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Merge; make prose nicer
author Adam Chlipala <adamc@hcoop.net>
date Wed, 03 Sep 2008 13:45:59 -0400
parents book/src/StackMachine.v@e5501b9c965d book/src/StackMachine.v@bcf375310f5f
children 8aed4562b62b
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@3 14 (* end hide *)
adamc@2 15
adamc@2 16
adamc@9 17 (** 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 18
adamc@11 19 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 20
adamc@11 21 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 22
adamc@9 23
adamc@2 24 (** * Arithmetic expressions over natural numbers *)
adamc@2 25
adamc@9 26 (** We will begin with that staple of compiler textbooks, arithemtic expressions over a single type of numbers. *)
adamc@9 27
adamc@9 28 (** ** Source language *)
adamc@9 29
adamc@9 30 (** We begin with the syntax of the source language. *)
adamc@2 31
adamc@4 32 Inductive binop : Set := Plus | Times.
adamc@2 33
adamc@9 34 (** 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 35
adamc@4 36 Inductive exp : Set :=
adamc@4 37 | Const : nat -> exp
adamc@4 38 | Binop : binop -> exp -> exp -> exp.
adamc@2 39
adamc@9 40 (** 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 41
adamc@9 42 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 43
adamc@9 44 %\medskip%
adamc@9 45
adamc@9 46 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 47
adamc@4 48 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 49 match b with
adamc@4 50 | Plus => plus
adamc@4 51 | Times => mult
adamc@4 52 end.
adamc@2 53
adamc@9 54 (** 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 55
adamc@9 56 [[
adamc@9 57 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
adamc@9 58 match b with
adamc@9 59 | Plus => plus
adamc@9 60 | Times => mult
adamc@9 61 end.
adamc@9 62
adamc@9 63 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 64
adamc@9 65 [[
adamc@9 66 Definition binopDenote := fun b =>
adamc@9 67 match b with
adamc@9 68 | Plus => plus
adamc@9 69 | Times => mult
adamc@9 70 end.
adamc@9 71
adamc@9 72 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 73
adamc@9 74 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 75
adamc@9 76 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 77
adamc@9 78 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 79
adamc@9 80 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 81
adamc@9 82 %\medskip%
adamc@9 83
adamc@9 84 We can give a simple definition of the meaning of an expression: *)
adamc@9 85
adamc@4 86 Fixpoint expDenote (e : exp) : nat :=
adamc@4 87 match e with
adamc@4 88 | Const n => n
adamc@4 89 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 90 end.
adamc@2 91
adamc@9 92 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 93
adamc@9 94
adamc@9 95 (** ** Target language *)
adamc@4 96
adamc@10 97 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 98
adamc@4 99 Inductive instr : Set :=
adamc@4 100 | IConst : nat -> instr
adamc@4 101 | IBinop : binop -> instr.
adamc@2 102
adamc@4 103 Definition prog := list instr.
adamc@4 104 Definition stack := list nat.
adamc@2 105
adamc@10 106 (** 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@10 107
adamc@10 108 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@10 109
adamc@4 110 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 111 match i with
adamc@4 112 | IConst n => Some (n :: s)
adamc@4 113 | IBinop b =>
adamc@4 114 match s with
adamc@4 115 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 116 | _ => None
adamc@4 117 end
adamc@4 118 end.
adamc@2 119
adamc@10 120 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@10 121
adamc@4 122 Fixpoint progDenote (p : prog) (s : stack) {struct p} : option stack :=
adamc@4 123 match p with
adamc@4 124 | nil => Some s
adamc@4 125 | i :: p' =>
adamc@4 126 match instrDenote i s with
adamc@4 127 | None => None
adamc@4 128 | Some s' => progDenote p' s'
adamc@4 129 end
adamc@4 130 end.
adamc@2 131
adamc@10 132 (** 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@2 133
adamc@4 134
adamc@9 135 (** ** Translation *)
adamc@4 136
adamc@10 137 (** Our compiler itself is now unsurprising. [++] is the list concatenation operator from the Coq standard library. *)
adamc@2 138
adamc@4 139 Fixpoint compile (e : exp) : prog :=
adamc@4 140 match e with
adamc@4 141 | Const n => IConst n :: nil
adamc@4 142 | Binop b e1 e2 => compile e2 ++ compile e1 ++ IBinop b :: nil
adamc@4 143 end.
adamc@2 144
adamc@2 145
adamc@9 146 (** ** Translation correctness *)
adamc@4 147
adamc@11 148 (** 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@11 149
adamc@11 150 Theorem compileCorrect : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 151 (* begin hide *)
adamc@11 152 Abort.
adamc@11 153 (* end hide *)
adamc@11 154
adamc@11 155 (** 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@11 156 *)
adamc@2 157
adamc@4 158 Lemma compileCorrect' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 159 progDenote p (expDenote e :: s).
adamc@11 160
adamc@11 161 (** 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@11 162
adamc@11 163 [[
adamc@11 164 1 subgoal
adamc@11 165
adamc@11 166 ============================
adamc@11 167 forall (e : exp) (s : stack) (p : list instr),
adamc@11 168 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@11 169 ]]
adamc@11 170
adamc@11 171 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@11 172
adamc@11 173 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@11 174
adamc@11 175 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@11 176 *)
adamc@11 177
adamc@4 178 induction e.
adamc@2 179
adamc@11 180 (** 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@11 181
adamc@11 182 [[
adamc@11 183 2 subgoals
adamc@11 184
adamc@11 185 n : nat
adamc@11 186 ============================
adamc@11 187 forall (s : stack) (p : list instr),
adamc@11 188 progDenote (compile (Const n) ++ p) s =
adamc@11 189 progDenote p (expDenote (Const n) :: s)
adamc@11 190 ]]
adamc@11 191 [[
adamc@11 192 subgoal 2 is:
adamc@11 193 forall (s : stack) (p : list instr),
adamc@11 194 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 195 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@11 196 ]]
adamc@11 197
adamc@11 198 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@11 199
adamc@11 200 We begin the first case with another very common tactic.
adamc@11 201 *)
adamc@11 202
adamc@4 203 intros.
adamc@11 204
adamc@11 205 (** The current subgoal changes to:
adamc@11 206 [[
adamc@11 207
adamc@11 208 n : nat
adamc@11 209 s : stack
adamc@11 210 p : list instr
adamc@11 211 ============================
adamc@11 212 progDenote (compile (Const n) ++ p) s =
adamc@11 213 progDenote p (expDenote (Const n) :: s)
adamc@11 214 ]]
adamc@11 215
adamc@11 216 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 217
adamc@11 218 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@11 219 *)
adamc@11 220
adamc@4 221 unfold compile.
adamc@11 222
adamc@11 223 (** [[
adamc@11 224
adamc@11 225 n : nat
adamc@11 226 s : stack
adamc@11 227 p : list instr
adamc@11 228 ============================
adamc@11 229 progDenote ((IConst n :: nil) ++ p) s =
adamc@11 230 progDenote p (expDenote (Const n) :: s)
adamc@11 231 ]] *)
adamc@11 232
adamc@4 233 unfold expDenote.
adamc@11 234
adamc@11 235 (** [[
adamc@11 236
adamc@11 237 n : nat
adamc@11 238 s : stack
adamc@11 239 p : list instr
adamc@11 240 ============================
adamc@11 241 progDenote ((IConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@11 242 ]]
adamc@11 243
adamc@11 244 We only need to unfold the first occurrence of [progDenote] to prove the goal: *)
adamc@11 245
adamc@11 246 unfold progDenote at 1.
adamc@11 247
adamc@11 248 (** [[
adamc@11 249
adamc@11 250 n : nat
adamc@11 251 s : stack
adamc@11 252 p : list instr
adamc@11 253 ============================
adamc@11 254 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 255 option stack :=
adamc@11 256 match p0 with
adamc@11 257 | nil => Some s0
adamc@11 258 | i :: p' =>
adamc@11 259 match instrDenote i s0 with
adamc@11 260 | Some s' => progDenote p' s'
adamc@11 261 | None => None (A:=stack)
adamc@11 262 end
adamc@11 263 end) ((IConst n :: nil) ++ p) s =
adamc@11 264 progDenote p (n :: s)
adamc@11 265 ]]
adamc@11 266
adamc@11 267 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@11 268 *)
adamc@11 269
adamc@4 270 simpl.
adamc@11 271
adamc@11 272 (** [[
adamc@11 273
adamc@11 274 n : nat
adamc@11 275 s : stack
adamc@11 276 p : list instr
adamc@11 277 ============================
adamc@11 278 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 279 option stack :=
adamc@11 280 match p0 with
adamc@11 281 | nil => Some s0
adamc@11 282 | i :: p' =>
adamc@11 283 match instrDenote i s0 with
adamc@11 284 | Some s' => progDenote p' s'
adamc@11 285 | None => None (A:=stack)
adamc@11 286 end
adamc@11 287 end) p (n :: s) = progDenote p (n :: s)
adamc@11 288 ]]
adamc@11 289
adamc@11 290 Now we can unexpand the definition of [progDenote]:
adamc@11 291 *)
adamc@11 292
adamc@11 293 fold progDenote.
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 progDenote p (n :: s) = progDenote p (n :: s)
adamc@11 302 ]]
adamc@11 303
adamc@11 304 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@11 305 *)
adamc@11 306
adamc@4 307 reflexivity.
adamc@2 308
adamc@11 309 (** On to the second inductive case:
adamc@11 310
adamc@11 311 [[
adamc@11 312
adamc@11 313 b : binop
adamc@11 314 e1 : exp
adamc@11 315 IHe1 : forall (s : stack) (p : list instr),
adamc@11 316 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 317 e2 : exp
adamc@11 318 IHe2 : forall (s : stack) (p : list instr),
adamc@11 319 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 320 ============================
adamc@11 321 forall (s : stack) (p : list instr),
adamc@11 322 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 323 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@11 324 ]]
adamc@11 325
adamc@11 326 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@11 327
adamc@11 328 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@11 329
adamc@4 330 intros.
adamc@4 331 unfold compile.
adamc@4 332 fold compile.
adamc@4 333 unfold expDenote.
adamc@4 334 fold expDenote.
adamc@11 335
adamc@11 336 (** Now we arrive at a point where the tactics we have seen so far are insufficient:
adamc@11 337
adamc@11 338 [[
adamc@11 339
adamc@11 340 b : binop
adamc@11 341 e1 : exp
adamc@11 342 IHe1 : forall (s : stack) (p : list instr),
adamc@11 343 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 344 e2 : exp
adamc@11 345 IHe2 : forall (s : stack) (p : list instr),
adamc@11 346 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 347 s : stack
adamc@11 348 p : list instr
adamc@11 349 ============================
adamc@11 350 progDenote ((compile e2 ++ compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 351 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 352 ]]
adamc@11 353
adamc@11 354 What we need is the associative law of list concatenation, available as a theorem [app_ass] in the standard library. *)
adamc@11 355
adamc@11 356 Check app_ass.
adamc@11 357
adamc@11 358 (** [[
adamc@11 359
adamc@11 360 app_ass
adamc@11 361 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@11 362 ]]
adamc@11 363
adamc@11 364 We use it to perform a rewrite: *)
adamc@11 365
adamc@4 366 rewrite app_ass.
adamc@11 367
adamc@11 368 (** changing the conclusion to: [[
adamc@11 369
adamc@11 370 progDenote (compile e2 ++ (compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 371 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 372 ]]
adamc@11 373
adamc@11 374 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@11 375
adamc@4 376 rewrite IHe2.
adamc@11 377
adamc@11 378 (** [[
adamc@11 379
adamc@11 380 progDenote ((compile e1 ++ IBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 381 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 382 ]]
adamc@11 383
adamc@11 384 The same process lets us apply the remaining hypothesis. *)
adamc@11 385
adamc@4 386 rewrite app_ass.
adamc@4 387 rewrite IHe1.
adamc@11 388
adamc@11 389 (** [[
adamc@11 390
adamc@11 391 progDenote ((IBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 392 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@11 393 ]]
adamc@11 394
adamc@11 395 Now we can apply a similar sequence of tactics to that that ended the proof of the first case.
adamc@11 396 *)
adamc@11 397
adamc@11 398 unfold progDenote at 1.
adamc@4 399 simpl.
adamc@11 400 fold progDenote.
adamc@4 401 reflexivity.
adamc@11 402
adamc@11 403 (** And the proof is completed, as indicated by the message:
adamc@11 404
adamc@11 405 [[
adamc@11 406 Proof completed.
adamc@11 407
adamc@11 408 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@11 409 *)
adamc@11 410
adamc@4 411 Abort.
adamc@2 412
adamc@4 413 Lemma compileCorrect' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 414 progDenote p (expDenote e :: s).
adamc@4 415 induction e; crush.
adamc@4 416 Qed.
adamc@2 417
adamc@11 418 (** 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@11 419
adamc@11 420 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@11 421
adamc@4 422 Theorem compileCorrect : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 423 intros.
adamc@11 424
adamc@11 425 (** [[
adamc@11 426
adamc@11 427 e : exp
adamc@11 428 ============================
adamc@11 429 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@11 430 ]]
adamc@11 431
adamc@11 432 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@11 433
adamc@11 434 Check app_nil_end.
adamc@11 435
adamc@11 436 (** [[
adamc@11 437
adamc@11 438 app_nil_end
adamc@11 439 : forall (A : Type) (l : list A), l = l ++ nil
adamc@11 440 ]] *)
adamc@11 441
adamc@4 442 rewrite (app_nil_end (compile e)).
adamc@11 443
adamc@11 444 (** 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@11 445
adamc@11 446 [[
adamc@11 447
adamc@11 448 e : exp
adamc@11 449 ============================
adamc@11 450 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@11 451 ]]
adamc@11 452
adamc@11 453 Now we can apply the lemma. *)
adamc@11 454
adamc@4 455 rewrite compileCorrect'.
adamc@11 456
adamc@11 457 (** [[
adamc@11 458
adamc@11 459 e : exp
adamc@11 460 ============================
adamc@11 461 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@11 462 ]]
adamc@11 463
adamc@11 464 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@11 465
adamc@4 466 reflexivity.
adamc@4 467 Qed.