802
adamc@20 803 (** We can state a correctness theorem similar to the last one. *)
adamc@20 804
adamc@26 805 Theorem tcompile_correct : forall t (e : texp t), tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 806 (* begin hide *)
adamc@20 807 Abort.
adamc@20 808 (* end hide *)
adamc@22 809 (* begin thide *)
adamc@20 810
adamc@20 811 (** 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@14 812
adamc@26 813 Lemma tcompile_correct' : forall t (e : texp t)
adamc@14 814 ts (s : vstack ts),
adamc@14 815 tprogDenote (tcompile e ts) s
adamc@14 816 = (texpDenote e, s).
adamc@20 817
adamc@26 818 (** While lemma [compile_correct'] 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@20 819
adamc@20 820 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 821
adamc@14 822 induction e; crush.
adamc@20 823
adamc@20 824 (** We are left with this unproved conclusion:
adamc@20 825
adamc@20 826 [[
adamc@20 827
adamc@20 828 tprogDenote
adamc@20 829 (tconcat (tcompile e2 ts)
adamc@20 830 (tconcat (tcompile e1 (arg2 :: ts))
adamc@20 831 (TCons (TIBinop ts t) (TNil (res :: ts))))) s =
adamc@20 832 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@20 833 ]]
adamc@20 834
adamc@20 835 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@20 836 *)
adamc@14 837 Abort.
adamc@14 838
adamc@26 839 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 840 (s : vstack ts),
adamc@14 841 tprogDenote (tconcat p p') s
adamc@14 842 = tprogDenote p' (tprogDenote p s).
adamc@14 843 induction p; crush.
adamc@14 844 Qed.
adamc@14 845
adamc@20 846 (** This one goes through completely automatically.
adamc@20 847
adamc@26 848 Some code behind the scenes registers [app_ass] for use by [crush]. We must register [tconcat_correct] similarly to get the same effect: *)
adamc@20 849
adamc@26 850 Hint Rewrite tconcat_correct : cpdt.
adamc@14 851
adamc@26 852 (** 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 [tcompile_correct'], proving it automatically this time. *)
adamc@20 853
adamc@26 854 Lemma tcompile_correct' : forall t (e : texp t)
adamc@14 855 ts (s : vstack ts),
adamc@14 856 tprogDenote (tcompile e ts) s
adamc@14 857 = (texpDenote e, s).
adamc@14 858 induction e; crush.
adamc@14 859 Qed.
adamc@14 860
adamc@20 861 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 862
adamc@26 863 Hint Rewrite tcompile_correct' : cpdt.
adamc@14 864
adamc@26 865 Theorem tcompile_correct : forall t (e : texp t), tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 866 crush.
adamc@14 867 Qed.
adamc@22 868 (* end thide *)