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Coq'Art exercises
author Adam Chlipala <adam@chlipala.net>
date Tue, 08 Jan 2013 16:20:01 -0500
parents 1fd4109f7b31
children fa2eb72446a8
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@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@447 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 is 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@475 45 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 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@475 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 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@475 73 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 74
adam@475 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 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 76
adam@475 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@475 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
adam@442 106 (** %\smallskip{}%Nothing too surprising goes on here, so we are ready to move on to the target language of our compiler. *)
adam@442 107
adam@442 108
adamc@20 109 (** ** Target Language *)
adamc@4 110
adamc@10 111 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 112
adamc@4 113 Inductive instr : Set :=
adam@311 114 | iConst : nat -> instr
adam@311 115 | iBinop : binop -> instr.
adamc@2 116
adamc@4 117 Definition prog := list instr.
adamc@4 118 Definition stack := list nat.
adamc@2 119
adamc@10 120 (** 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@10 121
adam@419 122 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@10 123
adamc@4 124 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 125 match i with
adam@311 126 | iConst n => Some (n :: s)
adam@311 127 | iBinop b =>
adamc@4 128 match s with
adamc@4 129 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 130 | _ => None
adamc@4 131 end
adamc@4 132 end.
adamc@2 133
adam@311 134 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@206 135
adamc@206 136 Fixpoint progDenote (p : prog) (s : stack) : option stack :=
adamc@206 137 match p with
adamc@206 138 | nil => Some s
adamc@206 139 | i :: p' =>
adamc@206 140 match instrDenote i s with
adamc@206 141 | None => None
adamc@206 142 | Some s' => progDenote p' s'
adamc@206 143 end
adamc@206 144 end.
adamc@2 145
adam@442 146 (** With the two programming languages defined, we can turn to the compiler definition. *)
adam@442 147
adamc@4 148
adamc@9 149 (** ** Translation *)
adamc@4 150
adam@471 151 (** Our compiler itself is now unsurprising. The list concatenation operator %\index{Gallina operators!++}\coqdocnotation{%#<tt>#++#</tt>#%}% comes from the Coq standard library. *)
adamc@2 152
adamc@4 153 Fixpoint compile (e : exp) : prog :=
adamc@4 154 match e with
adam@311 155 | Const n => iConst n :: nil
adam@311 156 | Binop b e1 e2 => compile e2 ++ compile e1 ++ iBinop b :: nil
adamc@4 157 end.
adamc@2 158
adamc@2 159
adamc@16 160 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 161
adamc@16 162 Eval simpl in compile (Const 42).
adam@311 163 (** [= iConst 42 :: nil : prog] *)
adamc@206 164
adamc@16 165 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adam@311 166 (** [= iConst 2 :: iConst 2 :: iBinop Plus :: nil : prog] *)
adamc@206 167
adamc@16 168 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adam@311 169 (** [= iConst 7 :: iConst 2 :: iConst 2 :: iBinop Plus :: iBinop Times :: nil : prog] *)
adamc@16 170
adam@442 171 (** %\smallskip{}%We can also run our compiled programs and check that they give the right results. *)
adamc@16 172
adamc@16 173 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@206 174 (** [= Some (42 :: nil) : option stack] *)
adamc@206 175
adamc@16 176 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@206 177 (** [= Some (4 :: nil) : option stack] *)
adamc@206 178
adam@311 179 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2))
adam@311 180 (Const 7))) nil.
adamc@206 181 (** [= Some (28 :: nil) : option stack] *)
adamc@16 182
adam@447 183 (** %\smallskip{}%So far so good, but how can we be sure the compiler operates correctly for _all_ input programs? *)
adamc@16 184
adamc@20 185 (** ** Translation Correctness *)
adamc@4 186
adam@311 187 (** 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@11 188
adamc@26 189 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@22 190 (* begin thide *)
adamc@11 191
adam@419 192 (** 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@2 193
adam@469 194 Abort.
adam@469 195
adamc@206 196 Lemma compile_correct' : forall e p s,
adamc@206 197 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 198
adam@399 199 (** 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@11 200
adamc@11 201 [[
adamc@11 202 1 subgoal
adamc@11 203
adamc@11 204 ============================
adamc@15 205 forall (e : exp) (p : list instr) (s : stack),
adamc@15 206 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@206 207
adamc@11 208 ]]
adamc@11 209
adam@311 210 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@11 211
adam@311 212 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@11 213
adam@399 214 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@11 215 *)
adamc@11 216
adamc@4 217 induction e.
adamc@2 218
adamc@11 219 (** 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@11 220
adam@439 221 [[
adam@439 222 2 subgoals
adam@311 223
adamc@11 224 n : nat
adamc@11 225 ============================
adamc@11 226 forall (s : stack) (p : list instr),
adamc@11 227 progDenote (compile (Const n) ++ p) s =
adamc@11 228 progDenote p (expDenote (Const n) :: s)
adam@439 229
adam@439 230 subgoal 2 is
adam@439 231
adamc@11 232 forall (s : stack) (p : list instr),
adamc@11 233 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 234 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 235
adamc@11 236 ]]
adamc@11 237
adam@311 238 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@11 239
adam@311 240 We begin the first case with another very common tactic.%\index{tactics!intros}%
adamc@11 241 *)
adamc@11 242
adamc@4 243 intros.
adamc@11 244
adamc@11 245 (** The current subgoal changes to:
adamc@11 246 [[
adamc@11 247
adamc@11 248 n : nat
adamc@11 249 s : stack
adamc@11 250 p : list instr
adamc@11 251 ============================
adamc@11 252 progDenote (compile (Const n) ++ p) s =
adamc@11 253 progDenote p (expDenote (Const n) :: s)
adamc@206 254
adamc@11 255 ]]
adamc@11 256
adamc@11 257 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 258
adam@311 259 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@11 260 *)
adamc@11 261
adamc@4 262 unfold compile.
adamc@11 263 (** [[
adamc@11 264 n : nat
adamc@11 265 s : stack
adamc@11 266 p : list instr
adamc@11 267 ============================
adam@311 268 progDenote ((iConst n :: nil) ++ p) s =
adamc@11 269 progDenote p (expDenote (Const n) :: s)
adamc@206 270
adam@302 271 ]]
adam@302 272 *)
adamc@11 273
adamc@4 274 unfold expDenote.
adamc@11 275 (** [[
adamc@11 276 n : nat
adamc@11 277 s : stack
adamc@11 278 p : list instr
adamc@11 279 ============================
adam@311 280 progDenote ((iConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@206 281
adamc@11 282 ]]
adamc@11 283
adam@311 284 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@11 285
adamc@11 286 unfold progDenote at 1.
adamc@11 287 (** [[
adamc@11 288 n : nat
adamc@11 289 s : stack
adamc@11 290 p : list instr
adamc@11 291 ============================
adamc@11 292 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 293 option stack :=
adamc@11 294 match p0 with
adamc@11 295 | nil => Some s0
adamc@11 296 | i :: p' =>
adamc@11 297 match instrDenote i s0 with
adamc@11 298 | Some s' => progDenote p' s'
adamc@11 299 | None => None (A:=stack)
adamc@11 300 end
adam@311 301 end) ((iConst n :: nil) ++ p) s =
adamc@11 302 progDenote p (n :: s)
adamc@206 303
adamc@11 304 ]]
adamc@11 305
adam@471 306 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@311 307
adam@311 308 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@11 309 *)
adamc@11 310
adamc@4 311 simpl.
adamc@11 312 (** [[
adamc@11 313 n : nat
adamc@11 314 s : stack
adamc@11 315 p : list instr
adamc@11 316 ============================
adamc@11 317 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 318 option stack :=
adamc@11 319 match p0 with
adamc@11 320 | nil => Some s0
adamc@11 321 | i :: p' =>
adamc@11 322 match instrDenote i s0 with
adamc@11 323 | Some s' => progDenote p' s'
adamc@11 324 | None => None (A:=stack)
adamc@11 325 end
adamc@11 326 end) p (n :: s) = progDenote p (n :: s)
adamc@206 327
adamc@11 328 ]]
adamc@11 329
adam@311 330 Now we can unexpand the definition of [progDenote]:%\index{tactics!fold}%
adamc@11 331 *)
adamc@11 332
adamc@11 333 fold progDenote.
adamc@11 334 (** [[
adamc@11 335 n : nat
adamc@11 336 s : stack
adamc@11 337 p : list instr
adamc@11 338 ============================
adamc@11 339 progDenote p (n :: s) = progDenote p (n :: s)
adamc@206 340
adamc@11 341 ]]
adamc@11 342
adam@311 343 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@11 344 *)
adamc@11 345
adamc@4 346 reflexivity.
adamc@2 347
adamc@11 348 (** On to the second inductive case:
adamc@11 349
adamc@11 350 [[
adamc@11 351 b : binop
adamc@11 352 e1 : exp
adamc@11 353 IHe1 : forall (s : stack) (p : list instr),
adamc@11 354 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 355 e2 : exp
adamc@11 356 IHe2 : forall (s : stack) (p : list instr),
adamc@11 357 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 358 ============================
adamc@11 359 forall (s : stack) (p : list instr),
adamc@11 360 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 361 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 362
adamc@11 363 ]]
adamc@11 364
adam@311 365 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@11 366
adam@399 367 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@11 368
adamc@4 369 intros.
adamc@4 370 unfold compile.
adamc@4 371 fold compile.
adamc@4 372 unfold expDenote.
adamc@4 373 fold expDenote.
adamc@11 374
adamc@44 375 (** 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@11 376
adamc@11 377 [[
adamc@11 378 b : binop
adamc@11 379 e1 : exp
adamc@11 380 IHe1 : forall (s : stack) (p : list instr),
adamc@11 381 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 382 e2 : exp
adamc@11 383 IHe2 : forall (s : stack) (p : list instr),
adamc@11 384 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 385 s : stack
adamc@11 386 p : list instr
adamc@11 387 ============================
adam@311 388 progDenote ((compile e2 ++ compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 389 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 390
adamc@11 391 ]]
adamc@11 392
adam@471 393 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@11 394
adam@469 395 Check app_assoc_reverse.
adam@439 396 (** %\vspace{-.15in}%[[
adam@311 397 app_assoc_reverse
adamc@11 398 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@206 399
adamc@11 400 ]]
adamc@11 401
adam@399 402 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@277 403
adam@277 404 SearchRewrite ((_ ++ _) ++ _).
adam@439 405 (** %\vspace{-.15in}%[[
adam@311 406 app_assoc_reverse:
adam@311 407 forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adam@311 408 ]]
adam@311 409 %\vspace{-.25in}%
adam@311 410 [[
adam@311 411 app_assoc: forall (A : Type) (l m n : list A), l ++ m ++ n = (l ++ m) ++ n
adam@277 412
adam@277 413 ]]
adam@277 414
adam@311 415 We use [app_assoc_reverse] to perform a rewrite: %\index{tactics!rewrite}% *)
adamc@11 416
adam@311 417 rewrite app_assoc_reverse.
adamc@11 418
adam@439 419 (** %\noindent{}%changing the conclusion to:
adamc@11 420
adamc@206 421 [[
adam@311 422 progDenote (compile e2 ++ (compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 423 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 424
adamc@11 425 ]]
adamc@11 426
adam@311 427 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@11 428
adamc@4 429 rewrite IHe2.
adamc@11 430 (** [[
adam@311 431 progDenote ((compile e1 ++ iBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 432 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 433
adamc@11 434 ]]
adamc@11 435
adam@311 436 The same process lets us apply the remaining hypothesis.%\index{tactics!rewrite}% *)
adamc@11 437
adam@311 438 rewrite app_assoc_reverse.
adamc@4 439 rewrite IHe1.
adamc@11 440 (** [[
adam@311 441 progDenote ((iBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 442 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 443
adamc@11 444 ]]
adamc@11 445
adam@311 446 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@11 447 *)
adamc@11 448
adamc@11 449 unfold progDenote at 1.
adamc@4 450 simpl.
adamc@11 451 fold progDenote.
adamc@4 452 reflexivity.
adamc@11 453
adam@311 454 (** And the proof is completed, as indicated by the message: *)
adamc@11 455
adam@399 456 (**
adam@399 457 <<
adam@399 458 Proof completed.
adam@399 459 >>
adam@399 460 *)
adamc@11 461
adam@311 462 (** 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@11 463 *)
adamc@11 464
adamc@4 465 Abort.
adamc@2 466
adam@311 467 (** %\index{tactics!induction}\index{tactics!crush}% *)
adam@311 468
adamc@26 469 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 470 progDenote p (expDenote e :: s).
adamc@4 471 induction e; crush.
adamc@4 472 Qed.
adamc@2 473
adam@328 474 (** 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@11 475
adam@399 476 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@210 477
adam@398 478 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@311 479
adam@311 480 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@11 481
adamc@26 482 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 483 intros.
adamc@11 484 (** [[
adamc@11 485 e : exp
adamc@11 486 ============================
adamc@11 487 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@206 488
adamc@11 489 ]]
adamc@11 490
adamc@26 491 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@11 492
adamc@11 493 Check app_nil_end.
adamc@11 494 (** [[
adamc@11 495 app_nil_end
adamc@11 496 : forall (A : Type) (l : list A), l = l ++ nil
adam@302 497 ]]
adam@311 498 %\index{tactics!rewrite}% *)
adamc@11 499
adamc@4 500 rewrite (app_nil_end (compile e)).
adamc@11 501
adam@417 502 (** This time, we explicitly specify the value of the variable [l] from the theorem statement, since multiple expressions of list type appear in the conclusion. The [rewrite] tactic might choose the wrong place to rewrite if we did not specify which we want.
adamc@11 503
adamc@11 504 [[
adamc@11 505 e : exp
adamc@11 506 ============================
adamc@11 507 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@206 508
adamc@11 509 ]]
adamc@11 510
adam@311 511 Now we can apply the lemma.%\index{tactics!rewrite}% *)
adamc@11 512
adamc@26 513 rewrite compile_correct'.
adamc@11 514 (** [[
adamc@11 515 e : exp
adamc@11 516 ============================
adamc@11 517 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@206 518
adamc@11 519 ]]
adamc@11 520
adam@311 521 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@11 522
adamc@4 523 reflexivity.
adamc@4 524 Qed.
adamc@22 525 (* end thide *)
adamc@14 526
adam@475 527 (** This proof can be shortened and automated, but we leave that task as an exercise for the reader. *)
adam@311 528
adamc@14 529
adamc@20 530 (** * Typed Expressions *)
adamc@14 531
adamc@14 532 (** 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@14 533
adamc@20 534 (** ** Source Language *)
adamc@14 535
adamc@15 536 (** We define a trivial language of types to classify our expressions: *)
adamc@15 537
adamc@14 538 Inductive type : Set := Nat | Bool.
adamc@14 539
adam@277 540 (** 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@277 541
adam@277 542 Now we define an expanded set of binary operators. *)
adamc@15 543
adamc@14 544 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 545 | TPlus : tbinop Nat Nat Nat
adamc@14 546 | TTimes : tbinop Nat Nat Nat
adamc@14 547 | TEq : forall t, tbinop t t Bool
adamc@14 548 | TLt : tbinop Nat Nat Bool.
adamc@14 549
adam@398 550 (** 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@15 551
adam@452 552 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@312 553
adamc@15 554 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@15 555
adam@469 556 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@15 557
adam@419 558 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@15 559 *)
adamc@15 560
adam@399 561 (** 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@15 562
adamc@14 563 Inductive texp : type -> Set :=
adamc@14 564 | TNConst : nat -> texp Nat
adamc@14 565 | TBConst : bool -> texp Bool
adam@312 566 | TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
adamc@14 567
adam@447 568 (** 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@15 569
adamc@14 570 Definition typeDenote (t : type) : Set :=
adamc@14 571 match t with
adamc@14 572 | Nat => nat
adamc@14 573 | Bool => bool
adamc@14 574 end.
adamc@14 575
adam@448 576 (** 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@15 577
adamc@207 578 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@207 579 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@207 580 match b with
adamc@207 581 | TPlus => plus
adamc@207 582 | TTimes => mult
adam@277 583 | TEq Nat => beq_nat
adam@277 584 | TEq Bool => eqb
adam@312 585 | TLt => leb
adamc@207 586 end.
adamc@207 587
adam@399 588 (** 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@312 589
adam@471 590 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@15 591
adamc@207 592 Fixpoint texpDenote t (e : texp t) : typeDenote t :=
adamc@207 593 match e with
adamc@14 594 | TNConst n => n
adamc@14 595 | TBConst b => b
adamc@14 596 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 597 end.
adamc@14 598
adamc@17 599 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 600
adamc@17 601 Eval simpl in texpDenote (TNConst 42).
adamc@207 602 (** [= 42 : typeDenote Nat] *)
adamc@207 603
adam@419 604 (* begin hide *)
adam@419 605 Eval simpl in texpDenote (TBConst false).
adam@419 606 (* end hide *)
adamc@17 607 Eval simpl in texpDenote (TBConst true).
adamc@207 608 (** [= true : typeDenote Bool] *)
adamc@207 609
adam@312 610 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 611 (TNConst 7)).
adamc@207 612 (** [= 28 : typeDenote Nat] *)
adamc@207 613
adam@312 614 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 615 (TNConst 7)).
adam@399 616 (** [= false : typeDenote Bool] *)
adamc@207 617
adam@312 618 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 619 (TNConst 7)).
adamc@207 620 (** [= true : typeDenote Bool] *)
adamc@17 621
adam@442 622 (** %\smallskip{}%Now we are ready to define a suitable stack machine target for compilation. *)
adam@442 623
adamc@14 624
adamc@20 625 (** ** Target Language *)
adamc@14 626
adam@442 627 (** 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@18 628
adamc@18 629 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@18 630
adamc@18 631 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 632
adamc@14 633 Definition tstack := list type.
adamc@14 634
adamc@18 635 (** 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@18 636
adamc@18 637 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@18 638
adamc@14 639 Inductive tinstr : tstack -> tstack -> Set :=
adam@312 640 | TiNConst : forall s, nat -> tinstr s (Nat :: s)
adam@312 641 | TiBConst : forall s, bool -> tinstr s (Bool :: s)
adam@311 642 | TiBinop : forall arg1 arg2 res s,
adamc@14 643 tbinop arg1 arg2 res
adamc@14 644 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 645
adamc@18 646 (** 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@18 647
adamc@14 648 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 649 | TNil : forall s, tprog s s
adamc@14 650 | TCons : forall s1 s2 s3,
adamc@14 651 tinstr s1 s2
adamc@14 652 -> tprog s2 s3
adamc@14 653 -> tprog s1 s3.
adamc@14 654
adamc@18 655 (** 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@18 656
adamc@14 657 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 658 match ts with
adamc@14 659 | nil => unit
adamc@14 660 | t :: ts' => typeDenote t * vstack ts'
adamc@14 661 end%type.
adamc@14 662
adam@312 663 (** 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@18 664
adam@312 665 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@18 666
adamc@14 667 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@207 668 match i with
adam@312 669 | TiNConst _ n => fun s => (n, s)
adam@312 670 | TiBConst _ b => fun s => (b, s)
adam@311 671 | TiBinop _ _ _ _ b => fun s =>
adam@312 672 let '(arg1, (arg2, s')) := s in
adam@312 673 ((tbinopDenote b) arg1 arg2, s')
adamc@14 674 end.
adamc@14 675
adamc@18 676 (** 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@18 677 [[
adamc@18 678 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@207 679 match i with
adam@312 680 | TiNConst _ n => (n, s)
adam@312 681 | TiBConst _ b => (b, s)
adam@311 682 | TiBinop _ _ _ _ b =>
adam@312 683 let '(arg1, (arg2, s')) := s in
adam@312 684 ((tbinopDenote b) arg1 arg2, s')
adamc@18 685 end.
adamc@205 686 ]]
adamc@205 687
adam@447 688 The Coq type checker complains that:
adamc@18 689
adam@312 690 <<
adamc@18 691 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@207 692 while it is expected to have type "vstack ?119".
adam@312 693 >>
adamc@207 694
adam@465 695 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@18 696 *)
adamc@18 697
adamc@18 698 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 699
adamc@207 700 Fixpoint tprogDenote ts ts' (p : tprog ts ts') : vstack ts -> vstack ts' :=
adamc@207 701 match p with
adamc@14 702 | TNil _ => fun s => s
adamc@14 703 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 704 end.
adamc@14 705
adam@447 706 (** The same argument-postponing trick is crucial for this definition. *)
adam@447 707
adamc@14 708
adamc@14 709 (** ** Translation *)
adamc@14 710
adamc@19 711 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 712
adamc@207 713 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') : tprog ts' ts'' -> tprog ts ts'' :=
adamc@207 714 match p with
adamc@14 715 | TNil _ => fun p' => p'
adamc@14 716 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 717 end.
adamc@14 718
adamc@19 719 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 720
adamc@207 721 Fixpoint tcompile t (e : texp t) (ts : tstack) : tprog ts (t :: ts) :=
adamc@207 722 match e with
adam@312 723 | TNConst n => TCons (TiNConst _ n) (TNil _)
adam@312 724 | TBConst b => TCons (TiBConst _ b) (TNil _)
adamc@14 725 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adam@311 726 (tconcat (tcompile e1 _) (TCons (TiBinop _ b) (TNil _)))
adamc@14 727 end.
adamc@14 728
adam@398 729 (** 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@19 730
adamc@19 731 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@19 732
adamc@14 733 Print tcompile.
adam@439 734 (** %\vspace{-.15in}%[[
adamc@19 735 tcompile =
adamc@19 736 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 737 tprog ts (t :: ts) :=
adamc@19 738 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adam@312 739 | TNConst n => TCons (TiNConst ts n) (TNil (Nat :: ts))
adam@312 740 | TBConst b => TCons (TiBConst ts b) (TNil (Bool :: ts))
adamc@19 741 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 742 tconcat (tcompile arg2 e2 ts)
adamc@19 743 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adam@311 744 (TCons (TiBinop ts b) (TNil (res :: ts))))
adamc@19 745 end
adamc@19 746 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adam@302 747 ]]
adam@302 748 *)
adamc@19 749
adamc@19 750
adamc@19 751 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 752
adamc@19 753 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adam@399 754 (** [= (42, tt) : vstack (Nat :: nil)] *)
adamc@207 755
adamc@19 756 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adam@399 757 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@207 758
adam@312 759 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2)
adam@312 760 (TNConst 2)) (TNConst 7)) nil) tt.
adam@399 761 (** [= (28, tt) : vstack (Nat :: nil)] *)
adamc@207 762
adam@312 763 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2)
adam@312 764 (TNConst 2)) (TNConst 7)) nil) tt.
adam@399 765 (** [= (false, tt) : vstack (Bool :: nil)] *)
adamc@207 766
adam@312 767 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 768 (TNConst 7)) nil) tt.
adam@399 769 (** [= (true, tt) : vstack (Bool :: nil)] *)
adamc@19 770
adam@442 771 (** %\smallskip{}%The compiler seems to be working, so let us turn to proving that it _always_ works. *)
adam@442 772
adamc@14 773
adamc@20 774 (** ** Translation Correctness *)
adamc@20 775
adamc@20 776 (** We can state a correctness theorem similar to the last one. *)
adamc@20 777
adamc@207 778 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 779 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 780 (* begin hide *)
adamc@20 781 Abort.
adamc@20 782 (* end hide *)
adamc@22 783 (* begin thide *)
adamc@20 784
adam@312 785 (** 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@14 786
adamc@207 787 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 788 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@20 789
adam@419 790 (** 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@20 791
adamc@20 792 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 793
adamc@14 794 induction e; crush.
adamc@20 795
adamc@20 796 (** We are left with this unproved conclusion:
adamc@20 797
adamc@20 798 [[
adamc@20 799 tprogDenote
adamc@20 800 (tconcat (tcompile e2 ts)
adamc@20 801 (tconcat (tcompile e1 (arg2 :: ts))
adam@311 802 (TCons (TiBinop ts t) (TNil (res :: ts))))) s =
adamc@20 803 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@207 804
adamc@20 805 ]]
adamc@20 806
adam@312 807 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@20 808 *)
adamc@207 809
adamc@14 810 Abort.
adamc@14 811
adamc@26 812 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 813 (s : vstack ts),
adamc@14 814 tprogDenote (tconcat p p') s
adamc@14 815 = tprogDenote p' (tprogDenote p s).
adamc@14 816 induction p; crush.
adamc@14 817 Qed.
adamc@14 818
adamc@20 819 (** This one goes through completely automatically.
adamc@20 820
adam@316 821 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@20 822
adam@375 823 Hint Rewrite tconcat_correct.
adamc@14 824
adam@419 825 (** 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@312 826
adam@312 827 Now we are ready to return to [tcompile_correct'], proving it automatically this time. *)
adamc@20 828
adamc@207 829 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 830 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@14 831 induction e; crush.
adamc@14 832 Qed.
adamc@14 833
adamc@20 834 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 835
adam@375 836 Hint Rewrite tcompile_correct'.
adamc@14 837
adamc@207 838 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 839 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 840 crush.
adamc@14 841 Qed.
adamc@22 842 (* end thide *)
adam@312 843
adam@399 844 (** 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@312 845
adam@312 846 Extraction tcompile.
adam@312 847
adam@312 848 (** <<
adam@312 849 let rec tcompile t e ts =
adam@312 850 match e with
adam@312 851 | TNConst n ->
adam@312 852 TCons (ts, (Cons (Nat, ts)), (Cons (Nat, ts)), (TiNConst (ts, n)), (TNil
adam@312 853 (Cons (Nat, ts))))
adam@312 854 | TBConst b ->
adam@312 855 TCons (ts, (Cons (Bool, ts)), (Cons (Bool, ts)), (TiBConst (ts, b)),
adam@312 856 (TNil (Cons (Bool, ts))))
adam@312 857 | TBinop (t1, t2, t0, b, e1, e2) ->
adam@312 858 tconcat ts (Cons (t2, ts)) (Cons (t0, ts)) (tcompile t2 e2 ts)
adam@312 859 (tconcat (Cons (t2, ts)) (Cons (t1, (Cons (t2, ts)))) (Cons (t0, ts))
adam@312 860 (tcompile t1 e1 (Cons (t2, ts))) (TCons ((Cons (t1, (Cons (t2,
adam@312 861 ts)))), (Cons (t0, ts)), (Cons (t0, ts)), (TiBinop (t1, t2, t0, ts,
adam@312 862 b)), (TNil (Cons (t0, ts))))))
adam@312 863 >>
adam@312 864
adam@312 865 We can compile this code with the usual OCaml compiler and obtain an executable program with halfway decent performance.
adam@312 866
adam@312 867 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. *)