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A pass of improvements to vertical spacing, up through end of InductiveTypes
author Adam Chlipala <adam@chlipala.net>
date Mon, 30 Jul 2012 13:21:36 -0400
parents 686cf945dd02
children 89c67796754e
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@399 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.
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@419 20 (** 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 21
adamc@9 22
adamc@20 23 (** * Arithmetic Expressions Over Natural Numbers *)
adamc@2 24
adamc@40 25 (** We will begin with that staple of compiler textbooks, arithmetic expressions over a single type of numbers. *)
adamc@9 26
adamc@20 27 (** ** Source Language *)
adamc@9 28
adam@311 29 (** We begin with the syntax of the source language.%\index{Vernacular commands!Inductive}% *)
adamc@2 30
adamc@4 31 Inductive binop : Set := Plus | Times.
adamc@2 32
adam@419 33 (** 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 bis useful in higher-order constructions. *)
adamc@9 34
adamc@4 35 Inductive exp : Set :=
adamc@4 36 | Const : nat -> exp
adamc@4 37 | Binop : binop -> exp -> exp -> exp.
adamc@2 38
adamc@9 39 (** 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 40
adam@419 41 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 42
adamc@9 43 %\medskip%
adamc@9 44
adam@419 45 Now we are ready to say what these programs 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 46
adamc@4 47 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 48 match b with
adamc@4 49 | Plus => plus
adamc@4 50 | Times => mult
adamc@4 51 end.
adamc@2 52
adam@419 53 (** 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 54 [[
adamc@9 55 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
adamc@9 56 match b with
adamc@9 57 | Plus => plus
adamc@9 58 | Times => mult
adamc@9 59 end.
adamc@205 60 ]]
adamc@205 61
adamc@9 62 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 63 [[
adamc@9 64 Definition binopDenote := fun b =>
adamc@9 65 match b with
adamc@9 66 | Plus => plus
adamc@9 67 | Times => mult
adamc@9 68 end.
adamc@205 69 ]]
adamc@205 70
adam@419 71 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 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 72
adam@419 73 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%\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 74
adam@399 75 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 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 76
adam@399 77 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 78
adam@399 79 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 80
adamc@9 81 %\medskip%
adamc@9 82
adam@311 83 We can give a simple definition of the meaning of an expression:%\index{Vernacular commands!Fixpoint}% *)
adamc@9 84
adamc@4 85 Fixpoint expDenote (e : exp) : nat :=
adamc@4 86 match e with
adamc@4 87 | Const n => n
adamc@4 88 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 89 end.
adamc@2 90
adamc@9 91 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 92
adam@419 93 (** 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 94
adam@311 95 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 96
adamc@16 97 Eval simpl in expDenote (Const 42).
adamc@205 98 (** [= 42 : nat] *)
adamc@205 99
adamc@16 100 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
adamc@205 101 (** [= 4 : nat] *)
adamc@205 102
adamc@16 103 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@205 104 (** [= 28 : nat] *)
adamc@9 105
adamc@20 106 (** ** Target Language *)
adamc@4 107
adamc@10 108 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 109
adamc@4 110 Inductive instr : Set :=
adam@311 111 | iConst : nat -> instr
adam@311 112 | iBinop : binop -> instr.
adamc@2 113
adamc@4 114 Definition prog := list instr.
adamc@4 115 Definition stack := list nat.
adamc@2 116
adamc@10 117 (** 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@10 118
adam@419 119 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@10 120
adamc@4 121 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 122 match i with
adam@311 123 | iConst n => Some (n :: s)
adam@311 124 | iBinop b =>
adamc@4 125 match s with
adamc@4 126 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 127 | _ => None
adamc@4 128 end
adamc@4 129 end.
adamc@2 130
adam@311 131 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@206 132
adamc@206 133 Fixpoint progDenote (p : prog) (s : stack) : option stack :=
adamc@206 134 match p with
adamc@206 135 | nil => Some s
adamc@206 136 | i :: p' =>
adamc@206 137 match instrDenote i s with
adamc@206 138 | None => None
adamc@206 139 | Some s' => progDenote p' s'
adamc@206 140 end
adamc@206 141 end.
adamc@2 142
adamc@4 143
adamc@9 144 (** ** Translation *)
adamc@4 145
adam@311 146 (** Our compiler itself is now unsurprising. The list concatenation operator %\index{Gallina operators!++}%[++] comes from the Coq standard library. *)
adamc@2 147
adamc@4 148 Fixpoint compile (e : exp) : prog :=
adamc@4 149 match e with
adam@311 150 | Const n => iConst n :: nil
adam@311 151 | Binop b e1 e2 => compile e2 ++ compile e1 ++ iBinop b :: nil
adamc@4 152 end.
adamc@2 153
adamc@2 154
adamc@16 155 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 156
adamc@16 157 Eval simpl in compile (Const 42).
adam@311 158 (** [= iConst 42 :: nil : prog] *)
adamc@206 159
adamc@16 160 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adam@311 161 (** [= iConst 2 :: iConst 2 :: iBinop Plus :: nil : prog] *)
adamc@206 162
adamc@16 163 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adam@311 164 (** [= iConst 7 :: iConst 2 :: iConst 2 :: iBinop Plus :: iBinop Times :: nil : prog] *)
adamc@16 165
adamc@40 166 (** We can also run our compiled programs and check that they give the right results. *)
adamc@16 167
adamc@16 168 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@206 169 (** [= Some (42 :: nil) : option stack] *)
adamc@206 170
adamc@16 171 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@206 172 (** [= Some (4 :: nil) : option stack] *)
adamc@206 173
adam@311 174 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2))
adam@311 175 (Const 7))) nil.
adamc@206 176 (** [= Some (28 :: nil) : option stack] *)
adamc@16 177
adamc@16 178
adamc@20 179 (** ** Translation Correctness *)
adamc@4 180
adam@311 181 (** 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@11 182
adamc@26 183 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 184 (* begin hide *)
adamc@11 185 Abort.
adamc@11 186 (* end hide *)
adamc@22 187 (* begin thide *)
adamc@11 188
adam@419 189 (** 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@2 190
adamc@206 191 Lemma compile_correct' : forall e p s,
adamc@206 192 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 193
adam@399 194 (** 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@11 195
adamc@11 196 [[
adamc@11 197 1 subgoal
adamc@11 198
adamc@11 199 ============================
adamc@15 200 forall (e : exp) (p : list instr) (s : stack),
adamc@15 201 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@206 202
adamc@11 203 ]]
adamc@11 204
adam@311 205 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@11 206
adam@311 207 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@11 208
adam@399 209 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@11 210 *)
adamc@11 211
adamc@4 212 induction e.
adamc@2 213
adamc@11 214 (** 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@11 215
adam@439 216 [[
adam@439 217 2 subgoals
adam@311 218
adamc@11 219 n : nat
adamc@11 220 ============================
adamc@11 221 forall (s : stack) (p : list instr),
adamc@11 222 progDenote (compile (Const n) ++ p) s =
adamc@11 223 progDenote p (expDenote (Const n) :: s)
adam@439 224
adam@439 225 subgoal 2 is
adam@439 226
adamc@11 227 forall (s : stack) (p : list instr),
adamc@11 228 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 229 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 230
adamc@11 231 ]]
adamc@11 232
adam@311 233 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@11 234
adam@311 235 We begin the first case with another very common tactic.%\index{tactics!intros}%
adamc@11 236 *)
adamc@11 237
adamc@4 238 intros.
adamc@11 239
adamc@11 240 (** The current subgoal changes to:
adamc@11 241 [[
adamc@11 242
adamc@11 243 n : nat
adamc@11 244 s : stack
adamc@11 245 p : list instr
adamc@11 246 ============================
adamc@11 247 progDenote (compile (Const n) ++ p) s =
adamc@11 248 progDenote p (expDenote (Const n) :: s)
adamc@206 249
adamc@11 250 ]]
adamc@11 251
adamc@11 252 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 253
adam@311 254 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@11 255 *)
adamc@11 256
adamc@4 257 unfold compile.
adamc@11 258 (** [[
adamc@11 259 n : nat
adamc@11 260 s : stack
adamc@11 261 p : list instr
adamc@11 262 ============================
adam@311 263 progDenote ((iConst n :: nil) ++ p) s =
adamc@11 264 progDenote p (expDenote (Const n) :: s)
adamc@206 265
adam@302 266 ]]
adam@302 267 *)
adamc@11 268
adamc@4 269 unfold expDenote.
adamc@11 270 (** [[
adamc@11 271 n : nat
adamc@11 272 s : stack
adamc@11 273 p : list instr
adamc@11 274 ============================
adam@311 275 progDenote ((iConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@206 276
adamc@11 277 ]]
adamc@11 278
adam@311 279 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@11 280
adamc@11 281 unfold progDenote at 1.
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 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 288 option stack :=
adamc@11 289 match p0 with
adamc@11 290 | nil => Some s0
adamc@11 291 | i :: p' =>
adamc@11 292 match instrDenote i s0 with
adamc@11 293 | Some s' => progDenote p' s'
adamc@11 294 | None => None (A:=stack)
adamc@11 295 end
adam@311 296 end) ((iConst n :: nil) ++ p) s =
adamc@11 297 progDenote p (n :: s)
adamc@206 298
adamc@11 299 ]]
adamc@11 300
adam@419 301 This last [unfold] has left us with an anonymous fixpoint version of [progDenote], 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@311 302
adam@311 303 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@11 304 *)
adamc@11 305
adamc@4 306 simpl.
adamc@11 307 (** [[
adamc@11 308 n : nat
adamc@11 309 s : stack
adamc@11 310 p : list instr
adamc@11 311 ============================
adamc@11 312 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 313 option stack :=
adamc@11 314 match p0 with
adamc@11 315 | nil => Some s0
adamc@11 316 | i :: p' =>
adamc@11 317 match instrDenote i s0 with
adamc@11 318 | Some s' => progDenote p' s'
adamc@11 319 | None => None (A:=stack)
adamc@11 320 end
adamc@11 321 end) p (n :: s) = progDenote p (n :: s)
adamc@206 322
adamc@11 323 ]]
adamc@11 324
adam@311 325 Now we can unexpand the definition of [progDenote]:%\index{tactics!fold}%
adamc@11 326 *)
adamc@11 327
adamc@11 328 fold progDenote.
adamc@11 329 (** [[
adamc@11 330 n : nat
adamc@11 331 s : stack
adamc@11 332 p : list instr
adamc@11 333 ============================
adamc@11 334 progDenote p (n :: s) = progDenote p (n :: s)
adamc@206 335
adamc@11 336 ]]
adamc@11 337
adam@311 338 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@11 339 *)
adamc@11 340
adamc@4 341 reflexivity.
adamc@2 342
adamc@11 343 (** On to the second inductive case:
adamc@11 344
adamc@11 345 [[
adamc@11 346 b : binop
adamc@11 347 e1 : exp
adamc@11 348 IHe1 : forall (s : stack) (p : list instr),
adamc@11 349 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 350 e2 : exp
adamc@11 351 IHe2 : forall (s : stack) (p : list instr),
adamc@11 352 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 353 ============================
adamc@11 354 forall (s : stack) (p : list instr),
adamc@11 355 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 356 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 357
adamc@11 358 ]]
adamc@11 359
adam@311 360 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@11 361
adam@399 362 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@11 363
adamc@4 364 intros.
adamc@4 365 unfold compile.
adamc@4 366 fold compile.
adamc@4 367 unfold expDenote.
adamc@4 368 fold expDenote.
adamc@11 369
adamc@44 370 (** 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@11 371
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 s : stack
adamc@11 381 p : list instr
adamc@11 382 ============================
adam@311 383 progDenote ((compile e2 ++ compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 384 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 385
adamc@11 386 ]]
adamc@11 387
adam@311 388 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}% *)
adamc@11 389
adam@311 390 Check app_assoc.
adam@439 391 (** %\vspace{-.15in}%[[
adam@311 392 app_assoc_reverse
adamc@11 393 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@206 394
adamc@11 395 ]]
adamc@11 396
adam@399 397 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@277 398
adam@277 399 SearchRewrite ((_ ++ _) ++ _).
adam@439 400 (** %\vspace{-.15in}%[[
adam@311 401 app_assoc_reverse:
adam@311 402 forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adam@311 403 ]]
adam@311 404 %\vspace{-.25in}%
adam@311 405 [[
adam@311 406 app_assoc: forall (A : Type) (l m n : list A), l ++ m ++ n = (l ++ m) ++ n
adam@277 407
adam@277 408 ]]
adam@277 409
adam@311 410 We use [app_assoc_reverse] to perform a rewrite: %\index{tactics!rewrite}% *)
adamc@11 411
adam@311 412 rewrite app_assoc_reverse.
adamc@11 413
adam@439 414 (** %\noindent{}%changing the conclusion to:
adamc@11 415
adamc@206 416 [[
adam@311 417 progDenote (compile e2 ++ (compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 418 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 419
adamc@11 420 ]]
adamc@11 421
adam@311 422 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@11 423
adamc@4 424 rewrite IHe2.
adamc@11 425 (** [[
adam@311 426 progDenote ((compile e1 ++ iBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 427 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 428
adamc@11 429 ]]
adamc@11 430
adam@311 431 The same process lets us apply the remaining hypothesis.%\index{tactics!rewrite}% *)
adamc@11 432
adam@311 433 rewrite app_assoc_reverse.
adamc@4 434 rewrite IHe1.
adamc@11 435 (** [[
adam@311 436 progDenote ((iBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 437 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 438
adamc@11 439 ]]
adamc@11 440
adam@311 441 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@11 442 *)
adamc@11 443
adamc@11 444 unfold progDenote at 1.
adamc@4 445 simpl.
adamc@11 446 fold progDenote.
adamc@4 447 reflexivity.
adamc@11 448
adam@311 449 (** And the proof is completed, as indicated by the message: *)
adamc@11 450
adam@399 451 (**
adam@399 452 <<
adam@399 453 Proof completed.
adam@399 454 >>
adam@399 455 *)
adamc@11 456
adam@311 457 (** 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@11 458 *)
adamc@11 459
adamc@4 460 Abort.
adamc@2 461
adam@311 462 (** %\index{tactics!induction}\index{tactics!crush}% *)
adam@311 463
adamc@26 464 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 465 progDenote p (expDenote e :: s).
adamc@4 466 induction e; crush.
adamc@4 467 Qed.
adamc@2 468
adam@328 469 (** 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@11 470
adam@399 471 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@210 472
adam@398 473 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@311 474
adam@311 475 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@11 476
adamc@26 477 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 478 intros.
adamc@11 479 (** [[
adamc@11 480 e : exp
adamc@11 481 ============================
adamc@11 482 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@206 483
adamc@11 484 ]]
adamc@11 485
adamc@26 486 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@11 487
adamc@11 488 Check app_nil_end.
adamc@11 489 (** [[
adamc@11 490 app_nil_end
adamc@11 491 : forall (A : Type) (l : list A), l = l ++ nil
adam@302 492 ]]
adam@311 493 %\index{tactics!rewrite}% *)
adamc@11 494
adamc@4 495 rewrite (app_nil_end (compile e)).
adamc@11 496
adam@417 497 (** 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@11 498
adamc@11 499 [[
adamc@11 500 e : exp
adamc@11 501 ============================
adamc@11 502 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@206 503
adamc@11 504 ]]
adamc@11 505
adam@311 506 Now we can apply the lemma.%\index{tactics!rewrite}% *)
adamc@11 507
adamc@26 508 rewrite compile_correct'.
adamc@11 509 (** [[
adamc@11 510 e : exp
adamc@11 511 ============================
adamc@11 512 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@206 513
adamc@11 514 ]]
adamc@11 515
adam@311 516 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@11 517
adamc@4 518 reflexivity.
adamc@4 519 Qed.
adamc@22 520 (* end thide *)
adamc@14 521
adam@311 522 (** This proof can be shortened and made automated, but we leave that as an exercise for the reader. *)
adam@311 523
adamc@14 524
adamc@20 525 (** * Typed Expressions *)
adamc@14 526
adamc@14 527 (** 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@14 528
adamc@20 529 (** ** Source Language *)
adamc@14 530
adamc@15 531 (** We define a trivial language of types to classify our expressions: *)
adamc@15 532
adamc@14 533 Inductive type : Set := Nat | Bool.
adamc@14 534
adam@277 535 (** 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@277 536
adam@277 537 Now we define an expanded set of binary operators. *)
adamc@15 538
adamc@14 539 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 540 | TPlus : tbinop Nat Nat Nat
adamc@14 541 | TTimes : tbinop Nat Nat Nat
adamc@14 542 | TEq : forall t, tbinop t t Bool
adamc@14 543 | TLt : tbinop Nat Nat Bool.
adamc@14 544
adam@398 545 (** 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@15 546
adam@398 547 The inuitive 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@312 548
adamc@15 549 ML and Haskell have indexed algebraic datatypes. For instance, their list types are indexed by the type of data that the list carries. However, compared to Coq, ML and Haskell 98 place two important restrictions on datatype definitions.
adamc@15 550
adam@417 551 First, the indices of the range of each data constructor must be type variables bound at the top level of the datatype definition. There is no way to do what we did here, where we, for instance, say that [TPlus] is a constructor building a [tbinop] whose indices are all fixed at [Nat]. %\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 and other languages that removes this first restriction.
adamc@15 552
adam@419 553 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@15 554 *)
adamc@15 555
adam@399 556 (** 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@15 557
adamc@14 558 Inductive texp : type -> Set :=
adamc@14 559 | TNConst : nat -> texp Nat
adamc@14 560 | TBConst : bool -> texp Bool
adam@312 561 | TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
adamc@14 562
adamc@15 563 (** Thanks to our use of dependent types, every well-typed [texp] represents a well-typed source expression, by construction. This turns out to be very convenient for many things we might want to do with expressions. For instance, it is easy to adapt our interpreter approach to defining semantics. We start by defining a function mapping the types of our languages into Coq types: *)
adamc@15 564
adamc@14 565 Definition typeDenote (t : type) : Set :=
adamc@14 566 match t with
adamc@14 567 | Nat => nat
adamc@14 568 | Bool => bool
adamc@14 569 end.
adamc@14 570
adam@312 571 (** It can take a few moments to come to terms with the fact that [Set], the type of types of programs, is itself a first-class type, and that we can write functions that return [Set]s. Past that wrinkle, the definition of [typeDenote] is trivial, relying on the [nat] and [bool] types from the Coq standard library. 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@15 572
adamc@207 573 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@207 574 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@207 575 match b with
adamc@207 576 | TPlus => plus
adamc@207 577 | TTimes => mult
adam@277 578 | TEq Nat => beq_nat
adam@277 579 | TEq Bool => eqb
adam@312 580 | TLt => leb
adamc@207 581 end.
adamc@207 582
adam@399 583 (** 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@312 584
adamc@15 585 The same tricks suffice to define an expression denotation function in an unsurprising way:
adamc@15 586 *)
adamc@15 587
adamc@207 588 Fixpoint texpDenote t (e : texp t) : typeDenote t :=
adamc@207 589 match e with
adamc@14 590 | TNConst n => n
adamc@14 591 | TBConst b => b
adamc@14 592 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 593 end.
adamc@14 594
adamc@17 595 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 596
adamc@17 597 Eval simpl in texpDenote (TNConst 42).
adamc@207 598 (** [= 42 : typeDenote Nat] *)
adamc@207 599
adam@419 600 (* begin hide *)
adam@419 601 Eval simpl in texpDenote (TBConst false).
adam@419 602 (* end hide *)
adamc@17 603 Eval simpl in texpDenote (TBConst true).
adamc@207 604 (** [= true : typeDenote Bool] *)
adamc@207 605
adam@312 606 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 607 (TNConst 7)).
adamc@207 608 (** [= 28 : typeDenote Nat] *)
adamc@207 609
adam@312 610 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 611 (TNConst 7)).
adam@399 612 (** [= false : typeDenote Bool] *)
adamc@207 613
adam@312 614 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 615 (TNConst 7)).
adamc@207 616 (** [= true : typeDenote Bool] *)
adamc@17 617
adamc@14 618
adamc@20 619 (** ** Target Language *)
adamc@14 620
adam@419 621 (** 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@18 622
adamc@18 623 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@18 624
adamc@18 625 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 626
adamc@14 627 Definition tstack := list type.
adamc@14 628
adamc@18 629 (** 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@18 630
adamc@18 631 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@18 632
adamc@14 633 Inductive tinstr : tstack -> tstack -> Set :=
adam@312 634 | TiNConst : forall s, nat -> tinstr s (Nat :: s)
adam@312 635 | TiBConst : forall s, bool -> tinstr s (Bool :: s)
adam@311 636 | TiBinop : forall arg1 arg2 res s,
adamc@14 637 tbinop arg1 arg2 res
adamc@14 638 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 639
adamc@18 640 (** 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@18 641
adamc@14 642 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 643 | TNil : forall s, tprog s s
adamc@14 644 | TCons : forall s1 s2 s3,
adamc@14 645 tinstr s1 s2
adamc@14 646 -> tprog s2 s3
adamc@14 647 -> tprog s1 s3.
adamc@14 648
adamc@18 649 (** 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@18 650
adamc@14 651 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 652 match ts with
adamc@14 653 | nil => unit
adamc@14 654 | t :: ts' => typeDenote t * vstack ts'
adamc@14 655 end%type.
adamc@14 656
adam@312 657 (** 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@18 658
adam@312 659 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@18 660
adamc@14 661 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@207 662 match i with
adam@312 663 | TiNConst _ n => fun s => (n, s)
adam@312 664 | TiBConst _ b => fun s => (b, s)
adam@311 665 | TiBinop _ _ _ _ b => fun s =>
adam@312 666 let '(arg1, (arg2, s')) := s in
adam@312 667 ((tbinopDenote b) arg1 arg2, s')
adamc@14 668 end.
adamc@14 669
adamc@18 670 (** 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@18 671 [[
adamc@18 672 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@207 673 match i with
adam@312 674 | TiNConst _ n => (n, s)
adam@312 675 | TiBConst _ b => (b, s)
adam@311 676 | TiBinop _ _ _ _ b =>
adam@312 677 let '(arg1, (arg2, s')) := s in
adam@312 678 ((tbinopDenote b) arg1 arg2, s')
adamc@18 679 end.
adamc@18 680
adamc@205 681 ]]
adamc@205 682
adamc@18 683 The Coq type-checker complains that:
adamc@18 684
adam@312 685 <<
adamc@18 686 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@207 687 while it is expected to have type "vstack ?119".
adam@312 688 >>
adamc@207 689
adam@312 690 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 treatement of Gallina's typing rules will explain why this helps.
adamc@18 691 *)
adamc@18 692
adamc@18 693 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 694
adamc@207 695 Fixpoint tprogDenote ts ts' (p : tprog ts ts') : vstack ts -> vstack ts' :=
adamc@207 696 match p with
adamc@14 697 | TNil _ => fun s => s
adamc@14 698 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 699 end.
adamc@14 700
adamc@14 701
adamc@14 702 (** ** Translation *)
adamc@14 703
adamc@19 704 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 705
adamc@207 706 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') : tprog ts' ts'' -> tprog ts ts'' :=
adamc@207 707 match p with
adamc@14 708 | TNil _ => fun p' => p'
adamc@14 709 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 710 end.
adamc@14 711
adamc@19 712 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 713
adamc@207 714 Fixpoint tcompile t (e : texp t) (ts : tstack) : tprog ts (t :: ts) :=
adamc@207 715 match e with
adam@312 716 | TNConst n => TCons (TiNConst _ n) (TNil _)
adam@312 717 | TBConst b => TCons (TiBConst _ b) (TNil _)
adamc@14 718 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adam@311 719 (tconcat (tcompile e1 _) (TCons (TiBinop _ b) (TNil _)))
adamc@14 720 end.
adamc@14 721
adam@398 722 (** 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@19 723
adamc@19 724 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@19 725
adamc@14 726 Print tcompile.
adam@439 727 (** %\vspace{-.15in}%[[
adamc@19 728 tcompile =
adamc@19 729 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 730 tprog ts (t :: ts) :=
adamc@19 731 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adam@312 732 | TNConst n => TCons (TiNConst ts n) (TNil (Nat :: ts))
adam@312 733 | TBConst b => TCons (TiBConst ts b) (TNil (Bool :: ts))
adamc@19 734 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 735 tconcat (tcompile arg2 e2 ts)
adamc@19 736 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adam@311 737 (TCons (TiBinop ts b) (TNil (res :: ts))))
adamc@19 738 end
adamc@19 739 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adam@302 740 ]]
adam@302 741 *)
adamc@19 742
adamc@19 743
adamc@19 744 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 745
adamc@19 746 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adam@399 747 (** [= (42, tt) : vstack (Nat :: nil)] *)
adamc@207 748
adamc@19 749 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adam@399 750 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@207 751
adam@312 752 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2)
adam@312 753 (TNConst 2)) (TNConst 7)) nil) tt.
adam@399 754 (** [= (28, tt) : vstack (Nat :: nil)] *)
adamc@207 755
adam@312 756 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2)
adam@312 757 (TNConst 2)) (TNConst 7)) nil) tt.
adam@399 758 (** [= (false, tt) : vstack (Bool :: nil)] *)
adamc@207 759
adam@312 760 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 761 (TNConst 7)) nil) tt.
adam@399 762 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@19 763
adamc@14 764
adamc@20 765 (** ** Translation Correctness *)
adamc@20 766
adamc@20 767 (** We can state a correctness theorem similar to the last one. *)
adamc@20 768
adamc@207 769 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 770 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 771 (* begin hide *)
adamc@20 772 Abort.
adamc@20 773 (* end hide *)
adamc@22 774 (* begin thide *)
adamc@20 775
adam@312 776 (** 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@14 777
adamc@207 778 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 779 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@20 780
adam@419 781 (** 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@20 782
adamc@20 783 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 784
adamc@14 785 induction e; crush.
adamc@20 786
adamc@20 787 (** We are left with this unproved conclusion:
adamc@20 788
adamc@20 789 [[
adamc@20 790 tprogDenote
adamc@20 791 (tconcat (tcompile e2 ts)
adamc@20 792 (tconcat (tcompile e1 (arg2 :: ts))
adam@311 793 (TCons (TiBinop ts t) (TNil (res :: ts))))) s =
adamc@20 794 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@207 795
adamc@20 796 ]]
adamc@20 797
adam@312 798 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@20 799 *)
adamc@207 800
adamc@14 801 Abort.
adamc@14 802
adamc@26 803 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 804 (s : vstack ts),
adamc@14 805 tprogDenote (tconcat p p') s
adamc@14 806 = tprogDenote p' (tprogDenote p s).
adamc@14 807 induction p; crush.
adamc@14 808 Qed.
adamc@14 809
adamc@20 810 (** This one goes through completely automatically.
adamc@20 811
adam@316 812 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@20 813
adam@375 814 Hint Rewrite tconcat_correct.
adamc@14 815
adam@419 816 (** 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@312 817
adam@312 818 Now we are ready to return to [tcompile_correct'], proving it automatically this time. *)
adamc@20 819
adamc@207 820 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 821 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@14 822 induction e; crush.
adamc@14 823 Qed.
adamc@14 824
adamc@20 825 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 826
adam@375 827 Hint Rewrite tcompile_correct'.
adamc@14 828
adamc@207 829 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 830 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 831 crush.
adamc@14 832 Qed.
adamc@22 833 (* end thide *)
adam@312 834
adam@399 835 (** 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@312 836
adam@312 837 Extraction tcompile.
adam@312 838
adam@312 839 (** <<
adam@312 840 let rec tcompile t e ts =
adam@312 841 match e with
adam@312 842 | TNConst n ->
adam@312 843 TCons (ts, (Cons (Nat, ts)), (Cons (Nat, ts)), (TiNConst (ts, n)), (TNil
adam@312 844 (Cons (Nat, ts))))
adam@312 845 | TBConst b ->
adam@312 846 TCons (ts, (Cons (Bool, ts)), (Cons (Bool, ts)), (TiBConst (ts, b)),
adam@312 847 (TNil (Cons (Bool, ts))))
adam@312 848 | TBinop (t1, t2, t0, b, e1, e2) ->
adam@312 849 tconcat ts (Cons (t2, ts)) (Cons (t0, ts)) (tcompile t2 e2 ts)
adam@312 850 (tconcat (Cons (t2, ts)) (Cons (t1, (Cons (t2, ts)))) (Cons (t0, ts))
adam@312 851 (tcompile t1 e1 (Cons (t2, ts))) (TCons ((Cons (t1, (Cons (t2,
adam@312 852 ts)))), (Cons (t0, ts)), (Cons (t0, ts)), (TiBinop (t1, t2, t0, ts,
adam@312 853 b)), (TNil (Cons (t0, ts))))))
adam@312 854 >>
adam@312 855
adam@312 856 We can compile this code with the usual OCaml compiler and obtain an executable program with halfway decent performance.
adam@312 857
adam@312 858 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. *)