Mercurial > cpdt > repo

annotate src/StackMachine.v @ 312:495153a41819

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Pass through second half of StackMachine
author Adam Chlipala <adam@chlipala.net>
date Thu, 01 Sep 2011 11:32:15 -0400
parents 4cb3ba8604bc
children d5787b70cf48
rev   line source
adam@298 1 (* Copyright (c) 2008-2011, Adam Chlipala
adamc@2 2 *
adamc@2 3 * This work is licensed under a
adamc@2 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@2 5 * Unported License.
adamc@2 6 * The license text is available at:
adamc@2 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@2 8 *)
adamc@2 9
adamc@3 10 (* begin hide *)
adam@312 11 Require Import Bool Arith List.
adamc@2 12
adamc@2 13 Require Import Tactics.
adamc@14 14
adamc@14 15 Set Implicit Arguments.
adamc@3 16 (* end hide *)
adamc@2 17
adamc@2 18
adamc@25 19 (** %\chapter{Some Quick Examples}% *)
adamc@25 20
adamc@25 21
adam@279 22 (** I will start off by jumping right in to a fully-worked set of examples, building certified compilers from increasingly complicated source languages to stack machines. We will meet a few useful tactics and see how they can be used in manual proofs, and we will also see how easily these proofs can be automated instead. This chapter is not meant to give full explanations of the features that are employed. Rather, it is meant more as an advertisement of what is possible. Later chapters will introduce all of the concepts in bottom-up fashion.
adam@279 23
adam@312 24 As always, you can step through the source file %\texttt{%#<tt>#StackMachine.v#</tt>#%}% for this chapter interactively in Proof General. Alternatively, to get a feel for the whole lifecycle of creating a Coq development, you can enter the pieces of source code in this chapter in a new %\texttt{%#<tt>#.v#</tt>#%}% file in an Emacs buffer. If you do the latter, include two lines %\index{Vernacular commands!Require}%[Require Import Bool] #<span class="inlinecode"><span class="id" type="var">#%\coqdocconstructor{%Arith%}%#</span></span># #<span class="inlinecode"><span class="id" type="var">#%\coqdocconstructor{%List%}%#</span></span># [Tactics.] and %\index{Vernacular commands!Set Implicit Arguments}%[Set Implicit] #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Arguments%}%#</span></span>#[.] at the start of the file, to match some code hidden in this rendering of the chapter source. In general, similar commands will be hidden in the book rendering of each chapter's source code, so you will need to insert them in from-scratch replayings of the code that is presented. To be more specific, every chapter begins with some imports of other modules, followed by [Set Implicit] #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Arguments%}%#</span></span>#[.], where the latter affects the default behavior of definitions regarding type inference.
adam@307 25 *)
adamc@9 26
adamc@9 27
adamc@20 28 (** * Arithmetic Expressions Over Natural Numbers *)
adamc@2 29
adamc@40 30 (** We will begin with that staple of compiler textbooks, arithmetic expressions over a single type of numbers. *)
adamc@9 31
adamc@20 32 (** ** Source Language *)
adamc@9 33
adam@311 34 (** We begin with the syntax of the source language.%\index{Vernacular commands!Inductive}% *)
adamc@2 35
adamc@4 36 Inductive binop : Set := Plus | Times.
adamc@2 37
adam@311 38 (** 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 %\texttt{%#<tt>#data#</tt>#%}%, %\texttt{%#<tt>#datatype#</tt>#%}%, or %\texttt{%#<tt>#type#</tt>#%}%. This is not just a trivial surface syntax difference; inductive types in Coq are much more expressive than garden variety algebraic datatypes, essentially enabling us to encode all of mathematics, though we begin humbly in this chapter. Second, there is the %\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 39
adamc@4 40 Inductive exp : Set :=
adamc@4 41 | Const : nat -> exp
adamc@4 42 | Binop : binop -> exp -> exp -> exp.
adamc@2 43
adamc@9 44 (** Now we define the type of arithmetic expressions. We write that a constant may be built from one argument, a natural number; and a binary operation may be built from a choice of operator and two operand expressions.
adamc@9 45
adam@311 46 A note for readers following along in the PDF version: %\index{coqdoc}%coqdoc supports pretty-printing of tokens in LaTeX or HTML. Where you see a right arrow character, the source contains the ASCII text %\texttt{%#<tt>#->#</tt>#%}%. Other examples of this substitution appearing in this chapter are a double right arrow for %\texttt{%#<tt>#=>#</tt>#%}%, the inverted %`%#'#A' symbol for %\texttt{%#<tt>#forall#</tt>#%}%, and the Cartesian product %`%#'#X' for %\texttt{%#<tt>#*#</tt>#%}%. When in doubt about the ASCII version of a symbol, you can consult the chapter source code.
adamc@9 47
adamc@9 48 %\medskip%
adamc@9 49
adam@311 50 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 51
adamc@4 52 Definition binopDenote (b : binop) : nat -> nat -> nat :=
adamc@4 53 match b with
adamc@4 54 | Plus => plus
adamc@4 55 | Times => mult
adamc@4 56 end.
adamc@2 57
adamc@9 58 (** The meaning of a binary operator is a binary function over naturals, defined with pattern-matching notation analogous to the %\texttt{%#<tt>#case#</tt>#%}% and %\texttt{%#<tt>#match#</tt>#%}% of ML and Haskell, and referring to the functions [plus] and [mult] from the Coq standard library. The keyword [Definition] is Coq's all-purpose notation for binding a term of the programming language to a name, with some associated syntactic sugar, like the notation we see here for defining a function. That sugar could be expanded to yield this definition:
adamc@9 59
adamc@9 60 [[
adamc@9 61 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
adamc@9 62 match b with
adamc@9 63 | Plus => plus
adamc@9 64 | Times => mult
adamc@9 65 end.
adamc@9 66
adamc@205 67 ]]
adamc@205 68
adamc@9 69 In this example, we could also omit all of the type annotations, arriving at:
adamc@9 70
adamc@9 71 [[
adamc@9 72 Definition binopDenote := fun b =>
adamc@9 73 match b with
adamc@9 74 | Plus => plus
adamc@9 75 | Times => mult
adamc@9 76 end.
adamc@9 77
adamc@205 78 ]]
adamc@205 79
adam@311 80 Languages like Haskell and ML have a convenient %\index{principal typing}\index{type inference}\emph{%#<i>#principal typing#</i>#%}% property, which gives us strong guarantees about how effective type inference will be. Unfortunately, Coq's type system is so expressive that any kind of %``%#"#complete#"#%''% type inference is impossible, and the task even seems to be hard heuristically in practice. Nonetheless, Coq includes some very helpful heuristics, many of them copying the workings of Haskell and ML type-checkers for programs that fall in simple fragments of Coq's language.
adamc@9 81
adam@311 82 This is as good a time as any to mention the preponderance of different languages associated with Coq. The theoretical foundation of Coq is a formal system called the %\index{Calculus of Inductive Constructions}\index{CIC|see{Calculus of Inductive Constructions}}\emph{%#<i>#Calculus of Inductive Constructions (CIC)#</i>#%}~\cite{CIC}%, which is an extension of the older %\index{Calculus of Constructions}\index{CoC|see{Calculus of Constructions}}\emph{%#<i>#Calculus of Constructions (CoC)#</i>#%}~\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}\emph{%#<i>#strong normalization#</i>#%}~\cite{CIC}%, meaning that every program (and, more importantly, every proof term) terminates; and %\index{relative consistency}\emph{%#<i>#relative consistency#</i>#%}~\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 83
adam@311 84 Coq is actually based on an extension of CIC called %\index{Gallina}\emph{%#<i>#Gallina#</i>#%}%. The text after the [:=] and before the period in the last code example is a term of Gallina. Gallina adds many useful features that are not compiled internally to more primitive CIC features. The important metatheorems about CIC have not been extended to the full breadth of these features, but most Coq users do not seem to lose much sleep over this omission.
adamc@9 85
adam@311 86 Next, there is %\index{Ltac}\emph{%#<i>#Ltac#</i>#%}%, Coq's domain-specific language for writing proofs and decision procedures. We will see some basic examples of Ltac later in this chapter, and much of this book is devoted to more involved Ltac examples.
adamc@9 87
adam@311 88 Finally, commands like [Inductive] and [Definition] are part of %\index{Vernacular commands}\emph{%#<i>#the Vernacular#</i>#%}%, which includes all sorts of useful queries and requests to the Coq system. Every Coq source file is a series of vernacular commands, where many command forms take arguments that are Gallina or Ltac programs. (Actually, Coq source files are more like %\emph{%#<i>#trees#</i>#%}% of vernacular commands, thanks to various nested scoping constructs.)
adamc@9 89
adamc@9 90 %\medskip%
adamc@9 91
adam@311 92 We can give a simple definition of the meaning of an expression:%\index{Vernacular commands!Fixpoint}% *)
adamc@9 93
adamc@4 94 Fixpoint expDenote (e : exp) : nat :=
adamc@4 95 match e with
adamc@4 96 | Const n => n
adamc@4 97 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
adamc@4 98 end.
adamc@2 99
adamc@9 100 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
adamc@2 101
adam@311 102 (** It is convenient to be able to test definitions before starting to prove things about them. We can verify that our semantics is sensible by evaluating some sample uses, using the command %\index{Vernacular commands!Eval}%[Eval]. This command takes an argument expressing a %\index{reduction strategy}\emph{%#<i>#reduction strategy#</i>#%}%, or an %``%#"#order of evaluation.#"#%''% Unlike with ML, which hardcodes an %\emph{%#<i>#eager#</i>#%}% reduction strategy, or Haskell, which hardcodes a %\emph{%#<i>#lazy#</i>#%}% 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.
adam@311 103
adam@311 104 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@16 105
adamc@16 106 Eval simpl in expDenote (Const 42).
adamc@205 107 (** [= 42 : nat] *)
adamc@205 108
adamc@16 109 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
adamc@205 110 (** [= 4 : nat] *)
adamc@205 111
adamc@16 112 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adamc@205 113 (** [= 28 : nat] *)
adamc@9 114
adamc@20 115 (** ** Target Language *)
adamc@4 116
adamc@10 117 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
adamc@2 118
adamc@4 119 Inductive instr : Set :=
adam@311 120 | iConst : nat -> instr
adam@311 121 | iBinop : binop -> instr.
adamc@2 122
adamc@4 123 Definition prog := list instr.
adamc@4 124 Definition stack := list nat.
adamc@2 125
adamc@10 126 (** An instruction either pushes a constant onto the stack or pops two arguments, applies a binary operator to them, and pushes the result onto the stack. A program is a list of instructions, and a stack is a list of natural numbers.
adamc@10 127
adam@311 128 We can give instructions meanings as functions from stacks to optional stacks, where running an instruction results in [None] in case of a stack underflow and results in [Some s'] when the result of execution is the new stack [s']. %\index{Gallina operators!::}%The infix operator [::] is %``%#"#list cons#"#%''% from the Coq standard library.%\index{Gallina terms!option}% *)
adamc@10 129
adamc@4 130 Definition instrDenote (i : instr) (s : stack) : option stack :=
adamc@4 131 match i with
adam@311 132 | iConst n => Some (n :: s)
adam@311 133 | iBinop b =>
adamc@4 134 match s with
adamc@4 135 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
adamc@4 136 | _ => None
adamc@4 137 end
adamc@4 138 end.
adamc@2 139
adam@311 140 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
adamc@206 141
adamc@206 142 Fixpoint progDenote (p : prog) (s : stack) : option stack :=
adamc@206 143 match p with
adamc@206 144 | nil => Some s
adamc@206 145 | i :: p' =>
adamc@206 146 match instrDenote i s with
adamc@206 147 | None => None
adamc@206 148 | Some s' => progDenote p' s'
adamc@206 149 end
adamc@206 150 end.
adamc@2 151
adamc@4 152
adamc@9 153 (** ** Translation *)
adamc@4 154
adam@311 155 (** Our compiler itself is now unsurprising. The list concatenation operator %\index{Gallina operators!++}%[++] comes from the Coq standard library. *)
adamc@2 156
adamc@4 157 Fixpoint compile (e : exp) : prog :=
adamc@4 158 match e with
adam@311 159 | Const n => iConst n :: nil
adam@311 160 | Binop b e1 e2 => compile e2 ++ compile e1 ++ iBinop b :: nil
adamc@4 161 end.
adamc@2 162
adamc@2 163
adamc@16 164 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
adamc@16 165
adamc@16 166 Eval simpl in compile (Const 42).
adam@311 167 (** [= iConst 42 :: nil : prog] *)
adamc@206 168
adamc@16 169 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
adam@311 170 (** [= iConst 2 :: iConst 2 :: iBinop Plus :: nil : prog] *)
adamc@206 171
adamc@16 172 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
adam@311 173 (** [= iConst 7 :: iConst 2 :: iConst 2 :: iBinop Plus :: iBinop Times :: nil : prog] *)
adamc@16 174
adamc@40 175 (** We can also run our compiled programs and check that they give the right results. *)
adamc@16 176
adamc@16 177 Eval simpl in progDenote (compile (Const 42)) nil.
adamc@206 178 (** [= Some (42 :: nil) : option stack] *)
adamc@206 179
adamc@16 180 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
adamc@206 181 (** [= Some (4 :: nil) : option stack] *)
adamc@206 182
adam@311 183 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2))
adam@311 184 (Const 7))) nil.
adamc@206 185 (** [= Some (28 :: nil) : option stack] *)
adamc@16 186
adamc@16 187
adamc@20 188 (** ** Translation Correctness *)
adamc@4 189
adam@311 190 (** 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@11 191
adamc@26 192 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 193 (* begin hide *)
adamc@11 194 Abort.
adamc@11 195 (* end hide *)
adamc@22 196 (* begin thide *)
adamc@11 197
adam@311 198 (** Though a pencil-and-paper proof might clock out at this point, writing %``%#"#by a routine induction on [e],#"#%''% it turns out not to make sense to attack this proof directly. We need to use the standard trick of %\index{strengthening the induction hypothesis}\emph{%#<i>#strengthening the induction hypothesis#</i>#%}%. We do that by proving an auxiliary lemma, using the command [Lemma] that is a synonym for [Theorem], conventionally used for less important theorems that appear in the proofs of primary theorems.%\index{Vernacular commands!Lemma}% *)
adamc@2 199
adamc@206 200 Lemma compile_correct' : forall e p s,
adamc@206 201 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
adamc@11 202
adam@311 203 (** After the period in the [Lemma] command, we are in %\index{interactive proof-editing mode}\emph{%#<i>#the interactive proof-editing mode#</i>#%}%. We find ourselves staring at this ominous screen of text:
adamc@11 204
adamc@11 205 [[
adamc@11 206 1 subgoal
adamc@11 207
adamc@11 208 ============================
adamc@15 209 forall (e : exp) (p : list instr) (s : stack),
adamc@15 210 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
adamc@206 211
adamc@11 212 ]]
adamc@11 213
adam@311 214 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@11 215
adam@311 216 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@11 217
adam@311 218 We manipulate the proof state by running commands called %\index{tactics}\emph{%#<i>#tactics#</i>#%}%. Let us start out by running one of the most important tactics:%\index{tactics!induction}%
adamc@11 219 *)
adamc@11 220
adamc@4 221 induction e.
adamc@2 222
adamc@11 223 (** 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@11 224
adam@311 225 %\vspace{.1in} \noindent 2 \coqdockw{subgoals}\vspace{-.1in}%#<tt>2 subgoals</tt>#
adam@311 226
adamc@11 227 [[
adamc@11 228 n : nat
adamc@11 229 ============================
adamc@11 230 forall (s : stack) (p : list instr),
adamc@11 231 progDenote (compile (Const n) ++ p) s =
adamc@11 232 progDenote p (expDenote (Const n) :: s)
adamc@11 233 ]]
adam@311 234 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
adamc@11 235 [[
adamc@11 236 forall (s : stack) (p : list instr),
adamc@11 237 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 238 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 239
adamc@11 240 ]]
adamc@11 241
adam@311 242 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@11 243
adam@311 244 We begin the first case with another very common tactic.%\index{tactics!intros}%
adamc@11 245 *)
adamc@11 246
adamc@4 247 intros.
adamc@11 248
adamc@11 249 (** The current subgoal changes to:
adamc@11 250 [[
adamc@11 251
adamc@11 252 n : nat
adamc@11 253 s : stack
adamc@11 254 p : list instr
adamc@11 255 ============================
adamc@11 256 progDenote (compile (Const n) ++ p) s =
adamc@11 257 progDenote p (expDenote (Const n) :: s)
adamc@206 258
adamc@11 259 ]]
adamc@11 260
adamc@11 261 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
adamc@11 262
adam@311 263 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@11 264 *)
adamc@11 265
adam@311 266 (* begin hide *)
adamc@4 267 unfold compile.
adam@311 268 (* end hide *)
adam@311 269 (** %\coqdockw{unfold} \coqdocdefinition{compile}.%#<tt>unfold compile.</tt># *)
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 =
adamc@11 276 progDenote p (expDenote (Const n) :: s)
adamc@206 277
adam@302 278 ]]
adam@302 279 *)
adamc@11 280
adam@311 281 (* begin hide *)
adamc@4 282 unfold expDenote.
adam@311 283 (* end hide *)
adam@311 284 (** %\coqdockw{unfold} \coqdocdefinition{expDenote}.%#<tt>unfold expDenote.</tt># *)
adamc@11 285 (** [[
adamc@11 286 n : nat
adamc@11 287 s : stack
adamc@11 288 p : list instr
adamc@11 289 ============================
adam@311 290 progDenote ((iConst n :: nil) ++ p) s = progDenote p (n :: s)
adamc@206 291
adamc@11 292 ]]
adamc@11 293
adam@311 294 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@11 295
adam@311 296 (* begin hide *)
adamc@11 297 unfold progDenote at 1.
adam@311 298 (* end hide *)
adam@311 299 (** %\coqdockw{unfold} \coqdocdefinition{progDenote} \coqdoctac{at} 1.%#<tt>unfold progDenote at 1.</tt># *)
adamc@11 300 (** [[
adamc@11 301 n : nat
adamc@11 302 s : stack
adamc@11 303 p : list instr
adamc@11 304 ============================
adamc@11 305 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 306 option stack :=
adamc@11 307 match p0 with
adamc@11 308 | nil => Some s0
adamc@11 309 | i :: p' =>
adamc@11 310 match instrDenote i s0 with
adamc@11 311 | Some s' => progDenote p' s'
adamc@11 312 | None => None (A:=stack)
adamc@11 313 end
adam@311 314 end) ((iConst n :: nil) ++ p) s =
adamc@11 315 progDenote p (n :: s)
adamc@206 316
adamc@11 317 ]]
adamc@11 318
adam@311 319 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 %\coqdocconstructor{None} (\coqdocvar{A}:=\coqdocdefinition{stack})%#<tt>None (A:=stack)</tt>#, which has an annotation specifying that the type of the term ought to be [option A]. This is phrased as an explicit instantiation of a named type parameter [A] from the definition of [option].
adam@311 320
adam@311 321 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@11 322 *)
adamc@11 323
adamc@4 324 simpl.
adamc@11 325 (** [[
adamc@11 326 n : nat
adamc@11 327 s : stack
adamc@11 328 p : list instr
adamc@11 329 ============================
adamc@11 330 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
adamc@11 331 option stack :=
adamc@11 332 match p0 with
adamc@11 333 | nil => Some s0
adamc@11 334 | i :: p' =>
adamc@11 335 match instrDenote i s0 with
adamc@11 336 | Some s' => progDenote p' s'
adamc@11 337 | None => None (A:=stack)
adamc@11 338 end
adamc@11 339 end) p (n :: s) = progDenote p (n :: s)
adamc@206 340
adamc@11 341 ]]
adamc@11 342
adam@311 343 Now we can unexpand the definition of [progDenote]:%\index{tactics!fold}%
adamc@11 344 *)
adamc@11 345
adam@311 346 (* begin hide *)
adamc@11 347 fold progDenote.
adam@311 348 (* end hide *)
adam@311 349 (** %\coqdockw{fold} \coqdocdefinition{progDenote}.%#<tt>fold progDenote.</tt># *)
adamc@11 350 (** [[
adamc@11 351 n : nat
adamc@11 352 s : stack
adamc@11 353 p : list instr
adamc@11 354 ============================
adamc@11 355 progDenote p (n :: s) = progDenote p (n :: s)
adamc@206 356
adamc@11 357 ]]
adamc@11 358
adam@311 359 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@11 360 *)
adamc@11 361
adamc@4 362 reflexivity.
adamc@2 363
adamc@11 364 (** On to the second inductive case:
adamc@11 365
adamc@11 366 [[
adamc@11 367 b : binop
adamc@11 368 e1 : exp
adamc@11 369 IHe1 : forall (s : stack) (p : list instr),
adamc@11 370 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 371 e2 : exp
adamc@11 372 IHe2 : forall (s : stack) (p : list instr),
adamc@11 373 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 374 ============================
adamc@11 375 forall (s : stack) (p : list instr),
adamc@11 376 progDenote (compile (Binop b e1 e2) ++ p) s =
adamc@11 377 progDenote p (expDenote (Binop b e1 e2) :: s)
adamc@206 378
adamc@11 379 ]]
adamc@11 380
adam@311 381 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@11 382
adam@311 383 We start out the same way as before, introducing new free variables and unfolding and folding the appropriate definitions. The seemingly frivolous [unfold]/%\coqdockw{fold}%#<tt>fold</tt># pairs are actually accomplishing useful work, because [unfold] will sometimes perform easy simplifications. %\index{tactics!intros}\index{tactics!unfold}\index{tactics!fold}% *)
adamc@11 384
adamc@4 385 intros.
adam@311 386 (* begin hide *)
adamc@4 387 unfold compile.
adamc@4 388 fold compile.
adamc@4 389 unfold expDenote.
adamc@4 390 fold expDenote.
adam@311 391 (* end hide *)
adam@311 392 (** %\coqdockw{unfold} \coqdocdefinition{compile}.
adam@311 393 \coqdockw{fold} \coqdocdefinition{compile}.
adam@311 394 \coqdockw{unfold} \coqdocdefinition{expDenote}.
adam@311 395 \coqdockw{fold} \coqdocdefinition{expDenote}.%
adam@311 396 #<tt>unfold compile. fold compile. unfold expDenote. fold expDenote.</tt># *)
adamc@11 397
adamc@44 398 (** 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@11 399
adamc@11 400 [[
adamc@11 401 b : binop
adamc@11 402 e1 : exp
adamc@11 403 IHe1 : forall (s : stack) (p : list instr),
adamc@11 404 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
adamc@11 405 e2 : exp
adamc@11 406 IHe2 : forall (s : stack) (p : list instr),
adamc@11 407 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
adamc@11 408 s : stack
adamc@11 409 p : list instr
adamc@11 410 ============================
adam@311 411 progDenote ((compile e2 ++ compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 412 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 413
adamc@11 414 ]]
adamc@11 415
adam@311 416 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@11 417
adam@311 418 Check app_assoc.
adam@311 419
adamc@11 420 (** [[
adam@311 421 app_assoc_reverse
adamc@11 422 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adamc@206 423
adamc@11 424 ]]
adamc@11 425
adam@311 426 If we did not already know the name of the theorem, we could use the %\index{Vernacular commands!SearchRewrite}\coqdockw{%#<tt>#SearchRewrite#</tt>#%}% command to find it, based on a pattern that we would like to rewrite: *)
adam@277 427
adam@311 428 (* begin hide *)
adam@277 429 SearchRewrite ((_ ++ _) ++ _).
adam@311 430 (* end hide *)
adam@311 431 (** %\coqdockw{%#<tt>#SearchRewrite#</tt>#%}% [((_ ++ _) ++ _).] *)
adam@277 432 (** [[
adam@311 433 app_assoc_reverse:
adam@311 434 forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
adam@311 435 ]]
adam@311 436 %\vspace{-.25in}%
adam@311 437 [[
adam@311 438 app_assoc: forall (A : Type) (l m n : list A), l ++ m ++ n = (l ++ m) ++ n
adam@277 439
adam@277 440 ]]
adam@277 441
adam@311 442 We use [app_assoc_reverse] to perform a rewrite: %\index{tactics!rewrite}% *)
adamc@11 443
adam@311 444 rewrite app_assoc_reverse.
adamc@11 445
adamc@206 446 (** changing the conclusion to:
adamc@11 447
adamc@206 448 [[
adam@311 449 progDenote (compile e2 ++ (compile e1 ++ iBinop b :: nil) ++ p) s =
adamc@11 450 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 451
adamc@11 452 ]]
adamc@11 453
adam@311 454 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@11 455
adamc@4 456 rewrite IHe2.
adamc@11 457 (** [[
adam@311 458 progDenote ((compile e1 ++ iBinop b :: nil) ++ p) (expDenote e2 :: s) =
adamc@11 459 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 460
adamc@11 461 ]]
adamc@11 462
adam@311 463 The same process lets us apply the remaining hypothesis.%\index{tactics!rewrite}% *)
adamc@11 464
adam@311 465 rewrite app_assoc_reverse.
adamc@4 466 rewrite IHe1.
adamc@11 467 (** [[
adam@311 468 progDenote ((iBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
adamc@11 469 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
adamc@206 470
adamc@11 471 ]]
adamc@11 472
adam@311 473 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@11 474 *)
adamc@11 475
adam@311 476 (* begin hide *)
adamc@11 477 unfold progDenote at 1.
adamc@4 478 simpl.
adamc@11 479 fold progDenote.
adamc@4 480 reflexivity.
adam@311 481 (* end hide *)
adam@311 482 (** %\coqdockw{unfold} \coqdocdefinition{progDenote} \coqdoctac{at} 1. \coqdockw{simpl}. \coqdockw{fold} \coqdocdefinition{progDenote}. \coqdockw{reflexivity}.%#<tt>unfold progDenote at 1. simpl. fold progDenote. reflexivity.</tt># *)
adamc@11 483
adam@311 484 (** And the proof is completed, as indicated by the message: *)
adamc@11 485
adam@311 486 (** %\coqdockw{Proof} \coqdockw{completed}.%#<tt>Proof completed.</tt># *)
adamc@11 487
adam@311 488 (** 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@11 489 *)
adamc@11 490
adamc@4 491 Abort.
adamc@2 492
adam@311 493 (** %\index{tactics!induction}\index{tactics!crush}% *)
adam@311 494
adamc@26 495 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
adamc@4 496 progDenote p (expDenote e :: s).
adamc@4 497 induction e; crush.
adamc@4 498 Qed.
adamc@2 499
adam@311 500 (** 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{tacticals!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@11 501
adamc@210 502 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@210 503
adam@312 504 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 %\emph{%#<i>#proof script#</i>#%}%, or a series of Ltac programs; while [Qed] uses the result of the script to generate a %\emph{%#<i>#proof term#</i>#%}%, 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@311 505
adam@311 506 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@11 507
adamc@26 508 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
adamc@11 509 intros.
adamc@11 510 (** [[
adamc@11 511 e : exp
adamc@11 512 ============================
adamc@11 513 progDenote (compile e) nil = Some (expDenote e :: nil)
adamc@206 514
adamc@11 515 ]]
adamc@11 516
adamc@26 517 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@11 518
adamc@11 519 Check app_nil_end.
adamc@11 520 (** [[
adamc@11 521 app_nil_end
adamc@11 522 : forall (A : Type) (l : list A), l = l ++ nil
adam@302 523 ]]
adam@311 524 %\index{tactics!rewrite}% *)
adamc@11 525
adamc@4 526 rewrite (app_nil_end (compile e)).
adamc@11 527
adamc@11 528 (** This time, we explicitly specify the value of the variable [l] from the theorem statement, since multiple expressions of list type appear in the conclusion. [rewrite] might choose the wrong place to rewrite if we did not specify which we want.
adamc@11 529
adamc@11 530 [[
adamc@11 531 e : exp
adamc@11 532 ============================
adamc@11 533 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
adamc@206 534
adamc@11 535 ]]
adamc@11 536
adam@311 537 Now we can apply the lemma.%\index{tactics!rewrite}% *)
adamc@11 538
adamc@26 539 rewrite compile_correct'.
adamc@11 540 (** [[
adamc@11 541 e : exp
adamc@11 542 ============================
adamc@11 543 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
adamc@206 544
adamc@11 545 ]]
adamc@11 546
adam@311 547 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@11 548
adamc@4 549 reflexivity.
adamc@4 550 Qed.
adamc@22 551 (* end thide *)
adamc@14 552
adam@311 553 (** This proof can be shortened and made automated, but we leave that as an exercise for the reader. *)
adam@311 554
adamc@14 555
adamc@20 556 (** * Typed Expressions *)
adamc@14 557
adamc@14 558 (** 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@14 559
adamc@20 560 (** ** Source Language *)
adamc@14 561
adamc@15 562 (** We define a trivial language of types to classify our expressions: *)
adamc@15 563
adamc@14 564 Inductive type : Set := Nat | Bool.
adamc@14 565
adam@277 566 (** 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@277 567
adam@277 568 Now we define an expanded set of binary operators. *)
adamc@15 569
adamc@14 570 Inductive tbinop : type -> type -> type -> Set :=
adamc@14 571 | TPlus : tbinop Nat Nat Nat
adamc@14 572 | TTimes : tbinop Nat Nat Nat
adamc@14 573 | TEq : forall t, tbinop t t Bool
adamc@14 574 | TLt : tbinop Nat Nat Bool.
adamc@14 575
adam@307 576 (** The definition of [tbinop] is different from [binop] in an important way. Where we declared that [binop] has type [Set], here we declare that [tbinop] has type [type -> type -> type -> Set]. We define [tbinop] as an %\emph{%#<i>#indexed type family#</i>#%}%. Indexed inductive types are at the heart of Coq's expressive power; almost everything else of interest is defined in terms of them.
adamc@15 577
adam@312 578 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 %\emph{%#<i>#same#</i>#%}% type.
adam@312 579
adamc@15 580 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@15 581
adam@312 582 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}}\emph{%#<i>#Generalized algebraic datatypes (GADTs)#</i>#%}~\cite{GADT}% are a popular feature in %\index{GHC Haskell}%GHC Haskell and other languages that removes this first restriction.
adamc@15 583
adam@312 584 The second restriction is not lifted by GADTs. In ML and Haskell, indices of types must be types and may not be %\emph{%#<i>#expressions#</i>#%}%. In Coq, types may be indexed by arbitrary Gallina terms. Type indices can live in the same universe as programs, and we can compute with them just like regular programs. Haskell supports a hobbled form of computation in type indices based on %\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@15 585 *)
adamc@15 586
adam@312 587 (** 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}\emph{%#<i>#meta language#</i>#%}% and a language being formalized the %\index{object language}\emph{%#<i>#object language#</i>#%}%.) *)
adamc@15 588
adamc@14 589 Inductive texp : type -> Set :=
adamc@14 590 | TNConst : nat -> texp Nat
adamc@14 591 | TBConst : bool -> texp Bool
adam@312 592 | TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
adamc@14 593
adamc@15 594 (** 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@15 595
adamc@14 596 Definition typeDenote (t : type) : Set :=
adamc@14 597 match t with
adamc@14 598 | Nat => nat
adamc@14 599 | Bool => bool
adamc@14 600 end.
adamc@14 601
adam@312 602 (** 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@15 603
adamc@207 604 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
adamc@207 605 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
adamc@207 606 match b with
adamc@207 607 | TPlus => plus
adamc@207 608 | TTimes => mult
adam@277 609 | TEq Nat => beq_nat
adam@277 610 | TEq Bool => eqb
adam@312 611 | TLt => leb
adamc@207 612 end.
adamc@207 613
adam@312 614 (** 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}\emph{%#<i>#dependent pattern match#</i>#%}%, where the necessary %\emph{%#<i>#type#</i>#%}% of each case body depends on the %\emph{%#<i>#value#</i>#%}% 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@312 615
adamc@15 616 The same tricks suffice to define an expression denotation function in an unsurprising way:
adamc@15 617 *)
adamc@15 618
adamc@207 619 Fixpoint texpDenote t (e : texp t) : typeDenote t :=
adamc@207 620 match e with
adamc@14 621 | TNConst n => n
adamc@14 622 | TBConst b => b
adamc@14 623 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
adamc@14 624 end.
adamc@14 625
adamc@17 626 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
adamc@17 627
adamc@17 628 Eval simpl in texpDenote (TNConst 42).
adamc@207 629 (** [= 42 : typeDenote Nat] *)
adamc@207 630
adamc@17 631 Eval simpl in texpDenote (TBConst true).
adamc@207 632 (** [= true : typeDenote Bool] *)
adamc@207 633
adam@312 634 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 635 (TNConst 7)).
adamc@207 636 (** [= 28 : typeDenote Nat] *)
adamc@207 637
adam@312 638 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 639 (TNConst 7)).
adam@312 640 (** [= ] %\coqdocconstructor{%#<tt>#false#</tt>#%}% [ : typeDenote Bool] *)
adamc@207 641
adam@312 642 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 643 (TNConst 7)).
adamc@207 644 (** [= true : typeDenote Bool] *)
adamc@17 645
adamc@14 646
adamc@20 647 (** ** Target Language *)
adamc@14 648
adam@292 649 (** 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@18 650
adamc@18 651 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@18 652
adamc@18 653 We start by defining stack types, which classify sets of possible stacks. *)
adamc@18 654
adamc@14 655 Definition tstack := list type.
adamc@14 656
adamc@18 657 (** 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@18 658
adamc@18 659 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@18 660
adamc@14 661 Inductive tinstr : tstack -> tstack -> Set :=
adam@312 662 | TiNConst : forall s, nat -> tinstr s (Nat :: s)
adam@312 663 | TiBConst : forall s, bool -> tinstr s (Bool :: s)
adam@311 664 | TiBinop : forall arg1 arg2 res s,
adamc@14 665 tbinop arg1 arg2 res
adamc@14 666 -> tinstr (arg1 :: arg2 :: s) (res :: s).
adamc@14 667
adamc@18 668 (** 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@18 669
adamc@14 670 Inductive tprog : tstack -> tstack -> Set :=
adamc@14 671 | TNil : forall s, tprog s s
adamc@14 672 | TCons : forall s1 s2 s3,
adamc@14 673 tinstr s1 s2
adamc@14 674 -> tprog s2 s3
adamc@14 675 -> tprog s1 s3.
adamc@14 676
adamc@18 677 (** 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@18 678
adamc@14 679 Fixpoint vstack (ts : tstack) : Set :=
adamc@14 680 match ts with
adamc@14 681 | nil => unit
adamc@14 682 | t :: ts' => typeDenote t * vstack ts'
adamc@14 683 end%type.
adamc@14 684
adam@312 685 (** 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@18 686
adam@312 687 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@18 688
adamc@14 689 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
adamc@207 690 match i with
adam@312 691 | TiNConst _ n => fun s => (n, s)
adam@312 692 | TiBConst _ b => fun s => (b, s)
adam@311 693 | TiBinop _ _ _ _ b => fun s =>
adam@312 694 let '(arg1, (arg2, s')) := s in
adam@312 695 ((tbinopDenote b) arg1 arg2, s')
adamc@14 696 end.
adamc@14 697
adamc@18 698 (** 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@18 699 [[
adamc@18 700 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
adamc@207 701 match i with
adam@312 702 | TiNConst _ n => (n, s)
adam@312 703 | TiBConst _ b => (b, s)
adam@311 704 | TiBinop _ _ _ _ b =>
adam@312 705 let '(arg1, (arg2, s')) := s in
adam@312 706 ((tbinopDenote b) arg1 arg2, s')
adamc@18 707 end.
adamc@18 708
adamc@205 709 ]]
adamc@205 710
adamc@18 711 The Coq type-checker complains that:
adamc@18 712
adam@312 713 <<
adamc@18 714 The term "(n, s)" has type "(nat * vstack ts)%type"
adamc@207 715 while it is expected to have type "vstack ?119".
adam@312 716 >>
adamc@207 717
adam@312 718 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@18 719 *)
adamc@18 720
adamc@18 721 (** We finish the semantics with a straightforward definition of program denotation. *)
adamc@18 722
adamc@207 723 Fixpoint tprogDenote ts ts' (p : tprog ts ts') : vstack ts -> vstack ts' :=
adamc@207 724 match p with
adamc@14 725 | TNil _ => fun s => s
adamc@14 726 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
adamc@14 727 end.
adamc@14 728
adamc@14 729
adamc@14 730 (** ** Translation *)
adamc@14 731
adamc@19 732 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
adamc@19 733
adamc@207 734 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') : tprog ts' ts'' -> tprog ts ts'' :=
adamc@207 735 match p with
adamc@14 736 | TNil _ => fun p' => p'
adamc@14 737 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
adamc@14 738 end.
adamc@14 739
adamc@19 740 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
adamc@19 741
adamc@207 742 Fixpoint tcompile t (e : texp t) (ts : tstack) : tprog ts (t :: ts) :=
adamc@207 743 match e with
adam@312 744 | TNConst n => TCons (TiNConst _ n) (TNil _)
adam@312 745 | TBConst b => TCons (TiBConst _ b) (TNil _)
adamc@14 746 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
adam@311 747 (tconcat (tcompile e1 _) (TCons (TiBinop _ b) (TNil _)))
adamc@14 748 end.
adamc@14 749
adam@307 750 (** One interesting feature of the definition is the underscores appearing to the right of [=>] arrows. Haskell and ML programmers are quite familiar with compilers that infer type parameters to polymorphic values. In Coq, it is possible to go even further and ask the system to infer arbitrary terms, by writing underscores in place of specific values. You may have noticed that we have been calling functions without specifying all of their arguments. For instance, the recursive calls here to [tcompile] omit the [t] argument. Coq's %\emph{%#<i>#implicit argument#</i>#%}% mechanism automatically inserts underscores for arguments that it will probably be able to infer. Inference of such values is far from complete, though; generally, it only works in cases similar to those encountered with polymorphic type instantiation in Haskell and ML.
adamc@19 751
adamc@19 752 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@19 753
adamc@14 754 Print tcompile.
adamc@19 755 (** [[
adamc@19 756 tcompile =
adamc@19 757 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
adamc@19 758 tprog ts (t :: ts) :=
adamc@19 759 match e in (texp t0) return (tprog ts (t0 :: ts)) with
adam@312 760 | TNConst n => TCons (TiNConst ts n) (TNil (Nat :: ts))
adam@312 761 | TBConst b => TCons (TiBConst ts b) (TNil (Bool :: ts))
adamc@19 762 | TBinop arg1 arg2 res b e1 e2 =>
adamc@19 763 tconcat (tcompile arg2 e2 ts)
adamc@19 764 (tconcat (tcompile arg1 e1 (arg2 :: ts))
adam@311 765 (TCons (TiBinop ts b) (TNil (res :: ts))))
adamc@19 766 end
adamc@19 767 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
adam@302 768 ]]
adam@302 769 *)
adamc@19 770
adamc@19 771
adamc@19 772 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
adamc@19 773
adamc@19 774 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
adam@312 775 (** [= (42, tt) : vstack (][Nat :: nil)] *)
adamc@207 776
adamc@19 777 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
adam@312 778 (** [= (][true][, tt) : vstack (][Bool :: nil)] *)
adamc@207 779
adam@312 780 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2)
adam@312 781 (TNConst 2)) (TNConst 7)) nil) tt.
adam@312 782 (** [= (28, tt) : vstack (][Nat :: nil)] *)
adamc@207 783
adam@312 784 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2)
adam@312 785 (TNConst 2)) (TNConst 7)) nil) tt.
adam@312 786 (** [= (]%\coqdocconstructor{%#<tt>#false#</tt>#%}%[, tt) : vstack (][Bool :: nil)] *)
adamc@207 787
adam@312 788 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
adam@312 789 (TNConst 7)) nil) tt.
adam@312 790 (** [= (][true][, tt) : vstack (][Bool :: nil)] *)
adamc@19 791
adamc@14 792
adamc@20 793 (** ** Translation Correctness *)
adamc@20 794
adamc@20 795 (** We can state a correctness theorem similar to the last one. *)
adamc@20 796
adamc@207 797 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 798 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@20 799 (* begin hide *)
adamc@20 800 Abort.
adamc@20 801 (* end hide *)
adamc@22 802 (* begin thide *)
adamc@20 803
adam@312 804 (** 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@14 805
adamc@207 806 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 807 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@20 808
adam@292 809 (** While lemma [compile_correct'] quantified over a program that is the %``%#"#continuation#"#%''% for the expression we are considering, here we avoid drawing in any extra syntactic elements. In addition to the source expression and its type, we also quantify over an initial stack type and a stack compatible with it. Running the compilation of the program starting from that stack, we should arrive at a stack that differs only in having the program's denotation pushed onto it.
adamc@20 810
adamc@20 811 Let us try to prove this theorem in the same way that we settled on in the last section. *)
adamc@20 812
adamc@14 813 induction e; crush.
adamc@20 814
adamc@20 815 (** We are left with this unproved conclusion:
adamc@20 816
adamc@20 817 [[
adamc@20 818 tprogDenote
adamc@20 819 (tconcat (tcompile e2 ts)
adamc@20 820 (tconcat (tcompile e1 (arg2 :: ts))
adam@311 821 (TCons (TiBinop ts t) (TNil (res :: ts))))) s =
adamc@20 822 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
adamc@207 823
adamc@20 824 ]]
adamc@20 825
adam@312 826 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@20 827 *)
adamc@207 828
adamc@14 829 Abort.
adamc@14 830
adamc@26 831 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
adamc@14 832 (s : vstack ts),
adamc@14 833 tprogDenote (tconcat p p') s
adamc@14 834 = tprogDenote p' (tprogDenote p s).
adamc@14 835 induction p; crush.
adamc@14 836 Qed.
adamc@14 837
adamc@20 838 (** This one goes through completely automatically.
adamc@20 839
adam@312 840 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{Verncular commands!Hint Rewrite}% *)
adamc@20 841
adam@312 842 (* begin hide *)
adamc@26 843 Hint Rewrite tconcat_correct : cpdt.
adam@312 844 (* end hide *)
adam@312 845 (** %\noindent%[Hint] %\coqdockw{%#<tt>#Rewrite#</tt>#%}% [tconcat_correct : cpdt.] *)
adamc@14 846
adam@312 847 (** Here we meet the pervasive concept of a %\emph{%#<i>#hint#</i>#%}%. 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, and we ask for the hint to be added to the hint database called [cpdt], which is the database used by [crush]. In general, segragating hints into different databases is helpful to control the performance of proof search, in cases where domain knowledge allows us to narrow the set of proof steps to be considered in brute force search. Part III of this book considers such pragmatic aspects of proof search in much more detail.
adam@312 848
adam@312 849 Now we are ready to return to [tcompile_correct'], proving it automatically this time. *)
adamc@20 850
adamc@207 851 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
adamc@207 852 tprogDenote (tcompile e ts) s = (texpDenote e, s).
adamc@14 853 induction e; crush.
adamc@14 854 Qed.
adamc@14 855
adamc@20 856 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
adamc@20 857
adam@312 858 (* begin hide *)
adamc@26 859 Hint Rewrite tcompile_correct' : cpdt.
adam@312 860 (* end hide *)
adam@312 861 (** %\noindent%[Hint ]%\coqdockw{%#<tt>#Rewrite#</tt>#%}%[ tcompile_correct' : cpdt.] *)
adamc@14 862
adamc@207 863 Theorem tcompile_correct : forall t (e : texp t),
adamc@207 864 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
adamc@14 865 crush.
adamc@14 866 Qed.
adamc@22 867 (* end thide *)
adam@312 868
adam@312 869 (** 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}\emph{%#<i>#program extraction#</i>#%}%, 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@312 870
adam@312 871 (* begin hide *)
adam@312 872 Extraction tcompile.
adam@312 873 (* end hide *)
adam@312 874 (** %\noindent\coqdockw{%#<tt>#Extraction#</tt>#%}%[ tcompile.] *)
adam@312 875
adam@312 876 (** <<
adam@312 877 let rec tcompile t e ts =
adam@312 878 match e with
adam@312 879 | TNConst n ->
adam@312 880 TCons (ts, (Cons (Nat, ts)), (Cons (Nat, ts)), (TiNConst (ts, n)), (TNil
adam@312 881 (Cons (Nat, ts))))
adam@312 882 | TBConst b ->
adam@312 883 TCons (ts, (Cons (Bool, ts)), (Cons (Bool, ts)), (TiBConst (ts, b)),
adam@312 884 (TNil (Cons (Bool, ts))))
adam@312 885 | TBinop (t1, t2, t0, b, e1, e2) ->
adam@312 886 tconcat ts (Cons (t2, ts)) (Cons (t0, ts)) (tcompile t2 e2 ts)
adam@312 887 (tconcat (Cons (t2, ts)) (Cons (t1, (Cons (t2, ts)))) (Cons (t0, ts))
adam@312 888 (tcompile t1 e1 (Cons (t2, ts))) (TCons ((Cons (t1, (Cons (t2,
adam@312 889 ts)))), (Cons (t0, ts)), (Cons (t0, ts)), (TiBinop (t1, t2, t0, ts,
adam@312 890 b)), (TNil (Cons (t0, ts))))))
adam@312 891 >>
adam@312 892
adam@312 893 We can compile this code with the usual OCaml compiler and obtain an executable program with halfway decent performance.
adam@312 894
adam@312 895 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. *)