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Fix HTML error on index page
author Adam Chlipala <adam@chlipala.net>
date Fri, 13 Dec 2013 10:45:46 -0500
parents a4b3386ae140
children ed829eaa91b2
rev   line source
adam@398 1 (* Copyright (c) 2008-2012, 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@2 10
adamc@25 11 (** %\chapter{Some Quick Examples}% *)
adamc@25 12
adam@447 13 (** 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. In other words, it is expected that most readers will not understand what exactly is going on here, but I hope this demo will whet your appetite for the remaining chapters!
adam@279 14
adam@419 15 As always, you can step through the source file <<StackMachine.v>> 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 <<.v>> file in an Emacs buffer. If you do the latter, include these two lines at the start of the file. *)
adam@314 16
adam@419 17 Require Import Bool Arith List CpdtTactics.
adam@419 18 Set Implicit Arguments.
adam@314 19
adam@513 20 (* begin hide *)
adam@513 21 (* begin thide *)
adam@513 22 Definition bleh := app_assoc.
adam@513 23 (* end thide *)
adam@513 24 (* end hide *)
adam@513 25
adam@419 26 (** 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 the above two lines, with the import list tweaked as appropriate, considering which definitions the chapter uses. The second command above affects the default behavior of definitions regarding type inference. *)
adamc@9 27
adamc@9 28
adamc@20 29 (** * Arithmetic Expressions Over Natural Numbers *)
adamc@2 30
adamc@40 31 (** We will begin with that staple of compiler textbooks, arithmetic expressions over a single type of numbers. *)
adamc@9 32
adamc@20 33 (** ** Source Language *)
adamc@9 34
adam@311 35 (** We begin with the syntax of the source language.%\index{Vernacular commands!Inductive}% *)
adamc@2 36
adamc@4 37 Inductive binop : Set := Plus | Times.
adamc@2 38
adam@447 39 (** Our first line of Coq code should be unsurprising to ML and Haskell programmers. We define an %\index{algebraic datatypes}%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 <<data>>, <<datatype>>, or <<type>>. 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 %\index{Gallina terms!Set}%[: 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 40
adamc@4 41 Inductive exp : Set :=
adamc@4 42 | Const : nat -> exp
adamc@4 43 | Binop : binop -> exp -> exp -> exp.
adamc@2 44
adamc@9 45 (** 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 46
adam@419 47 A note for readers following along in the PDF version: %\index{coqdoc}%coqdoc supports pretty-printing of tokens in %\LaTeX{}%#LaTeX# or HTML. Where you see a right arrow character, the source contains the ASCII text <<->>>. Other examples of this substitution appearing in this chapter are a double right arrow for <<=>>>, the inverted %`%#'#A' symbol for <<forall>>, and the Cartesian product %`%#'#X' for <<*>>. When in doubt about the ASCII version of a symbol, you can consult the chapter source code.
adamc@9 48
adamc@9 49 %\medskip%
adamc@9 50
adam@475 51 Now we are ready to say what programs in our expression language mean. We will do this by writing an %\index{interpreters}%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.)%\index{Vernacular commands!Definition}% *)
adamc@9 52
adamc@4 53 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 54 match b with
adamc@4 55 | Plus => plus
adamc@4 56 | Times => mult
adamc@4 57 end.
adamc@2 58
adam@419 59 (** The meaning of a binary operator is a binary function over naturals, defined with pattern-matching notation analogous to the <<case>> and <<match>> 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 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@205 66 ]]
adamc@205 67
adamc@9 68 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 69 [[
adamc@9 70 Definition binopDenote := fun b =>
adamc@9 71 match b with
adamc@9 72 | Plus => plus
adamc@9 73 | Times => mult
adamc@9 74 end.
adamc@205 75 ]]
adamc@205 76
adam@475 77 Languages like Haskell and ML have a convenient%\index{principal types}\index{type inference}% _principal types_ 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 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 78
adam@475 79 This is as good a time as any to mention the profusion of different languages associated with Coq. The theoretical foundation of Coq is a formal system called the%\index{Calculus of Inductive Constructions}\index{CIC|see{Calculus of Inductive Constructions}}% _Calculus of Inductive Constructions_ (CIC)%~\cite{CIC}%, which is an extension of the older%\index{Calculus of Constructions}\index{CoC|see{Calculus of Constructions}}% _Calculus of Constructions_ (CoC)%~\cite{CoC}%. 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%\index{strong normalization}% _strong normalization_ %\cite{CIC}%, meaning that every program (and, more importantly, every proof term) terminates; and%\index{relative consistency}% _relative consistency_ %\cite{SetsInTypes}% with systems like versions of %\index{Zermelo-Fraenkel set theory}%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 80
adam@475 81 Coq is actually based on an extension of CIC called %\index{Gallina}%Gallina. The text after the [:=] and before the period in the last code example is a term of Gallina. Gallina includes several useful features that must be considered as extensions to CIC. The important metatheorems about CIC have not been extended to the full breadth of the features that go beyond the formalized language, but most Coq users do not seem to lose much sleep over this omission.
adamc@9 82
adam@475 83 Next, there is %\index{Ltac}%Ltac, 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 84
adam@475 85 Finally, commands like [Inductive] and [Definition] are part of %\index{Vernacular commands}%the Vernacular, 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 _trees_ of vernacular commands, thanks to various nested scoping constructs.)
adamc@9 86
adamc@9 87 %\medskip%
adamc@9 88
adam@311 89 We can give a simple definition of the meaning of an expression:%\index{Vernacular commands!Fixpoint}% *)
adamc@9 90
adamc@4 91 Fixpoint expDenote (e : exp) : nat :=
adamc@4 92 match e with
adamc@4 93 | Const n => n
adamc@4 94 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 95 end.
adamc@2 96
adamc@9 97 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 98
adam@419 99 (** 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, using the command %\index{Vernacular commands!Eval}%[Eval]. This command takes an argument expressing a%\index{reduction strategy}% _reduction strategy_, or an "order of evaluation." Unlike with ML, which hardcodes an _eager_ reduction strategy, or Haskell, which hardcodes a _lazy_ strategy, in Coq we are free to choose between these and many other orders of evaluation, because all Coq programs terminate. In fact, Coq silently checked %\index{termination checking}%termination of our [Fixpoint] definition above, using a simple heuristic based on monotonically decreasing size of arguments across recursive calls. Specifically, recursive calls must be made on arguments that were pulled out of the original recursive argument with [match] expressions. (In Chapter 7, we will see some ways of getting around this restriction, though simply removing the restriction would leave Coq useless as a theorem proving tool, for reasons we will start to learn about in the next chapter.)
adam@311 100
adam@311 101 To return to our test evaluations, we run the [Eval] command using the [simpl] evaluation strategy, whose definition is best postponed until we have learned more about Coq's foundations, but which usually gets the job done. *)
adamc@16 102
adamc@16 103 Eval simpl in expDenote (Const 42).
adamc@205 104 (** [= 42 : nat] *)
adamc@205 105
adamc@16 106 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
adamc@205 107 (** [= 4 : nat] *)
adamc@205 108
adamc@16 109 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@205 110 (** [= 28 : nat] *)
adamc@9 111
adam@442 112 (** %\smallskip{}%Nothing too surprising goes on here, so we are ready to move on to the target language of our compiler. *)
adam@442 113
adam@442 114
adamc@20 115 (** ** Target Language *)
adamc@4 116
adamc@10 117 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 118
adamc@4 119 Inductive instr : Set :=
adam@311 120 | iConst : nat -> instr
adam@311 121 | iBinop : binop -> instr.
adamc@2 122
adamc@4 123 Definition prog := list instr.
adamc@4 124 Definition stack := list nat.
adamc@2 125
adamc@10 126 (** 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@10 127
adam@419 128 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']. %\index{Gallina operators!::}%The infix operator [::] is "list cons" from the Coq standard library.%\index{Gallina terms!option}% *)
adamc@10 129
adamc@4 130 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 131 match i with
adam@311 132 | iConst n => Some (n :: s)
adam@311 133 | iBinop b =>
adamc@4 134 match s with
adamc@4 135 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 136 | _ => None
adamc@4 137 end
adamc@4 138 end.
adamc@2 139
adam@311 140 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@206 141
adamc@206 142 Fixpoint progDenote (p : prog) (s : stack) : option stack :=
adamc@206 143 match p with
adamc@206 144 | nil => Some s
adamc@206 145 | i :: p' =>
adamc@206 146 match instrDenote i s with
adamc@206 147 | None => None
adamc@206 148 | Some s' => progDenote p' s'
adamc@206 149 end
adamc@206 150 end.
adamc@2 151
adam@442 152 (** With the two programming languages defined, we can turn to the compiler definition. *)
adam@442 153
adamc@4 154
adamc@9 155 (** ** Translation *)
adamc@4 156
adam@471 157 (** Our compiler itself is now unsurprising. The list concatenation operator %\index{Gallina operators!++}\coqdocnotation{%#<tt>#++#</tt>#%}% comes from the Coq standard library. *)
adamc@2 158
adamc@4 159 Fixpoint compile (e : exp) : prog :=
adamc@4 160 match e with
adam@311 161 | Const n => iConst n :: nil
adam@311 162 | Binop b e1 e2 => compile e2 ++ compile e1 ++ iBinop b :: nil
adamc@4 163 end.
adamc@2 164
adamc@2 165
adamc@16 166 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 167
adamc@16 168 Eval simpl in compile (Const 42).
adam@311 169 (** [= iConst 42 :: nil : prog] *)
adamc@206 170
adamc@16 171 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adam@311 172 (** [= iConst 2 :: iConst 2 :: iBinop Plus :: nil : prog] *)
adamc@206 173
adamc@16 174 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adam@311 175 (** [= iConst 7 :: iConst 2 :: iConst 2 :: iBinop Plus :: iBinop Times :: nil : prog] *)
adamc@16 176
adam@442 177 (** %\smallskip{}%We can also run our compiled programs and check that they give the right results. *)
adamc@16 178
adamc@16 179 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@206 180 (** [= Some (42 :: nil) : option stack] *)
adamc@206 181
adamc@16 182 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@206 183 (** [= Some (4 :: nil) : option stack] *)
adamc@206 184
adam@311 185 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2))
adam@311 186 (Const 7))) nil.
adamc@206 187 (** [= Some (28 :: nil) : option stack] *)
adamc@16 188
adam@447 189 (** %\smallskip{}%So far so good, but how can we be sure the compiler operates correctly for _all_ input programs? *)
adamc@16 190
adamc@20 191 (** ** Translation Correctness *)
adamc@4 192
adam@311 193 (** 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:%\index{Vernacular commands!Theorem}% *)
adamc@11 194
adamc@26 195 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@22 196 (* begin thide *)
adamc@11 197
adam@419 198 (** 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%\index{strengthening the induction hypothesis}% _strengthening the induction hypothesis_. We do that by proving an auxiliary lemma, using the command [Lemma] that is a synonym for [Theorem], conventionally used for less important theorems that appear in the proofs of primary theorems.%\index{Vernacular commands!Lemma}% *)
adamc@2 199
adam@469 200 Abort.
adam@469 201
adamc@206 202 Lemma compile_correct' : forall e p s,
adamc@206 203 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 204
adam@399 205 (** After the period in the [Lemma] command, we are in%\index{interactive proof-editing mode}% _the interactive proof-editing mode_. We find ourselves staring at this ominous screen of text:
adamc@11 206
adamc@11 207 [[
adamc@11 208 1 subgoal
adamc@11 209
adamc@11 210 ============================
adamc@15 211 forall (e : exp) (p : list instr) (s : stack),
adamc@15 212 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@206 213
adamc@11 214 ]]
adamc@11 215
adam@311 216 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 %\index{subgoals}%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@11 217
adam@311 218 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 %\index{hypotheses}%hypotheses, if we had any. Below the line is the %\index{conclusion}%conclusion, which, in general, is to be proved from the hypotheses.
adamc@11 219
adam@399 220 We manipulate the proof state by running commands called%\index{tactics}% _tactics_. Let us start out by running one of the most important tactics:%\index{tactics!induction}%
adamc@11 221 *)
adamc@11 222
adamc@4 223 induction e.
adamc@2 224
adamc@11 225 (** 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@11 226
adam@439 227 [[
adam@439 228 2 subgoals
adam@311 229
adamc@11 230 n : nat
adamc@11 231 ============================
adamc@11 232 forall (s : stack) (p : list instr),
adamc@11 233 progDenote (compile (Const n) ++ p) s =
adamc@11 234 progDenote p (expDenote (Const n) :: s)
adam@439 235
adam@439 236 subgoal 2 is
adam@439 237
adamc@11 238 forall (s : stack) (p : list instr),
adamc@11 239 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 240 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 241
adamc@11 242 ]]
adamc@11 243
adam@311 244 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 %\index{free variable}%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 %\index{structural induction}%structural induction.
adamc@11 245
adam@311 246 We begin the first case with another very common tactic.%\index{tactics!intros}%
adamc@11 247 *)
adamc@11 248
adamc@4 249 intros.
adamc@11 250
adamc@11 251 (** The current subgoal changes to:
adamc@11 252 [[
adamc@11 253
adamc@11 254 n : nat
adamc@11 255 s : stack
adamc@11 256 p : list instr
adamc@11 257 ============================
adamc@11 258 progDenote (compile (Const n) ++ p) s =
adamc@11 259 progDenote p (expDenote (Const n) :: s)
adamc@206 260
adamc@11 261 ]]
adamc@11 262
adamc@11 263 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 264
adam@311 265 To progress further, we need to use the definitions of some of the functions appearing in the goal. The [unfold] tactic replaces an identifier with its definition.%\index{tactics!unfold}%
adamc@11 266 *)
adamc@11 267
adamc@4 268 unfold compile.
adamc@11 269 (** [[
adamc@11 270 n : nat
adamc@11 271 s : stack
adamc@11 272 p : list instr
adamc@11 273 ============================
adam@311 274 progDenote ((iConst n :: nil) ++ p) s =
adamc@11 275 progDenote p (expDenote (Const n) :: s)
adamc@206 276
adam@302 277 ]]
adam@302 278 *)
adamc@11 279
adamc@4 280 unfold expDenote.
adamc@11 281 (** [[
adamc@11 282 n : nat
adamc@11 283 s : stack
adamc@11 284 p : list instr
adamc@11 285 ============================
adam@311 286 progDenote ((iConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@206 287
adamc@11 288 ]]
adamc@11 289
adam@311 290 We only need to unfold the first occurrence of [progDenote] to prove the goal. An [at] clause used with [unfold] specifies a particular occurrence of an identifier to unfold, where we count occurrences from left to right.%\index{tactics!unfold}% *)
adamc@11 291
adamc@11 292 unfold progDenote at 1.
adamc@11 293 (** [[
adamc@11 294 n : nat
adamc@11 295 s : stack
adamc@11 296 p : list instr
adamc@11 297 ============================
adamc@11 298 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 299 option stack :=
adamc@11 300 match p0 with
adamc@11 301 | nil => Some s0
adamc@11 302 | i :: p' =>
adamc@11 303 match instrDenote i s0 with
adamc@11 304 | Some s' => progDenote p' s'
adamc@11 305 | None => None (A:=stack)
adamc@11 306 end
adam@311 307 end) ((iConst n :: nil) ++ p) s =
adamc@11 308 progDenote p (n :: s)
adamc@206 309
adamc@11 310 ]]
adamc@11 311
adam@471 312 This last [unfold] has left us with an anonymous recursive definition of [progDenote] (similarly to how [fun] or "lambda" constructs in general allow anonymous non-recursive functions), which will generally happen when unfolding recursive definitions. Note that Coq has automatically renamed the [fix] arguments [p] and [s] to [p0] and [s0], to avoid clashes with our local free variables. There is also a subterm [None (A:=stack)], which has an annotation specifying that the type of the term ought to be [option stack]. This is phrased as an explicit instantiation of a named type parameter [A] from the definition of [option].
adam@311 313
adam@311 314 Fortunately, in this case, we can eliminate the complications of anonymous recursion 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, which applies the same reduction strategy that we used earlier with [Eval] (and whose details we still postpone).%\index{tactics!simpl}%
adamc@11 315 *)
adamc@11 316
adamc@4 317 simpl.
adamc@11 318 (** [[
adamc@11 319 n : nat
adamc@11 320 s : stack
adamc@11 321 p : list instr
adamc@11 322 ============================
adamc@11 323 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 324 option stack :=
adamc@11 325 match p0 with
adamc@11 326 | nil => Some s0
adamc@11 327 | i :: p' =>
adamc@11 328 match instrDenote i s0 with
adamc@11 329 | Some s' => progDenote p' s'
adamc@11 330 | None => None (A:=stack)
adamc@11 331 end
adamc@11 332 end) p (n :: s) = progDenote p (n :: s)
adamc@206 333
adamc@11 334 ]]
adamc@11 335
adam@311 336 Now we can unexpand the definition of [progDenote]:%\index{tactics!fold}%
adamc@11 337 *)
adamc@11 338
adamc@11 339 fold progDenote.
adamc@11 340 (** [[
adamc@11 341 n : nat
adamc@11 342 s : stack
adamc@11 343 p : list instr
adamc@11 344 ============================
adamc@11 345 progDenote p (n :: s) = progDenote p (n :: s)
adamc@206 346
adamc@11 347 ]]
adamc@11 348
adam@311 349 It looks like we are at the end of this case, since we have a trivial equality. Indeed, a single tactic finishes us off:%\index{tactics!reflexivity}%
adamc@11 350 *)
adamc@11 351
adamc@4 352 reflexivity.
adamc@2 353
adamc@11 354 (** On to the second inductive case:
adamc@11 355
adamc@11 356 [[
adamc@11 357 b : binop
adamc@11 358 e1 : exp
adamc@11 359 IHe1 : forall (s : stack) (p : list instr),
adamc@11 360 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 361 e2 : exp
adamc@11 362 IHe2 : forall (s : stack) (p : list instr),
adamc@11 363 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 364 ============================
adamc@11 365 forall (s : stack) (p : list instr),
adamc@11 366 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 367 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 368
adamc@11 369 ]]
adamc@11 370
adam@311 371 We see our first example of %\index{hypotheses}%hypotheses above the double-dashed line. They are the inductive hypotheses [IHe1] and [IHe2] corresponding to the subterms [e1] and [e2], respectively.
adamc@11 372
adam@399 373 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. %\index{tactics!intros}\index{tactics!unfold}\index{tactics!fold}% *)
adamc@11 374
adamc@4 375 intros.
adamc@4 376 unfold compile.
adamc@4 377 fold compile.
adamc@4 378 unfold expDenote.
adamc@4 379 fold expDenote.
adamc@11 380
adamc@44 381 (** 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@11 382
adamc@11 383 [[
adamc@11 384 b : binop
adamc@11 385 e1 : exp
adamc@11 386 IHe1 : forall (s : stack) (p : list instr),
adamc@11 387 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 388 e2 : exp
adamc@11 389 IHe2 : forall (s : stack) (p : list instr),
adamc@11 390 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 391 s : stack
adamc@11 392 p : list instr
adamc@11 393 ============================
adam@311 394 progDenote ((compile e2 ++ compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 395 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 396
adamc@11 397 ]]
adamc@11 398
adam@471 399 What we need is the associative law of list concatenation, which is available as a theorem [app_assoc_reverse] in the standard library.%\index{Vernacular commands!Check}% (Here and elsewhere, it is possible to tell the difference between inputs and outputs to Coq by periods at the ends of the inputs.) *)
adamc@11 400
adam@469 401 Check app_assoc_reverse.
adam@439 402 (** %\vspace{-.15in}%[[
adam@311 403 app_assoc_reverse
adamc@11 404 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@206 405
adamc@11 406 ]]
adamc@11 407
adam@399 408 If we did not already know the name of the theorem, we could use the %\index{Vernacular commands!SearchRewrite}%[SearchRewrite] command to find it, based on a pattern that we would like to rewrite: *)
adam@277 409
adam@277 410 SearchRewrite ((_ ++ _) ++ _).
adam@439 411 (** %\vspace{-.15in}%[[
adam@311 412 app_assoc_reverse:
adam@311 413 forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adam@311 414 ]]
adam@311 415 %\vspace{-.25in}%
adam@311 416 [[
adam@311 417 app_assoc: forall (A : Type) (l m n : list A), l ++ m ++ n = (l ++ m) ++ n
adam@277 418
adam@277 419 ]]
adam@277 420
adam@311 421 We use [app_assoc_reverse] to perform a rewrite: %\index{tactics!rewrite}% *)
adamc@11 422
adam@311 423 rewrite app_assoc_reverse.
adamc@11 424
adam@439 425 (** %\noindent{}%changing the conclusion to:
adamc@11 426
adamc@206 427 [[
adam@311 428 progDenote (compile e2 ++ (compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 429 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 430
adamc@11 431 ]]
adamc@11 432
adam@311 433 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.%\index{tactics!rewrite}% *)
adamc@11 434
adamc@4 435 rewrite IHe2.
adamc@11 436 (** [[
adam@311 437 progDenote ((compile e1 ++ iBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 438 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 439
adamc@11 440 ]]
adamc@11 441
adam@311 442 The same process lets us apply the remaining hypothesis.%\index{tactics!rewrite}% *)
adamc@11 443
adam@311 444 rewrite app_assoc_reverse.
adamc@4 445 rewrite IHe1.
adamc@11 446 (** [[
adam@311 447 progDenote ((iBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 448 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 449
adamc@11 450 ]]
adamc@11 451
adam@311 452 Now we can apply a similar sequence of tactics to the one that ended the proof of the first case.%\index{tactics!unfold}\index{tactics!simpl}\index{tactics!fold}\index{tactics!reflexivity}%
adamc@11 453 *)
adamc@11 454
adamc@11 455 unfold progDenote at 1.
adamc@4 456 simpl.
adamc@11 457 fold progDenote.
adamc@4 458 reflexivity.
adamc@11 459
adam@311 460 (** And the proof is completed, as indicated by the message: *)
adamc@11 461
adam@399 462 (**
adam@399 463 <<
adam@399 464 Proof completed.
adam@399 465 >>
adam@399 466 *)
adamc@11 467
adam@311 468 (** 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.%\index{Vernacular commands!Abort}%
adamc@11 469 *)
adamc@11 470
adamc@4 471 Abort.
adamc@2 472
adam@311 473 (** %\index{tactics!induction}\index{tactics!crush}% *)
adam@311 474
adamc@26 475 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 476 progDenote p (expDenote e :: s).
adamc@4 477 induction e; crush.
adamc@4 478 Qed.
adamc@2 479
adam@328 480 (** 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 %\index{tactics!semicolon}%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@11 481
adam@399 482 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@210 483
adam@398 484 The %\index{Vernacular commands!Qed}%[Qed] command checks that the proof is finished and, if so, saves it. The tactic commands we have written above are an example of a _proof script_, or a series of Ltac programs; while [Qed] uses the result of the script to generate a _proof term_, a well-typed term of Gallina. To believe that a theorem is true, we only need to trust that the (relatively simple) checker for proof terms is correct; the use of proof scripts is immaterial. Part I of this book will introduce the principles behind encoding all proofs as terms of Gallina.
adam@311 485
adam@311 486 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.%\index{tactics!intros}% *)
adamc@11 487
adamc@26 488 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 489 intros.
adamc@11 490 (** [[
adamc@11 491 e : exp
adamc@11 492 ============================
adamc@11 493 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@206 494
adamc@11 495 ]]
adamc@11 496
adamc@26 497 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@11 498
adamc@11 499 Check app_nil_end.
adamc@11 500 (** [[
adamc@11 501 app_nil_end
adamc@11 502 : forall (A : Type) (l : list A), l = l ++ nil
adam@302 503 ]]
adam@311 504 %\index{tactics!rewrite}% *)
adamc@11 505
adamc@4 506 rewrite (app_nil_end (compile e)).
adamc@11 507
adam@417 508 (** 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. The [rewrite] tactic might choose the wrong place to rewrite if we did not specify which we want.
adamc@11 509
adamc@11 510 [[
adamc@11 511 e : exp
adamc@11 512 ============================
adamc@11 513 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@206 514
adamc@11 515 ]]
adamc@11 516
adam@311 517 Now we can apply the lemma.%\index{tactics!rewrite}% *)
adamc@11 518
adamc@26 519 rewrite compile_correct'.
adamc@11 520 (** [[
adamc@11 521 e : exp
adamc@11 522 ============================
adamc@11 523 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@206 524
adamc@11 525 ]]
adamc@11 526
adam@311 527 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 %\index{tactics!reflexivity}%[reflexivity] does the normalization and checks that the two results are syntactically equal.%\index{tactics!reflexivity}% *)
adamc@11 528
adamc@4 529 reflexivity.
adamc@4 530 Qed.
adamc@22 531 (* end thide *)
adamc@14 532
adam@475 533 (** This proof can be shortened and automated, but we leave that task as an exercise for the reader. *)
adam@311 534
adamc@14 535
adamc@20 536 (** * Typed Expressions *)
adamc@14 537
adamc@14 538 (** 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@14 539
adamc@20 540 (** ** Source Language *)
adamc@14 541
adamc@15 542 (** We define a trivial language of types to classify our expressions: *)
adamc@15 543
adamc@14 544 Inductive type : Set := Nat | Bool.
adamc@14 545
adam@277 546 (** 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@277 547
adam@277 548 Now we define an expanded set of binary operators. *)
adamc@15 549
adamc@14 550 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 551 | TPlus : tbinop Nat Nat Nat
adamc@14 552 | TTimes : tbinop Nat Nat Nat
adamc@14 553 | TEq : forall t, tbinop t t Bool
adamc@14 554 | TLt : tbinop Nat Nat Bool.
adamc@14 555
adam@398 556 (** 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 _indexed type family_. Indexed inductive types are at the heart of Coq's expressive power; almost everything else of interest is defined in terms of them.
adamc@15 557
adam@452 558 The intuitive explanation of [tbinop] is that a [tbinop t1 t2 t] is a binary operator whose operands should have types [t1] and [t2], and whose result has type [t]. For instance, constructor [TLt] (for less-than comparison of numbers) is assigned type [tbinop Nat Nat Bool], meaning the operator's arguments are naturals and its result is Boolean. The type of [TEq] introduces a small bit of additional complication via polymorphism: we want to allow equality comparison of any two values of any type, as long as they have the _same_ type.
adam@312 559
adamc@15 560 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@15 561
adam@469 562 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]. %\index{generalized algebraic datatypes}\index{GADTs|see{generalized algebraic datatypes}}% _Generalized algebraic datatypes_ (GADTs)%~\cite{GADT}% are a popular feature in %\index{GHC Haskell}%GHC Haskell, OCaml 4, and other languages that removes this first restriction.
adamc@15 563
adam@419 564 The second restriction is not lifted by GADTs. In ML and Haskell, indices of types must be types and may not be _expressions_. 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 %\index{Haskell}%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@15 565 *)
adamc@15 566
adam@399 567 (** We can define a similar type family for typed expressions, where a term of type [texp t] can be assigned object language type [t]. (It is conventional in the world of interactive theorem proving to call the language of the proof assistant the%\index{meta language}% _meta language_ and a language being formalized the%\index{object language}% _object language_.) *)
adamc@15 568
adamc@14 569 Inductive texp : type -> Set :=
adamc@14 570 | TNConst : nat -> texp Nat
adamc@14 571 | TBConst : bool -> texp Bool
adam@312 572 | TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
adamc@14 573
adam@447 574 (** 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 object language into Coq types: *)
adamc@15 575
adamc@14 576 Definition typeDenote (t : type) : Set :=
adamc@14 577 match t with
adamc@14 578 | Nat => nat
adamc@14 579 | Bool => bool
adamc@14 580 end.
adamc@14 581
adam@448 582 (** 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. We can interpret binary operators by relying on standard-library equality test functions [eqb] and [beq_nat] for Booleans and naturals, respectively, along with a less-than test [leb]: *)
adamc@15 583
adamc@207 584 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@207 585 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@207 586 match b with
adamc@207 587 | TPlus => plus
adamc@207 588 | TTimes => mult
adam@277 589 | TEq Nat => beq_nat
adam@277 590 | TEq Bool => eqb
adam@312 591 | TLt => leb
adamc@207 592 end.
adamc@207 593
adam@399 594 (** 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%\index{dependent pattern matching}% _dependent pattern match_, where the necessary _type_ of each case body depends on the _value_ that has been matched. At this early stage, we will not go into detail on the many subtle aspects of Gallina that support dependent pattern-matching, but the subject is central to Part II of the book.
adam@312 595
adam@471 596 The same tricks suffice to define an expression denotation function in an unsurprising way. Note that the [type] arguments to the [TBinop] constructor must be included explicitly in pattern-matching, but here we write underscores because we do not need to refer to those arguments directly. *)
adamc@15 597
adamc@207 598 Fixpoint texpDenote t (e : texp t) : typeDenote t :=
adamc@207 599 match e with
adamc@14 600 | TNConst n => n
adamc@14 601 | TBConst b => b
adamc@14 602 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 603 end.
adamc@14 604
adamc@17 605 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 606
adamc@17 607 Eval simpl in texpDenote (TNConst 42).
adamc@207 608 (** [= 42 : typeDenote Nat] *)
adamc@207 609
adam@419 610 (* begin hide *)
adam@419 611 Eval simpl in texpDenote (TBConst false).
adam@419 612 (* end hide *)
adamc@17 613 Eval simpl in texpDenote (TBConst true).
adamc@207 614 (** [= true : typeDenote Bool] *)
adamc@207 615
adam@312 616 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 617 (TNConst 7)).
adamc@207 618 (** [= 28 : typeDenote Nat] *)
adamc@207 619
adam@312 620 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 621 (TNConst 7)).
adam@399 622 (** [= false : typeDenote Bool] *)
adamc@207 623
adam@312 624 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 625 (TNConst 7)).
adamc@207 626 (** [= true : typeDenote Bool] *)
adamc@17 627
adam@442 628 (** %\smallskip{}%Now we are ready to define a suitable stack machine target for compilation. *)
adam@442 629
adamc@14 630
adamc@20 631 (** ** Target Language *)
adamc@14 632
adam@442 633 (** 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@18 634
adamc@18 635 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@18 636
adamc@18 637 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 638
adamc@14 639 Definition tstack := list type.
adamc@14 640
adamc@18 641 (** 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@18 642
adamc@18 643 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@18 644
adamc@14 645 Inductive tinstr : tstack -> tstack -> Set :=
adam@312 646 | TiNConst : forall s, nat -> tinstr s (Nat :: s)
adam@312 647 | TiBConst : forall s, bool -> tinstr s (Bool :: s)
adam@311 648 | TiBinop : forall arg1 arg2 res s,
adamc@14 649 tbinop arg1 arg2 res
adamc@14 650 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 651
adamc@18 652 (** 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@18 653
adamc@14 654 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 655 | TNil : forall s, tprog s s
adamc@14 656 | TCons : forall s1 s2 s3,
adamc@14 657 tinstr s1 s2
adamc@14 658 -> tprog s2 s3
adamc@14 659 -> tprog s1 s3.
adamc@14 660
adamc@18 661 (** 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@18 662
adamc@14 663 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 664 match ts with
adamc@14 665 | nil => unit
adamc@14 666 | t :: ts' => typeDenote t * vstack ts'
adamc@14 667 end%type.
adamc@14 668
adam@312 669 (** 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 [%]%\index{notation scopes}\coqdocvar{%#<tt>#type#</tt>#%}% as an instruction to Coq's extensible parser. In particular, this directive applies to the whole [match] expression, which we ask to be parsed as though it were a type, so that the operator [*] is interpreted as Cartesian product instead of, say, multiplication. (Note that this use of %\coqdocvar{%#<tt>#type#</tt>#%}% has no connection to the inductive type [type] that we have defined.)
adamc@18 670
adam@312 671 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. We use a special form of [let] to destructure a multi-level tuple. *)
adamc@18 672
adamc@14 673 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@207 674 match i with
adam@312 675 | TiNConst _ n => fun s => (n, s)
adam@312 676 | TiBConst _ b => fun s => (b, s)
adam@311 677 | TiBinop _ _ _ _ b => fun s =>
adam@312 678 let '(arg1, (arg2, s')) := s in
adam@312 679 ((tbinopDenote b) arg1 arg2, s')
adamc@14 680 end.
adamc@14 681
adamc@18 682 (** 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@18 683 [[
adamc@18 684 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@207 685 match i with
adam@312 686 | TiNConst _ n => (n, s)
adam@312 687 | TiBConst _ b => (b, s)
adam@311 688 | TiBinop _ _ _ _ b =>
adam@312 689 let '(arg1, (arg2, s')) := s in
adam@312 690 ((tbinopDenote b) arg1 arg2, s')
adamc@18 691 end.
adamc@205 692 ]]
adamc@205 693
adam@447 694 The Coq type checker complains that:
adamc@18 695
adam@312 696 <<
adamc@18 697 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@207 698 while it is expected to have type "vstack ?119".
adam@312 699 >>
adamc@207 700
adam@465 701 This and other mysteries of Coq dependent typing we postpone until Part II of the book. The upshot of our later discussion is that it is often useful to push inside of [match] branches those function parameters whose types depend on the type of the value being matched. Our later, more complete treatment of Gallina's typing rules will explain why this helps.
adamc@18 702 *)
adamc@18 703
adamc@18 704 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 705
adamc@207 706 Fixpoint tprogDenote ts ts' (p : tprog ts ts') : vstack ts -> vstack ts' :=
adamc@207 707 match p with
adamc@14 708 | TNil _ => fun s => s
adamc@14 709 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 710 end.
adamc@14 711
adam@447 712 (** The same argument-postponing trick is crucial for this definition. *)
adam@447 713
adamc@14 714
adamc@14 715 (** ** Translation *)
adamc@14 716
adamc@19 717 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 718
adamc@207 719 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') : tprog ts' ts'' -> tprog ts ts'' :=
adamc@207 720 match p with
adamc@14 721 | TNil _ => fun p' => p'
adamc@14 722 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 723 end.
adamc@14 724
adamc@19 725 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 726
adamc@207 727 Fixpoint tcompile t (e : texp t) (ts : tstack) : tprog ts (t :: ts) :=
adamc@207 728 match e with
adam@312 729 | TNConst n => TCons (TiNConst _ n) (TNil _)
adam@312 730 | TBConst b => TCons (TiBConst _ b) (TNil _)
adamc@14 731 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adam@311 732 (tconcat (tcompile e1 _) (TCons (TiBinop _ b) (TNil _)))
adamc@14 733 end.
adamc@14 734
adam@398 735 (** 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 _implicit argument_ 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@19 736
adamc@19 737 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@19 738
adamc@14 739 Print tcompile.
adam@439 740 (** %\vspace{-.15in}%[[
adamc@19 741 tcompile =
adamc@19 742 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 743 tprog ts (t :: ts) :=
adamc@19 744 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adam@312 745 | TNConst n => TCons (TiNConst ts n) (TNil (Nat :: ts))
adam@312 746 | TBConst b => TCons (TiBConst ts b) (TNil (Bool :: ts))
adamc@19 747 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 748 tconcat (tcompile arg2 e2 ts)
adamc@19 749 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adam@311 750 (TCons (TiBinop ts b) (TNil (res :: ts))))
adamc@19 751 end
adamc@19 752 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adam@302 753 ]]
adam@302 754 *)
adamc@19 755
adamc@19 756
adamc@19 757 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 758
adamc@19 759 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adam@399 760 (** [= (42, tt) : vstack (Nat :: nil)] *)
adamc@207 761
adamc@19 762 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adam@399 763 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@207 764
adam@312 765 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2)
adam@312 766 (TNConst 2)) (TNConst 7)) nil) tt.
adam@399 767 (** [= (28, tt) : vstack (Nat :: nil)] *)
adamc@207 768
adam@312 769 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2)
adam@312 770 (TNConst 2)) (TNConst 7)) nil) tt.
adam@399 771 (** [= (false, tt) : vstack (Bool :: nil)] *)
adamc@207 772
adam@312 773 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 774 (TNConst 7)) nil) tt.
adam@399 775 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@19 776
adam@442 777 (** %\smallskip{}%The compiler seems to be working, so let us turn to proving that it _always_ works. *)
adam@442 778
adamc@14 779
adamc@20 780 (** ** Translation Correctness *)
adamc@20 781
adamc@20 782 (** We can state a correctness theorem similar to the last one. *)
adamc@20 783
adamc@207 784 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 785 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 786 (* begin hide *)
adamc@20 787 Abort.
adamc@20 788 (* end hide *)
adamc@22 789 (* begin thide *)
adamc@20 790
adam@312 791 (** Again, we need to strengthen the theorem statement so that the induction will go through. This time, to provide an excuse to demonstrate different tactics, I will develop an alternative approach to this kind of proof, stating the key lemma as: *)
adamc@14 792
adamc@207 793 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 794 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@20 795
adam@419 796 (** While lemma [compile_correct'] quantified over a program that is the "continuation"%~\cite{continuations}% 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@20 797
adamc@20 798 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 799
adamc@14 800 induction e; crush.
adamc@20 801
adamc@20 802 (** We are left with this unproved conclusion:
adamc@20 803
adamc@20 804 [[
adamc@20 805 tprogDenote
adamc@20 806 (tconcat (tcompile e2 ts)
adamc@20 807 (tconcat (tcompile e1 (arg2 :: ts))
adam@311 808 (TCons (TiBinop ts t) (TNil (res :: ts))))) s =
adamc@20 809 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@207 810
adamc@20 811 ]]
adamc@20 812
adam@312 813 We need an analogue to the [app_assoc_reverse] 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@20 814 *)
adamc@207 815
adamc@14 816 Abort.
adamc@14 817
adamc@26 818 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 819 (s : vstack ts),
adamc@14 820 tprogDenote (tconcat p p') s
adamc@14 821 = tprogDenote p' (tprogDenote p s).
adamc@14 822 induction p; crush.
adamc@14 823 Qed.
adamc@14 824
adamc@20 825 (** This one goes through completely automatically.
adamc@20 826
adam@316 827 Some code behind the scenes registers [app_assoc_reverse] for use by [crush]. We must register [tconcat_correct] similarly to get the same effect:%\index{Vernacular commands!Hint Rewrite}% *)
adamc@20 828
adam@375 829 Hint Rewrite tconcat_correct.
adamc@14 830
adam@419 831 (** Here we meet the pervasive concept of a _hint_. Many proofs can be found through exhaustive enumerations of combinations of possible proof steps; hints provide the set of steps to consider. The tactic [crush] is applying such brute force search for us silently, and it will consider more possibilities as we add more hints. This particular hint asks that the lemma be used for left-to-right rewriting.
adam@312 832
adam@312 833 Now we are ready to return to [tcompile_correct'], proving it automatically this time. *)
adamc@20 834
adamc@207 835 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 836 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@14 837 induction e; crush.
adamc@14 838 Qed.
adamc@14 839
adamc@20 840 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 841
adam@375 842 Hint Rewrite tcompile_correct'.
adamc@14 843
adamc@207 844 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 845 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 846 crush.
adamc@14 847 Qed.
adamc@22 848 (* end thide *)
adam@312 849
adam@399 850 (** It is probably worth emphasizing that we are doing more than building mathematical models. Our compilers are functional programs that can be executed efficiently. One strategy for doing so is based on%\index{program extraction}% _program extraction_, which generates OCaml code from Coq developments. For instance, we run a command to output the OCaml version of [tcompile]:%\index{Vernacular commands!Extraction}% *)
adam@312 851
adam@312 852 Extraction tcompile.
adam@312 853
adam@312 854 (** <<
adam@312 855 let rec tcompile t e ts =
adam@312 856 match e with
adam@312 857 | TNConst n ->
adam@312 858 TCons (ts, (Cons (Nat, ts)), (Cons (Nat, ts)), (TiNConst (ts, n)), (TNil
adam@312 859 (Cons (Nat, ts))))
adam@312 860 | TBConst b ->
adam@312 861 TCons (ts, (Cons (Bool, ts)), (Cons (Bool, ts)), (TiBConst (ts, b)),
adam@312 862 (TNil (Cons (Bool, ts))))
adam@312 863 | TBinop (t1, t2, t0, b, e1, e2) ->
adam@312 864 tconcat ts (Cons (t2, ts)) (Cons (t0, ts)) (tcompile t2 e2 ts)
adam@312 865 (tconcat (Cons (t2, ts)) (Cons (t1, (Cons (t2, ts)))) (Cons (t0, ts))
adam@312 866 (tcompile t1 e1 (Cons (t2, ts))) (TCons ((Cons (t1, (Cons (t2,
adam@312 867 ts)))), (Cons (t0, ts)), (Cons (t0, ts)), (TiBinop (t1, t2, t0, ts,
adam@312 868 b)), (TNil (Cons (t0, ts))))))
adam@312 869 >>
adam@312 870
adam@312 871 We can compile this code with the usual OCaml compiler and obtain an executable program with halfway decent performance.
adam@312 872
adam@312 873 This chapter has been a whirlwind tour through two examples of the style of Coq development that I advocate. Parts II and III of the book focus on the key elements of that style, namely dependent types and scripted proof automation, respectively. Before we get there, we will spend some time in Part I on more standard foundational material. Part I may still be of interest to seasoned Coq hackers, since I follow the highly automated proof style even at that early stage. *)