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Finished 2011 pass through Intro
author Adam Chlipala <adam@chlipala.net>
date Thu, 25 Aug 2011 14:41:49 -0400
parents 690796f4690d
children 4cb3ba8604bc
rev   line source
adam@298 1 (* Copyright (c) 2008-2011, 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 *)
adam@277 11 Require Import Arith Bool 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
adam@279 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. This chapter is not meant to give full explanations of the features that are employed. Rather, it is meant more as an advertisement of what is possible. Later chapters will introduce all of the concepts in bottom-up fashion.
adam@279 23
adam@307 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 two lines %\index{Vernacular commands!Require}%[Require Import] #<span class="inlinecode"><span class="id" type="var">#%\coqdocconstructor{%Arith%}%#</span></span># [Bool] #<span class="inlinecode"><span class="id" type="var">#%\coqdocconstructor{%List%}%#</span></span># [Tactics.] and %\index{Vernacular commands!Set Implicit Arguments}%[Set Implicit] #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Arguments%}%#</span></span>#[.] at the start of the file, to match some code hidden in this rendering of the chapter source. In general, similar commands will be hidden in the book rendering of each chapter's source code, so you will need to insert them in from-scratch replayings of the code that is presented. To be more specific, every chapter begins with some imports of other modules, followed by [Set Implicit] #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Arguments%}%#</span></span>#[.], where the latter affects the default behavior of definitions regarding type inference.
adam@307 25 *)
adamc@9 26
adamc@9 27
adamc@20 28 (** * Arithmetic Expressions Over Natural Numbers *)
adamc@2 29
adamc@40 30 (** We will begin with that staple of compiler textbooks, arithmetic expressions over a single type of numbers. *)
adamc@9 31
adamc@20 32 (** ** Source Language *)
adamc@9 33
adamc@9 34 (** We begin with the syntax of the source language. *)
adamc@2 35
adamc@4 36 Inductive binop : Set := Plus | Times.
adamc@2 37
adamc@9 38 (** 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 39
adamc@4 40 Inductive exp : Set :=
adamc@4 41 | Const : nat -> exp
adamc@4 42 | Binop : binop -> exp -> exp -> exp.
adamc@2 43
adamc@9 44 (** 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 45
adam@277 46 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>#%}%, the inverted 'A' symbol for %\texttt{%#<tt>#forall#</tt>#%}%, and the Cartesian product 'X' for %\texttt{%#<tt>#*#</tt>#%}%. When in doubt about the ASCII version of a symbol, you can consult the chapter source code.
adamc@9 47
adamc@9 48 %\medskip%
adamc@9 49
adam@292 50 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 51
adamc@4 52 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 53 match b with
adamc@4 54 | Plus => plus
adamc@4 55 | Times => mult
adamc@4 56 end.
adamc@2 57
adamc@9 58 (** 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 59
adamc@9 60 [[
adamc@9 61 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
adamc@9 62 match b with
adamc@9 63 | Plus => plus
adamc@9 64 | Times => mult
adamc@9 65 end.
adamc@9 66
adamc@205 67 ]]
adamc@205 68
adamc@9 69 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 70
adamc@9 71 [[
adamc@9 72 Definition binopDenote := fun b =>
adamc@9 73 match b with
adamc@9 74 | Plus => plus
adamc@9 75 | Times => mult
adamc@9 76 end.
adamc@9 77
adamc@205 78 ]]
adamc@205 79
adam@307 80 Languages like Haskell and ML have a convenient %\emph{%#<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 81
adam@307 82 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 %\emph{%#<i>#Calculus of Inductive Constructions (CIC)#</i>#%}%, which is an extension of the older %\emph{%#<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 %\emph{%#<i>#strong normalization#</i>#%}%, meaning that every program (and, more importantly, every proof term) terminates; and %\emph{%#<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 83
adam@307 84 Coq is actually based on an extension of CIC called %\emph{%#<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 85
adam@307 86 Next, there is %\emph{%#<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 87
adam@307 88 Finally, commands like [Inductive] and [Definition] are part of %\emph{%#<i>#the vernacular#</i>#%}%, which includes all sorts of useful queries and requests to the Coq system. Every Coq source file is a series of vernacular commands, where many command forms take arguments that are Gallina or Ltac programs. (Actually, Coq source files are more like %\emph{%#<i>#trees#</i>#%}% of vernacular commands, thanks to various nested scoping constructs.)
adamc@9 89
adamc@9 90 %\medskip%
adamc@9 91
adamc@9 92 We can give a simple definition of the meaning of an expression: *)
adamc@9 93
adamc@4 94 Fixpoint expDenote (e : exp) : nat :=
adamc@4 95 match e with
adamc@4 96 | Const n => n
adamc@4 97 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 98 end.
adamc@2 99
adamc@9 100 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 101
adamc@16 102 (** 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@16 103
adamc@16 104 Eval simpl in expDenote (Const 42).
adamc@205 105 (** [= 42 : nat] *)
adamc@205 106
adamc@16 107 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
adamc@205 108 (** [= 4 : nat] *)
adamc@205 109
adamc@16 110 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@205 111 (** [= 28 : nat] *)
adamc@9 112
adamc@20 113 (** ** Target Language *)
adamc@4 114
adamc@10 115 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 116
adamc@4 117 Inductive instr : Set :=
adamc@4 118 | IConst : nat -> instr
adamc@4 119 | IBinop : binop -> instr.
adamc@2 120
adamc@4 121 Definition prog := list instr.
adamc@4 122 Definition stack := list nat.
adamc@2 123
adamc@10 124 (** 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@10 125
adam@292 126 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@10 127
adamc@4 128 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 129 match i with
adamc@4 130 | IConst n => Some (n :: s)
adamc@4 131 | IBinop b =>
adamc@4 132 match s with
adamc@4 133 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 134 | _ => None
adamc@4 135 end
adamc@4 136 end.
adamc@2 137
adamc@206 138 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program.
adamc@10 139
adamc@206 140 [[
adamc@4 141 Fixpoint progDenote (p : prog) (s : stack) {struct p} : option stack :=
adamc@4 142 match p with
adamc@4 143 | nil => Some s
adamc@4 144 | i :: p' =>
adamc@4 145 match instrDenote i s with
adamc@4 146 | None => None
adamc@4 147 | Some s' => progDenote p' s'
adamc@4 148 end
adamc@4 149 end.
adamc@2 150
adamc@206 151 ]]
adamc@206 152
adamc@206 153 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@206 154
adamc@206 155 Recent versions of Coq will also infer a termination argument, so that we may write simply: *)
adamc@206 156
adamc@206 157 Fixpoint progDenote (p : prog) (s : stack) : option stack :=
adamc@206 158 match p with
adamc@206 159 | nil => Some s
adamc@206 160 | i :: p' =>
adamc@206 161 match instrDenote i s with
adamc@206 162 | None => None
adamc@206 163 | Some s' => progDenote p' s'
adamc@206 164 end
adamc@206 165 end.
adamc@2 166
adamc@4 167
adamc@9 168 (** ** Translation *)
adamc@4 169
adamc@10 170 (** Our compiler itself is now unsurprising. [++] is the list concatenation operator from the Coq standard library. *)
adamc@2 171
adamc@4 172 Fixpoint compile (e : exp) : prog :=
adamc@4 173 match e with
adamc@4 174 | Const n => IConst n :: nil
adamc@4 175 | Binop b e1 e2 => compile e2 ++ compile e1 ++ IBinop b :: nil
adamc@4 176 end.
adamc@2 177
adamc@2 178
adamc@16 179 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 180
adamc@16 181 Eval simpl in compile (Const 42).
adamc@206 182 (** [= IConst 42 :: nil : prog] *)
adamc@206 183
adamc@16 184 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adamc@206 185 (** [= IConst 2 :: IConst 2 :: IBinop Plus :: nil : prog] *)
adamc@206 186
adamc@16 187 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@206 188 (** [= IConst 7 :: IConst 2 :: IConst 2 :: IBinop Plus :: IBinop Times :: nil : prog] *)
adamc@16 189
adamc@40 190 (** We can also run our compiled programs and check that they give the right results. *)
adamc@16 191
adamc@16 192 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@206 193 (** [= Some (42 :: nil) : option stack] *)
adamc@206 194
adamc@16 195 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@206 196 (** [= Some (4 :: nil) : option stack] *)
adamc@206 197
adamc@16 198 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7))) nil.
adamc@206 199 (** [= Some (28 :: nil) : option stack] *)
adamc@16 200
adamc@16 201
adamc@20 202 (** ** Translation Correctness *)
adamc@4 203
adamc@11 204 (** 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@11 205
adamc@26 206 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 207 (* begin hide *)
adamc@11 208 Abort.
adamc@11 209 (* end hide *)
adamc@22 210 (* begin thide *)
adamc@11 211
adam@307 212 (** 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 %\emph{%#<i>#strengthening the induction hypothesis#</i>#%}%. We do that by proving an auxiliary lemma:
adamc@11 213 *)
adamc@2 214
adamc@206 215 Lemma compile_correct' : forall e p s,
adamc@206 216 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 217
adam@307 218 (** After the period in the [Lemma] command, we are in %\emph{%#<i>#the interactive proof-editing mode#</i>#%}%. We find ourselves staring at this ominous screen of text:
adamc@11 219
adamc@11 220 [[
adamc@11 221 1 subgoal
adamc@11 222
adamc@11 223 ============================
adamc@15 224 forall (e : exp) (p : list instr) (s : stack),
adamc@15 225 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@206 226
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
adam@307 233 We manipulate the proof state by running commands called %\emph{%#<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@206 254
adamc@11 255 ]]
adamc@11 256
adamc@11 257 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@11 258
adamc@11 259 We begin the first case with another very common tactic.
adamc@11 260 *)
adamc@11 261
adamc@4 262 intros.
adamc@11 263
adamc@11 264 (** The current subgoal changes to:
adamc@11 265 [[
adamc@11 266
adamc@11 267 n : nat
adamc@11 268 s : stack
adamc@11 269 p : list instr
adamc@11 270 ============================
adamc@11 271 progDenote (compile (Const n) ++ p) s =
adamc@11 272 progDenote p (expDenote (Const n) :: s)
adamc@206 273
adamc@11 274 ]]
adamc@11 275
adamc@11 276 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 277
adamc@11 278 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@11 279 *)
adamc@11 280
adamc@4 281 unfold compile.
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@206 289
adam@302 290 ]]
adam@302 291 *)
adamc@11 292
adamc@4 293 unfold expDenote.
adam@302 294
adamc@11 295 (** [[
adamc@11 296 n : nat
adamc@11 297 s : stack
adamc@11 298 p : list instr
adamc@11 299 ============================
adamc@11 300 progDenote ((IConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@206 301
adamc@11 302 ]]
adamc@11 303
adamc@11 304 We only need to unfold the first occurrence of [progDenote] to prove the goal: *)
adamc@11 305
adamc@11 306 unfold progDenote at 1.
adamc@11 307
adamc@11 308 (** [[
adamc@11 309
adamc@11 310 n : nat
adamc@11 311 s : stack
adamc@11 312 p : list instr
adamc@11 313 ============================
adamc@11 314 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 315 option stack :=
adamc@11 316 match p0 with
adamc@11 317 | nil => Some s0
adamc@11 318 | i :: p' =>
adamc@11 319 match instrDenote i s0 with
adamc@11 320 | Some s' => progDenote p' s'
adamc@11 321 | None => None (A:=stack)
adamc@11 322 end
adamc@11 323 end) ((IConst n :: nil) ++ p) s =
adamc@11 324 progDenote p (n :: s)
adamc@206 325
adamc@11 326 ]]
adamc@11 327
adamc@11 328 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@11 329 *)
adamc@11 330
adamc@4 331 simpl.
adamc@11 332 (** [[
adamc@11 333 n : nat
adamc@11 334 s : stack
adamc@11 335 p : list instr
adamc@11 336 ============================
adamc@11 337 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 338 option stack :=
adamc@11 339 match p0 with
adamc@11 340 | nil => Some s0
adamc@11 341 | i :: p' =>
adamc@11 342 match instrDenote i s0 with
adamc@11 343 | Some s' => progDenote p' s'
adamc@11 344 | None => None (A:=stack)
adamc@11 345 end
adamc@11 346 end) p (n :: s) = progDenote p (n :: s)
adamc@206 347
adamc@11 348 ]]
adamc@11 349
adamc@11 350 Now we can unexpand the definition of [progDenote]:
adamc@11 351 *)
adamc@11 352
adamc@11 353 fold progDenote.
adamc@11 354
adamc@11 355 (** [[
adamc@11 356
adamc@11 357 n : nat
adamc@11 358 s : stack
adamc@11 359 p : list instr
adamc@11 360 ============================
adamc@11 361 progDenote p (n :: s) = progDenote p (n :: s)
adamc@206 362
adamc@11 363 ]]
adamc@11 364
adamc@11 365 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@11 366 *)
adamc@11 367
adamc@4 368 reflexivity.
adamc@2 369
adamc@11 370 (** On to the second inductive case:
adamc@11 371
adamc@11 372 [[
adamc@11 373 b : binop
adamc@11 374 e1 : exp
adamc@11 375 IHe1 : forall (s : stack) (p : list instr),
adamc@11 376 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 377 e2 : exp
adamc@11 378 IHe2 : forall (s : stack) (p : list instr),
adamc@11 379 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 380 ============================
adamc@11 381 forall (s : stack) (p : list instr),
adamc@11 382 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 383 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 384
adamc@11 385 ]]
adamc@11 386
adamc@11 387 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@11 388
adamc@11 389 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@11 390
adamc@4 391 intros.
adamc@4 392 unfold compile.
adamc@4 393 fold compile.
adamc@4 394 unfold expDenote.
adamc@4 395 fold expDenote.
adamc@11 396
adamc@44 397 (** 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@11 398
adamc@11 399 [[
adamc@11 400 b : binop
adamc@11 401 e1 : exp
adamc@11 402 IHe1 : forall (s : stack) (p : list instr),
adamc@11 403 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 404 e2 : exp
adamc@11 405 IHe2 : forall (s : stack) (p : list instr),
adamc@11 406 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 407 s : stack
adamc@11 408 p : list instr
adamc@11 409 ============================
adamc@11 410 progDenote ((compile e2 ++ compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 411 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 412
adamc@11 413 ]]
adamc@11 414
adam@277 415 What we need is the associative law of list concatenation, which is available as a theorem [app_ass] in the standard library. *)
adamc@11 416
adamc@11 417 Check app_ass.
adamc@11 418 (** [[
adamc@11 419 app_ass
adamc@11 420 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@206 421
adamc@11 422 ]]
adamc@11 423
adam@277 424 If we did not already know the name of the theorem, we could use the [SearchRewrite] command to find it, based on a pattern that we would like to rewrite: *)
adam@277 425
adam@277 426 SearchRewrite ((_ ++ _) ++ _).
adam@277 427 (** [[
adam@277 428 app_ass: forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adam@277 429 ass_app: forall (A : Type) (l m n : list A), l ++ m ++ n = (l ++ m) ++ n
adam@277 430
adam@277 431 ]]
adam@277 432
adamc@11 433 We use it to perform a rewrite: *)
adamc@11 434
adamc@4 435 rewrite app_ass.
adamc@11 436
adamc@206 437 (** changing the conclusion to:
adamc@11 438
adamc@206 439 [[
adamc@11 440 progDenote (compile e2 ++ (compile e1 ++ IBinop b :: nil) ++ p) s =
adamc@11 441 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 442
adamc@11 443 ]]
adamc@11 444
adamc@11 445 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@11 446
adamc@4 447 rewrite IHe2.
adamc@11 448 (** [[
adamc@11 449 progDenote ((compile e1 ++ IBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 450 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 451
adamc@11 452 ]]
adamc@11 453
adamc@11 454 The same process lets us apply the remaining hypothesis. *)
adamc@11 455
adamc@4 456 rewrite app_ass.
adamc@4 457 rewrite IHe1.
adamc@11 458 (** [[
adamc@11 459 progDenote ((IBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 460 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 461
adamc@11 462 ]]
adamc@11 463
adamc@11 464 Now we can apply a similar sequence of tactics to that that ended the proof of the first case.
adamc@11 465 *)
adamc@11 466
adamc@11 467 unfold progDenote at 1.
adamc@4 468 simpl.
adamc@11 469 fold progDenote.
adamc@4 470 reflexivity.
adamc@11 471
adamc@11 472 (** And the proof is completed, as indicated by the message:
adamc@11 473
adamc@11 474 [[
adamc@11 475 Proof completed.
adamc@11 476
adamc@205 477 ]]
adamc@205 478
adamc@11 479 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@11 480 *)
adamc@11 481
adamc@4 482 Abort.
adamc@2 483
adamc@26 484 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 485 progDenote p (expDenote e :: s).
adamc@4 486 induction e; crush.
adamc@4 487 Qed.
adamc@2 488
adamc@11 489 (** 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@11 490
adamc@210 491 The [crush] tactic comes from the library associated with this book and is not part of the Coq standard library. The book's library contains a number of other tactics that are especially helpful in highly-automated proofs.
adamc@210 492
adamc@11 493 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@11 494
adamc@26 495 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 496 intros.
adamc@11 497 (** [[
adamc@11 498 e : exp
adamc@11 499 ============================
adamc@11 500 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@206 501
adamc@11 502 ]]
adamc@11 503
adamc@26 504 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@11 505
adamc@11 506 Check app_nil_end.
adamc@11 507 (** [[
adamc@11 508 app_nil_end
adamc@11 509 : forall (A : Type) (l : list A), l = l ++ nil
adam@302 510 ]]
adam@302 511 *)
adamc@11 512
adamc@4 513 rewrite (app_nil_end (compile e)).
adamc@11 514
adamc@11 515 (** 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@11 516
adamc@11 517 [[
adamc@11 518 e : exp
adamc@11 519 ============================
adamc@11 520 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@206 521
adamc@11 522 ]]
adamc@11 523
adamc@11 524 Now we can apply the lemma. *)
adamc@11 525
adamc@26 526 rewrite compile_correct'.
adamc@11 527 (** [[
adamc@11 528 e : exp
adamc@11 529 ============================
adamc@11 530 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@206 531
adamc@11 532 ]]
adamc@11 533
adamc@11 534 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@11 535
adamc@4 536 reflexivity.
adamc@4 537 Qed.
adamc@22 538 (* end thide *)
adamc@14 539
adamc@14 540
adamc@20 541 (** * Typed Expressions *)
adamc@14 542
adamc@14 543 (** 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@14 544
adamc@20 545 (** ** Source Language *)
adamc@14 546
adamc@15 547 (** We define a trivial language of types to classify our expressions: *)
adamc@15 548
adamc@14 549 Inductive type : Set := Nat | Bool.
adamc@14 550
adam@277 551 (** Like most programming languages, Coq uses case-sensitive variable names, so that our user-defined type [type] is distinct from the [Type] keyword that we have already seen appear in the statement of a polymorphic theorem (and that we will meet in more detail later), and our constructor names [Nat] and [Bool] are distinct from the types [nat] and [bool] in the standard library.
adam@277 552
adam@277 553 Now we define an expanded set of binary operators. *)
adamc@15 554
adamc@14 555 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 556 | TPlus : tbinop Nat Nat Nat
adamc@14 557 | TTimes : tbinop Nat Nat Nat
adamc@14 558 | TEq : forall t, tbinop t t Bool
adamc@14 559 | TLt : tbinop Nat Nat Bool.
adamc@14 560
adam@307 561 (** 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 %\emph{%#<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@15 562
adamc@15 563 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@15 564
adam@307 565 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]. %\emph{%#<i>#Generalized algebraic datatypes (GADTs)#</i>#%}% are a popular feature in GHC Haskell and other languages that removes this first restriction.
adamc@15 566
adam@307 567 The second restriction is not lifted by GADTs. In ML and Haskell, indices of types must be types and may not be %\emph{%#<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@15 568 *)
adamc@15 569
adamc@15 570 (** We can define a similar type family for typed expressions. *)
adamc@15 571
adamc@14 572 Inductive texp : type -> Set :=
adamc@14 573 | TNConst : nat -> texp Nat
adamc@14 574 | TBConst : bool -> texp Bool
adamc@14 575 | TBinop : forall arg1 arg2 res, tbinop arg1 arg2 res -> texp arg1 -> texp arg2 -> texp res.
adamc@14 576
adamc@15 577 (** 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@15 578
adamc@14 579 Definition typeDenote (t : type) : Set :=
adamc@14 580 match t with
adamc@14 581 | Nat => nat
adamc@14 582 | Bool => bool
adamc@14 583 end.
adamc@14 584
adamc@15 585 (** 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@15 586
adam@292 587 We need to define one auxiliary function, implementing a boolean binary %``%#"#less-than#"#%''% operator, which only appears in the standard library with a type fancier than what we are prepared to deal with here. The code is 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@15 588 *)
adamc@15 589
adam@279 590 Fixpoint lessThan (n1 n2 : nat) : bool :=
adamc@14 591 match n1, n2 with
adamc@14 592 | O, S _ => true
adam@279 593 | S n1', S n2' => lessThan n1' n2'
adamc@14 594 | _, _ => false
adamc@14 595 end.
adamc@14 596
adam@277 597 (** Now we can interpret binary operators, relying on standard-library equality test functions [eqb] and [beq_nat] for booleans and naturals, respectively: *)
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@207 601 match b in (tbinop arg1 arg2 res)
adamc@207 602 return (typeDenote arg1 -> typeDenote arg2 -> typeDenote res) with
adamc@14 603 | TPlus => plus
adamc@14 604 | TTimes => mult
adam@277 605 | TEq Nat => beq_nat
adam@277 606 | TEq Bool => eqb
adam@279 607 | TLt => lessThan
adamc@14 608 end.
adamc@14 609
adam@307 610 (** 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 %\emph{%#<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, so it is often necessary to write annotations, like we see above on the line with [match].
adamc@15 611
adam@307 612 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 %\emph{%#<i>#binding occurrences#</i>#%}%. Their scope is the [return] clause. That is, [arg1], [arg2], and [res] 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@15 613
adamc@207 614 In fact, recent Coq versions use some heuristics that can save us the trouble of writing [match] annotations, and those heuristics get the job done in this case. We can get away with writing just: *)
adamc@207 615
adamc@207 616 (* begin hide *)
adamc@207 617 Reset tbinopDenote.
adamc@207 618 (* end hide *)
adamc@207 619 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@207 620 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@207 621 match b with
adamc@207 622 | TPlus => plus
adamc@207 623 | TTimes => mult
adam@277 624 | TEq Nat => beq_nat
adam@277 625 | TEq Bool => eqb
adam@279 626 | TLt => lessThan
adamc@207 627 end.
adamc@207 628
adamc@207 629 (**
adamc@15 630 The same tricks suffice to define an expression denotation function in an unsurprising way:
adamc@15 631 *)
adamc@15 632
adamc@207 633 Fixpoint texpDenote t (e : texp t) : typeDenote t :=
adamc@207 634 match e with
adamc@14 635 | TNConst n => n
adamc@14 636 | TBConst b => b
adamc@14 637 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 638 end.
adamc@14 639
adamc@17 640 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 641
adamc@17 642 Eval simpl in texpDenote (TNConst 42).
adamc@207 643 (** [= 42 : typeDenote Nat] *)
adamc@207 644
adamc@17 645 Eval simpl in texpDenote (TBConst true).
adamc@207 646 (** [= true : typeDenote Bool] *)
adamc@207 647
adamc@17 648 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@207 649 (** [= 28 : typeDenote Nat] *)
adamc@207 650
adamc@17 651 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@207 652 (** [= false : typeDenote Bool] *)
adamc@207 653
adamc@17 654 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
adamc@207 655 (** [= true : typeDenote Bool] *)
adamc@17 656
adamc@14 657
adamc@20 658 (** ** Target Language *)
adamc@14 659
adam@292 660 (** 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@18 661
adamc@18 662 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@18 663
adamc@18 664 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 665
adamc@14 666 Definition tstack := list type.
adamc@14 667
adamc@18 668 (** 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@18 669
adamc@18 670 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@18 671
adamc@14 672 Inductive tinstr : tstack -> tstack -> Set :=
adamc@14 673 | TINConst : forall s, nat -> tinstr s (Nat :: s)
adamc@14 674 | TIBConst : forall s, bool -> tinstr s (Bool :: s)
adamc@14 675 | TIBinop : forall arg1 arg2 res s,
adamc@14 676 tbinop arg1 arg2 res
adamc@14 677 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 678
adamc@18 679 (** 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@18 680
adamc@14 681 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 682 | TNil : forall s, tprog s s
adamc@14 683 | TCons : forall s1 s2 s3,
adamc@14 684 tinstr s1 s2
adamc@14 685 -> tprog s2 s3
adamc@14 686 -> tprog s1 s3.
adamc@14 687
adamc@18 688 (** 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@18 689
adamc@14 690 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 691 match ts with
adamc@14 692 | nil => unit
adamc@14 693 | t :: ts' => typeDenote t * vstack ts'
adamc@14 694 end%type.
adamc@14 695
adamc@210 696 (** 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. We write [%type] so that Coq knows to interpret [*] as Cartesian product rather than multiplication.
adamc@18 697
adamc@207 698 This idea of programming with types can take a while to internalize, but it enables a very simple definition of instruction denotation. Our definition 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@18 699
adamc@14 700 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@207 701 match i with
adamc@14 702 | TINConst _ n => fun s => (n, s)
adamc@14 703 | TIBConst _ b => fun s => (b, s)
adamc@14 704 | TIBinop _ _ _ _ b => fun s =>
adamc@14 705 match s with
adamc@14 706 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@14 707 end
adamc@14 708 end.
adamc@14 709
adamc@18 710 (** 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@18 711
adamc@18 712 [[
adamc@18 713 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@207 714 match i with
adamc@18 715 | TINConst _ n => (n, s)
adamc@18 716 | TIBConst _ b => (b, s)
adamc@18 717 | TIBinop _ _ _ _ b =>
adamc@18 718 match s with
adamc@18 719 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@18 720 end
adamc@18 721 end.
adamc@18 722
adamc@205 723 ]]
adamc@205 724
adamc@18 725 The Coq type-checker complains that:
adamc@18 726
adamc@18 727 [[
adamc@18 728 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@207 729 while it is expected to have type "vstack ?119".
adamc@207 730
adamc@207 731 ]]
adamc@207 732
adamc@207 733 The text [?119] stands for a unification variable. We can try to help Coq figure out the value of this variable with an explicit annotation on our [match] expression.
adamc@207 734
adamc@207 735 [[
adamc@207 736 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@207 737 match i in tinstr ts ts' return vstack ts' with
adamc@207 738 | TINConst _ n => (n, s)
adamc@207 739 | TIBConst _ b => (b, s)
adamc@207 740 | TIBinop _ _ _ _ b =>
adamc@207 741 match s with
adamc@207 742 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
adamc@207 743 end
adamc@207 744 end.
adamc@207 745
adamc@207 746 ]]
adamc@207 747
adamc@207 748 Now the error message changes.
adamc@207 749
adamc@207 750 [[
adamc@207 751 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@207 752 while it is expected to have type "vstack (Nat :: t)".
adamc@207 753
adamc@18 754 ]]
adamc@18 755
adamc@18 756 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@18 757
adam@307 758 There %\emph{%#<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@18 759
adamc@18 760 *)
adamc@18 761
adamc@18 762 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 763
adamc@207 764 Fixpoint tprogDenote ts ts' (p : tprog ts ts') : vstack ts -> vstack ts' :=
adamc@207 765 match p with
adamc@14 766 | TNil _ => fun s => s
adamc@14 767 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 768 end.
adamc@14 769
adamc@14 770
adamc@14 771 (** ** Translation *)
adamc@14 772
adamc@19 773 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 774
adamc@207 775 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') : tprog ts' ts'' -> tprog ts ts'' :=
adamc@207 776 match p with
adamc@14 777 | TNil _ => fun p' => p'
adamc@14 778 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 779 end.
adamc@14 780
adamc@19 781 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 782
adamc@207 783 Fixpoint tcompile t (e : texp t) (ts : tstack) : tprog ts (t :: ts) :=
adamc@207 784 match e with
adamc@14 785 | TNConst n => TCons (TINConst _ n) (TNil _)
adamc@14 786 | TBConst b => TCons (TIBConst _ b) (TNil _)
adamc@14 787 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adamc@14 788 (tconcat (tcompile e1 _) (TCons (TIBinop _ b) (TNil _)))
adamc@14 789 end.
adamc@14 790
adam@307 791 (** 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 %\emph{%#<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@19 792
adamc@19 793 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@19 794
adamc@14 795 Print tcompile.
adamc@19 796 (** [[
adamc@19 797 tcompile =
adamc@19 798 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 799 tprog ts (t :: ts) :=
adamc@19 800 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adamc@19 801 | TNConst n => TCons (TINConst ts n) (TNil (Nat :: ts))
adamc@19 802 | TBConst b => TCons (TIBConst ts b) (TNil (Bool :: ts))
adamc@19 803 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 804 tconcat (tcompile arg2 e2 ts)
adamc@19 805 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adamc@19 806 (TCons (TIBinop ts b) (TNil (res :: ts))))
adamc@19 807 end
adamc@19 808 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adam@302 809 ]]
adam@302 810 *)
adamc@19 811
adamc@19 812
adamc@19 813 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 814
adamc@19 815 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adamc@207 816 (** [= (42, tt) : vstack (Nat :: nil)] *)
adamc@207 817
adamc@19 818 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adamc@207 819 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@207 820
adamc@19 821 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@207 822 (** [= (28, tt) : vstack (Nat :: nil)] *)
adamc@207 823
adamc@19 824 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@207 825 (** [= (false, tt) : vstack (Bool :: nil)] *)
adamc@207 826
adamc@19 827 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
adamc@207 828 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@19 829
adamc@14 830
adamc@20 831 (** ** Translation Correctness *)
adamc@20 832
adamc@20 833 (** We can state a correctness theorem similar to the last one. *)
adamc@20 834
adamc@207 835 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 836 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 837 (* begin hide *)
adamc@20 838 Abort.
adamc@20 839 (* end hide *)
adamc@22 840 (* begin thide *)
adamc@20 841
adamc@20 842 (** 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@14 843
adamc@207 844 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 845 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@20 846
adam@292 847 (** 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@20 848
adamc@20 849 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 850
adamc@14 851 induction e; crush.
adamc@20 852
adamc@20 853 (** We are left with this unproved conclusion:
adamc@20 854
adamc@20 855 [[
adamc@20 856 tprogDenote
adamc@20 857 (tconcat (tcompile e2 ts)
adamc@20 858 (tconcat (tcompile e1 (arg2 :: ts))
adamc@20 859 (TCons (TIBinop ts t) (TNil (res :: ts))))) s =
adamc@20 860 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@207 861
adamc@20 862 ]]
adamc@20 863
adamc@20 864 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@20 865 *)
adamc@207 866
adamc@14 867 Abort.
adamc@14 868
adamc@26 869 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 870 (s : vstack ts),
adamc@14 871 tprogDenote (tconcat p p') s
adamc@14 872 = tprogDenote p' (tprogDenote p s).
adamc@14 873 induction p; crush.
adamc@14 874 Qed.
adamc@14 875
adamc@20 876 (** This one goes through completely automatically.
adamc@20 877
adamc@26 878 Some code behind the scenes registers [app_ass] for use by [crush]. We must register [tconcat_correct] similarly to get the same effect: *)
adamc@20 879
adamc@26 880 Hint Rewrite tconcat_correct : cpdt.
adamc@14 881
adamc@26 882 (** 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@20 883
adamc@207 884 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 885 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@14 886 induction e; crush.
adamc@14 887 Qed.
adamc@14 888
adamc@20 889 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 890
adamc@26 891 Hint Rewrite tcompile_correct' : cpdt.
adamc@14 892
adamc@207 893 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 894 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 895 crush.
adamc@14 896 Qed.
adamc@22 897 (* end thide *)