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author Adam Chlipala <adam@chlipala.net>
date Wed, 13 Feb 2013 10:24:28 -0500
parents fd6ec9b2dccb
children ed829eaa91b2
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adam@381 1 (* Copyright (c) 2011-2012, Adam Chlipala
adam@381 2 *
adam@381 3 * This work is licensed under a
adam@381 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adam@381 5 * Unported License.
adam@381 6 * The license text is available at:
adam@381 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adam@381 8 *)
adam@381 9
adam@381 10 (* begin hide *)
adam@381 11 Require Import FunctionalExtensionality List.
adam@381 12
adam@381 13 Require Import CpdtTactics DepList.
adam@381 14
adam@381 15 Set Implicit Arguments.
adam@381 16 (* end hide *)
adam@381 17
adam@381 18 (** %\chapter{A Taste of Reasoning About Programming Language Syntax}% *)
adam@381 19
adam@434 20 (** Reasoning about the syntax and semantics of programming languages is a popular application of proof assistants. Before proving the first theorem of this kind, it is necessary to choose a formal encoding of the informal notions of syntax, dealing with such issues as %\index{variable binding}%variable binding conventions. I believe the pragmatic questions in this domain are far from settled and remain as important open research problems. However, in this chapter, I will demonstrate two underused encoding approaches. Note that I am not recommending either approach as a silver bullet! Mileage will vary across concrete problems, and I expect there to be significant future advances in our knowledge of encoding techniques. For a broader introduction to programming language formalization, using more elementary techniques, see %\emph{%{{http://www.cis.upenn.edu/~bcpierce/sf/}Software Foundations}%}% by Pierce et al.
adam@381 21
adam@381 22 This chapter is also meant as a case study, bringing together what we have learned in the previous chapters. We will see a concrete example of the importance of representation choices; translating mathematics from paper to Coq is not a deterministic process, and different creative choices can have big impacts. We will also see dependent types and scripted proof automation in action, applied to solve a particular problem as well as possible, rather than to demonstrate new Coq concepts.
adam@381 23
adam@381 24 I apologize in advance to those readers not familiar with the theory of programming language semantics. I will make a few remarks intended to relate the material here with common ideas in semantics, but these remarks should be safe for others to skip.
adam@381 25
adam@381 26 We will define a small programming language and reason about its semantics, expressed as an interpreter into Coq terms, much as we have done in examples throughout the book. It will be helpful to build a slight extension of [crush] that tries to apply %\index{functional extensionality}%functional extensionality, an axiom we met in Chapter 12, which says that two functions are equal if they map equal inputs to equal outputs. *)
adam@381 27
adam@381 28 Ltac ext := let x := fresh "x" in extensionality x.
adam@497 29 Ltac pl := crush; repeat (ext || f_equal; crush).
adam@381 30
adam@381 31 (** At this point in the book source, some auxiliary proofs also appear. *)
adam@381 32
adam@381 33 (* begin hide *)
adam@381 34 Section hmap.
adam@381 35 Variable A : Type.
adam@381 36 Variables B1 B2 B3 : A -> Type.
adam@381 37
adam@381 38 Variable f1 : forall x, B1 x -> B2 x.
adam@381 39 Variable f2 : forall x, B2 x -> B3 x.
adam@381 40
adam@381 41 Theorem hmap_hmap : forall ls (hl : hlist B1 ls), hmap f2 (hmap f1 hl) = hmap (fun i (x : B1 i) => f2 (f1 x)) hl.
adam@381 42 induction hl; crush.
adam@381 43 Qed.
adam@381 44 End hmap.
adam@381 45
adam@381 46 Section Forall.
adam@381 47 Variable A : Type.
adam@381 48 Variable P : A -> Prop.
adam@381 49
adam@381 50 Theorem Forall_In : forall ls, Forall P ls -> forall x, In x ls -> P x.
adam@381 51 induction 1; crush.
adam@381 52 Qed.
adam@381 53
adam@381 54 Theorem Forall_In' : forall ls, (forall x, In x ls -> P x) -> Forall P ls.
adam@381 55 induction ls; crush.
adam@381 56 Qed.
adam@381 57
adam@381 58 Variable P' : A -> Prop.
adam@381 59
adam@381 60 Theorem Forall_weaken : forall ls, Forall P ls
adam@381 61 -> (forall x, P x -> P' x)
adam@381 62 -> Forall P' ls.
adam@381 63 induction 1; crush.
adam@381 64 Qed.
adam@381 65 End Forall.
adam@381 66 (* end hide *)
adam@381 67
adam@381 68 (** Here is a definition of the type system we will use throughout the chapter. It is for simply typed lambda calculus with natural numbers as the base type. *)
adam@381 69
adam@381 70 Inductive type : Type :=
adam@381 71 | Nat : type
adam@381 72 | Func : type -> type -> type.
adam@381 73
adam@381 74 Fixpoint typeDenote (t : type) : Type :=
adam@381 75 match t with
adam@381 76 | Nat => nat
adam@381 77 | Func t1 t2 => typeDenote t1 -> typeDenote t2
adam@381 78 end.
adam@381 79
adam@381 80 (** Now we have some choices as to how we represent the syntax of programs. The two sections of the chapter explore two such choices, demonstrating the effect the choice has on proof complexity. *)
adam@381 81
adam@381 82
adam@381 83 (** * Dependent de Bruijn Indices *)
adam@381 84
adam@465 85 (** The first encoding is one we met first in Chapter 9, the _dependent de Bruijn index_ encoding. We represent program syntax terms in a type family parameterized by a list of types, representing the _typing context_, or information on which free variables are in scope and what their types are. Variables are represented in a way isomorphic to the natural numbers, where number 0 represents the first element in the context, number 1 the second element, and so on. Actually, instead of numbers, we use the [member] dependent type family from Chapter 9. *)
adam@381 86
adam@381 87 Module FirstOrder.
adam@381 88
adam@381 89 (** Here is the definition of the [term] type, including variables, constants, addition, function abstraction and application, and let binding of local variables. *)
adam@381 90
adam@381 91 Inductive term : list type -> type -> Type :=
adam@381 92 | Var : forall G t, member t G -> term G t
adam@381 93
adam@381 94 | Const : forall G, nat -> term G Nat
adam@381 95 | Plus : forall G, term G Nat -> term G Nat -> term G Nat
adam@381 96
adam@381 97 | Abs : forall G dom ran, term (dom :: G) ran -> term G (Func dom ran)
adam@381 98 | App : forall G dom ran, term G (Func dom ran) -> term G dom -> term G ran
adam@381 99
adam@381 100 | Let : forall G t1 t2, term G t1 -> term (t1 :: G) t2 -> term G t2.
adam@381 101
adam@381 102 Implicit Arguments Const [G].
adam@381 103
adam@381 104 (** Here are two example term encodings, the first of addition packaged as a two-argument curried function, and the second of a sample application of addition to constants. *)
adam@381 105
adam@381 106 Example add : term nil (Func Nat (Func Nat Nat)) :=
adam@381 107 Abs (Abs (Plus (Var (HNext HFirst)) (Var HFirst))).
adam@381 108
adam@381 109 Example three_the_hard_way : term nil Nat :=
adam@381 110 App (App add (Const 1)) (Const 2).
adam@381 111
adam@381 112 (** Since dependent typing ensures that any term is well-formed in its context and has a particular type, it is easy to translate syntactic terms into Coq values. *)
adam@381 113
adam@381 114 Fixpoint termDenote G t (e : term G t) : hlist typeDenote G -> typeDenote t :=
adam@381 115 match e with
adam@381 116 | Var _ _ x => fun s => hget s x
adam@381 117
adam@381 118 | Const _ n => fun _ => n
adam@381 119 | Plus _ e1 e2 => fun s => termDenote e1 s + termDenote e2 s
adam@381 120
adam@381 121 | Abs _ _ _ e1 => fun s => fun x => termDenote e1 (x ::: s)
adam@381 122 | App _ _ _ e1 e2 => fun s => (termDenote e1 s) (termDenote e2 s)
adam@381 123
adam@381 124 | Let _ _ _ e1 e2 => fun s => termDenote e2 (termDenote e1 s ::: s)
adam@381 125 end.
adam@381 126
adam@398 127 (** With this term representation, some program transformations are easy to implement and prove correct. Certainly we would be worried if this were not the the case for the _identity_ transformation, which takes a term apart and reassembles it. *)
adam@381 128
adam@381 129 Fixpoint ident G t (e : term G t) : term G t :=
adam@381 130 match e with
adam@381 131 | Var _ _ x => Var x
adam@381 132
adam@381 133 | Const _ n => Const n
adam@381 134 | Plus _ e1 e2 => Plus (ident e1) (ident e2)
adam@381 135
adam@381 136 | Abs _ _ _ e1 => Abs (ident e1)
adam@381 137 | App _ _ _ e1 e2 => App (ident e1) (ident e2)
adam@381 138
adam@381 139 | Let _ _ _ e1 e2 => Let (ident e1) (ident e2)
adam@381 140 end.
adam@381 141
adam@381 142 Theorem identSound : forall G t (e : term G t) s,
adam@381 143 termDenote (ident e) s = termDenote e s.
adam@497 144 induction e; pl.
adam@381 145 Qed.
adam@381 146
adam@398 147 (** A slightly more ambitious transformation belongs to the family of _constant folding_ optimizations we have used as examples in other chapters. *)
adam@398 148
adam@381 149 Fixpoint cfold G t (e : term G t) : term G t :=
adam@381 150 match e with
adam@381 151 | Plus G e1 e2 =>
adam@381 152 let e1' := cfold e1 in
adam@381 153 let e2' := cfold e2 in
adam@398 154 let maybeOpt := match e1' return _ with
adam@398 155 | Const _ n1 =>
adam@398 156 match e2' return _ with
adam@398 157 | Const _ n2 => Some (Const (n1 + n2))
adam@398 158 | _ => None
adam@398 159 end
adam@398 160 | _ => None
adam@398 161 end in
adam@398 162 match maybeOpt with
adam@398 163 | None => Plus e1' e2'
adam@398 164 | Some e' => e'
adam@398 165 end
adam@381 166
adam@381 167 | Abs _ _ _ e1 => Abs (cfold e1)
adam@381 168 | App _ _ _ e1 e2 => App (cfold e1) (cfold e2)
adam@381 169
adam@381 170 | Let _ _ _ e1 e2 => Let (cfold e1) (cfold e2)
adam@381 171
adam@381 172 | e => e
adam@381 173 end.
adam@381 174
adam@381 175 (** The correctness proof is more complex, but only slightly so. *)
adam@381 176
adam@381 177 Theorem cfoldSound : forall G t (e : term G t) s,
adam@381 178 termDenote (cfold e) s = termDenote e s.
adam@497 179 induction e; pl;
adam@381 180 repeat (match goal with
adam@414 181 | [ |- context[match ?E with Var _ _ _ => _ | _ => _ end] ] =>
adam@414 182 dep_destruct E
adam@497 183 end; pl).
adam@381 184 Qed.
adam@381 185
adam@414 186 (** The transformations we have tried so far have been straightforward because they do not have interesting effects on the variable binding structure of terms. The dependent de Bruijn representation is called%\index{first-order syntax}% _first-order_ because it encodes variable identity explicitly; all such representations incur bookkeeping overheads in transformations that rearrange binding structure.
adam@381 187
adam@434 188 As an example of a tricky transformation, consider one that removes all uses of "[let x = e1 in e2]" by substituting [e1] for [x] in [e2]. We will implement the translation by pairing the "compile-time" typing environment with a "run-time" value environment or _substitution_, mapping each variable to a value to be substituted for it. Such a substitute term may be placed within a program in a position with a larger typing environment than applied at the point where the substitute term was chosen. To support such context transplantation, we need _lifting_, a standard de Bruijn indices operation. With dependent typing, lifting corresponds to weakening for typing judgments.
adam@381 189
adam@381 190 The fundamental goal of lifting is to add a new variable to a typing context, maintaining the validity of a term in the expanded context. To express the operation of adding a type to a context, we use a helper function [insertAt]. *)
adam@381 191
adam@381 192 Fixpoint insertAt (t : type) (G : list type) (n : nat) {struct n} : list type :=
adam@381 193 match n with
adam@381 194 | O => t :: G
adam@381 195 | S n' => match G with
adam@381 196 | nil => t :: G
adam@381 197 | t' :: G' => t' :: insertAt t G' n'
adam@381 198 end
adam@381 199 end.
adam@381 200
adam@381 201 (** Another function lifts bound variable instances, which we represent with [member] values. *)
adam@381 202
adam@381 203 Fixpoint liftVar t G (x : member t G) t' n : member t (insertAt t' G n) :=
adam@381 204 match x with
adam@381 205 | HFirst G' => match n return member t (insertAt t' (t :: G') n) with
adam@381 206 | O => HNext HFirst
adam@381 207 | _ => HFirst
adam@381 208 end
adam@381 209 | HNext t'' G' x' => match n return member t (insertAt t' (t'' :: G') n) with
adam@381 210 | O => HNext (HNext x')
adam@381 211 | S n' => HNext (liftVar x' t' n')
adam@381 212 end
adam@381 213 end.
adam@381 214
adam@381 215 (** The final helper function for lifting allows us to insert a new variable anywhere in a typing context. *)
adam@381 216
adam@381 217 Fixpoint lift' G t' n t (e : term G t) : term (insertAt t' G n) t :=
adam@381 218 match e with
adam@381 219 | Var _ _ x => Var (liftVar x t' n)
adam@381 220
adam@381 221 | Const _ n => Const n
adam@381 222 | Plus _ e1 e2 => Plus (lift' t' n e1) (lift' t' n e2)
adam@381 223
adam@381 224 | Abs _ _ _ e1 => Abs (lift' t' (S n) e1)
adam@381 225 | App _ _ _ e1 e2 => App (lift' t' n e1) (lift' t' n e2)
adam@381 226
adam@381 227 | Let _ _ _ e1 e2 => Let (lift' t' n e1) (lift' t' (S n) e2)
adam@381 228 end.
adam@381 229
adam@398 230 (** In the [Let] removal transformation, we only need to apply lifting to add a new variable at the _beginning_ of a typing context, so we package lifting into this final, simplified form. *)
adam@381 231
adam@381 232 Definition lift G t' t (e : term G t) : term (t' :: G) t :=
adam@381 233 lift' t' O e.
adam@381 234
adam@381 235 (** Finally, we can implement [Let] removal. The argument of type [hlist (term G') G] represents a substitution mapping each variable from context [G] into a term that is valid in context [G']. Note how the [Abs] case (1) extends via lifting the substitution [s] to hold in the broader context of the abstraction body [e1] and (2) maps the new first variable to itself. It is only the [Let] case that maps a variable to any substitute beside itself. *)
adam@381 236
adam@381 237 Fixpoint unlet G t (e : term G t) G' : hlist (term G') G -> term G' t :=
adam@381 238 match e with
adam@381 239 | Var _ _ x => fun s => hget s x
adam@381 240
adam@381 241 | Const _ n => fun _ => Const n
adam@381 242 | Plus _ e1 e2 => fun s => Plus (unlet e1 s) (unlet e2 s)
adam@381 243
adam@381 244 | Abs _ _ _ e1 => fun s => Abs (unlet e1 (Var HFirst ::: hmap (lift _) s))
adam@381 245 | App _ _ _ e1 e2 => fun s => App (unlet e1 s) (unlet e2 s)
adam@381 246
adam@381 247 | Let _ t1 _ e1 e2 => fun s => unlet e2 (unlet e1 s ::: s)
adam@381 248 end.
adam@381 249
adam@381 250 (** We have finished defining the transformation, but the parade of helper functions is not over. To prove correctness, we will use one more helper function and a few lemmas. First, we need an operation to insert a new value into a substitution at a particular position. *)
adam@381 251
adam@381 252 Fixpoint insertAtS (t : type) (x : typeDenote t) (G : list type) (n : nat) {struct n}
adam@381 253 : hlist typeDenote G -> hlist typeDenote (insertAt t G n) :=
adam@381 254 match n with
adam@381 255 | O => fun s => x ::: s
adam@381 256 | S n' => match G return hlist typeDenote G
adam@381 257 -> hlist typeDenote (insertAt t G (S n')) with
adam@381 258 | nil => fun s => x ::: s
adam@381 259 | t' :: G' => fun s => hhd s ::: insertAtS t x n' (htl s)
adam@381 260 end
adam@381 261 end.
adam@381 262
adam@381 263 Implicit Arguments insertAtS [t G].
adam@381 264
adam@381 265 (** Next we prove that [liftVar] is correct. That is, a lifted variable retains its value with respect to a substitution when we perform an analogue to lifting by inserting a new mapping into the substitution. *)
adam@381 266
adam@381 267 Lemma liftVarSound : forall t' (x : typeDenote t') t G (m : member t G) s n,
adam@381 268 hget s m = hget (insertAtS x n s) (liftVar m t' n).
adam@497 269 induction m; destruct n; dep_destruct s; pl.
adam@381 270 Qed.
adam@381 271
adam@381 272 Hint Resolve liftVarSound.
adam@381 273
adam@381 274 (** An analogous lemma establishes correctness of [lift']. *)
adam@381 275
adam@381 276 Lemma lift'Sound : forall G t' (x : typeDenote t') t (e : term G t) n s,
adam@381 277 termDenote e s = termDenote (lift' t' n e) (insertAtS x n s).
adam@497 278 induction e; pl;
adam@381 279 repeat match goal with
adam@381 280 | [ IH : forall n s, _ = termDenote (lift' _ n ?E) _
adam@381 281 |- context[lift' _ (S ?N) ?E] ] => specialize (IH (S N))
adam@497 282 end; pl.
adam@381 283 Qed.
adam@381 284
adam@381 285 (** Correctness of [lift] itself is an easy corollary. *)
adam@381 286
adam@381 287 Lemma liftSound : forall G t' (x : typeDenote t') t (e : term G t) s,
adam@381 288 termDenote (lift t' e) (x ::: s) = termDenote e s.
adam@381 289 unfold lift; intros; rewrite (lift'Sound _ x e O); trivial.
adam@381 290 Qed.
adam@381 291
adam@381 292 Hint Rewrite hget_hmap hmap_hmap liftSound.
adam@381 293
adam@381 294 (** Finally, we can prove correctness of [unletSound] for terms in arbitrary typing environments. *)
adam@381 295
adam@381 296 Lemma unletSound' : forall G t (e : term G t) G' (s : hlist (term G') G) s1,
adam@381 297 termDenote (unlet e s) s1
adam@381 298 = termDenote e (hmap (fun t' (e' : term G' t') => termDenote e' s1) s).
adam@497 299 induction e; pl.
adam@381 300 Qed.
adam@381 301
adam@510 302 (** The lemma statement is a mouthful, with all its details of typing contexts and substitutions. It is usually prudent to state a final theorem in as simple a way as possible, to help your readers believe that you have proved what they expect. We follow that advice here for the simple case of terms with empty typing contexts. *)
adam@381 303
adam@381 304 Theorem unletSound : forall t (e : term nil t),
adam@381 305 termDenote (unlet e HNil) HNil = termDenote e HNil.
adam@381 306 intros; apply unletSound'.
adam@381 307 Qed.
adam@381 308
adam@381 309 End FirstOrder.
adam@381 310
adam@381 311 (** The [Let] removal optimization is a good case study of a simple transformation that may turn out to be much more work than expected, based on representation choices. In the second part of this chapter, we consider an alternate choice that produces a more pleasant experience. *)
adam@381 312
adam@381 313
adam@381 314 (** * Parametric Higher-Order Abstract Syntax *)
adam@381 315
adam@510 316 (** In contrast to first-order encodings,%\index{higher-order syntax}% _higher-order_ encodings avoid explicit modeling of variable identity. Instead, the binding constructs of an%\index{object language}% _object language_ (the language being formalized) can be represented using the binding constructs of the%\index{meta language}% _meta language_ (the language in which the formalization is done). The best known higher-order encoding is called%\index{higher-order abstract syntax}% _higher-order abstract syntax_ (HOAS) %\cite{HOAS}%, and we can start by attempting to apply it directly in Coq. *)
adam@381 317
adam@381 318 Module HigherOrder.
adam@381 319
adam@398 320 (** With HOAS, each object language binding construct is represented with a _function_ of the meta language. Here is what we get if we apply that idea within an inductive definition of term syntax. *)
adam@381 321
adam@381 322 (** %\vspace{-.15in}%[[
adam@381 323 Inductive term : type -> Type :=
adam@381 324 | Const : nat -> term Nat
adam@381 325 | Plus : term Nat -> term Nat -> term Nat
adam@381 326
adam@381 327 | Abs : forall dom ran, (term dom -> term ran) -> term (Func dom ran)
adam@381 328 | App : forall dom ran, term (Func dom ran) -> term dom -> term ran
adam@381 329
adam@381 330 | Let : forall t1 t2, term t1 -> (term t1 -> term t2) -> term t2.
adam@381 331 ]]
adam@381 332
adam@464 333 However, Coq rejects this definition for failing to meet the %\index{strict positivity requirement}%strict positivity restriction. For instance, the constructor [Abs] takes an argument that is a function over the same type family [term] that we are defining. Inductive definitions of this kind can be used to write non-terminating Gallina programs, which breaks the consistency of Coq's logic.
adam@381 334
adam@510 335 An alternate higher-order encoding is%\index{parametric higher-order abstract syntax}\index{PHOAS|see{parametric higher-order abstract syntax}}% _parametric HOAS_, as introduced by Washburn and Weirich%~\cite{BGB}% for Haskell and tweaked by me%~\cite{PhoasICFP08}% for use in Coq. Here the idea is to parameterize the syntax type by a type family standing for a _representation of variables_. *)
adam@381 336
adam@381 337 Section var.
adam@381 338 Variable var : type -> Type.
adam@381 339
adam@381 340 Inductive term : type -> Type :=
adam@381 341 | Var : forall t, var t -> term t
adam@381 342
adam@381 343 | Const : nat -> term Nat
adam@381 344 | Plus : term Nat -> term Nat -> term Nat
adam@381 345
adam@381 346 | Abs : forall dom ran, (var dom -> term ran) -> term (Func dom ran)
adam@381 347 | App : forall dom ran, term (Func dom ran) -> term dom -> term ran
adam@381 348
adam@381 349 | Let : forall t1 t2, term t1 -> (var t1 -> term t2) -> term t2.
adam@381 350 End var.
adam@381 351
adam@381 352 Implicit Arguments Var [var t].
adam@381 353 Implicit Arguments Const [var].
adam@381 354 Implicit Arguments Abs [var dom ran].
adam@381 355
adam@398 356 (** Coq accepts this definition because our embedded functions now merely take _variables_ as arguments, instead of arbitrary terms. One might wonder whether there is an easy loophole to exploit here, instantiating the parameter [var] as [term] itself. However, to do that, we would need to choose a variable representation for this nested mention of [term], and so on through an infinite descent into [term] arguments.
adam@381 357
adam@381 358 We write the final type of a closed term using polymorphic quantification over all possible choices of [var] type family. *)
adam@381 359
adam@381 360 Definition Term t := forall var, term var t.
adam@381 361
adam@398 362 (** Here are the new representations of the example terms from the last section. Note how each is written as a function over a [var] choice, such that the specific choice has no impact on the _structure_ of the term. *)
adam@381 363
adam@381 364 Example add : Term (Func Nat (Func Nat Nat)) := fun var =>
adam@381 365 Abs (fun x => Abs (fun y => Plus (Var x) (Var y))).
adam@381 366
adam@381 367 Example three_the_hard_way : Term Nat := fun var =>
adam@381 368 App (App (add var) (Const 1)) (Const 2).
adam@381 369
adam@510 370 (** The argument [var] does not even appear in the function body for [add]. How can that be? By giving our terms expressive types, we allow Coq to infer many arguments for us. In fact, we do not even need to name the [var] argument! *)
adam@381 371
adam@381 372 Example add' : Term (Func Nat (Func Nat Nat)) := fun _ =>
adam@381 373 Abs (fun x => Abs (fun y => Plus (Var x) (Var y))).
adam@381 374
adam@381 375 Example three_the_hard_way' : Term Nat := fun _ =>
adam@381 376 App (App (add' _) (Const 1)) (Const 2).
adam@381 377
adam@510 378 (** Even though the [var] formal parameters appear as underscores, they _are_ mentioned in the function bodies that type inference calculates. *)
adam@510 379
adam@381 380
adam@381 381 (** ** Functional Programming with PHOAS *)
adam@381 382
adam@398 383 (** It may not be at all obvious that the PHOAS representation admits the crucial computable operations. The key to effective deconstruction of PHOAS terms is one principle: treat the [var] parameter as an unconstrained choice of _which data should be annotated on each variable_. We will begin with a simple example, that of counting how many variable nodes appear in a PHOAS term. This operation requires no data annotated on variables, so we simply annotate variables with [unit] values. Note that, when we go under binders in the cases for [Abs] and [Let], we must provide the data value to annotate on the new variable we pass beneath. For our current choice of [unit] data, we always pass [tt]. *)
adam@381 384
adam@381 385 Fixpoint countVars t (e : term (fun _ => unit) t) : nat :=
adam@381 386 match e with
adam@381 387 | Var _ _ => 1
adam@381 388
adam@381 389 | Const _ => 0
adam@381 390 | Plus e1 e2 => countVars e1 + countVars e2
adam@381 391
adam@381 392 | Abs _ _ e1 => countVars (e1 tt)
adam@381 393 | App _ _ e1 e2 => countVars e1 + countVars e2
adam@381 394
adam@381 395 | Let _ _ e1 e2 => countVars e1 + countVars (e2 tt)
adam@381 396 end.
adam@381 397
adam@381 398 (** The above definition may seem a bit peculiar. What gave us the right to represent variables as [unit] values? Recall that our final representation of closed terms is as polymorphic functions. We merely specialize a closed term to exactly the right variable representation for the transformation we wish to perform. *)
adam@381 399
adam@381 400 Definition CountVars t (E : Term t) := countVars (E (fun _ => unit)).
adam@381 401
adam@381 402 (** It is easy to test that [CountVars] operates properly. *)
adam@381 403
adam@381 404 Eval compute in CountVars three_the_hard_way.
adam@381 405 (** %\vspace{-.15in}%[[
adam@381 406 = 2
adam@381 407 ]]
adam@381 408 *)
adam@381 409
adam@381 410 (** In fact, PHOAS can be used anywhere that first-order representations can. We will not go into all the details here, but the intuition is that it is possible to interconvert between PHOAS and any reasonable first-order representation. Here is a suggestive example, translating PHOAS terms into strings giving a first-order rendering. To implement this translation, the key insight is to tag variables with strings, giving their names. The function takes as an additional input a string giving the name to be assigned to the next variable introduced. We evolve this name by adding a prime to its end. To avoid getting bogged down in orthogonal details, we render all constants as the string ["N"]. *)
adam@381 411
adam@381 412 Require Import String.
adam@381 413 Open Scope string_scope.
adam@381 414
adam@381 415 Fixpoint pretty t (e : term (fun _ => string) t) (x : string) : string :=
adam@381 416 match e with
adam@381 417 | Var _ s => s
adam@381 418
adam@381 419 | Const _ => "N"
adam@381 420 | Plus e1 e2 => "(" ++ pretty e1 x ++ " + " ++ pretty e2 x ++ ")"
adam@381 421
adam@381 422 | Abs _ _ e1 => "(fun " ++ x ++ " => " ++ pretty (e1 x) (x ++ "'") ++ ")"
adam@381 423 | App _ _ e1 e2 => "(" ++ pretty e1 x ++ " " ++ pretty e2 x ++ ")"
adam@381 424
adam@381 425 | Let _ _ e1 e2 => "(let " ++ x ++ " = " ++ pretty e1 x ++ " in "
adam@381 426 ++ pretty (e2 x) (x ++ "'") ++ ")"
adam@381 427 end.
adam@381 428
adam@381 429 Definition Pretty t (E : Term t) := pretty (E (fun _ => string)) "x".
adam@381 430
adam@381 431 Eval compute in Pretty three_the_hard_way.
adam@381 432 (** %\vspace{-.15in}%[[
adam@381 433 = "(((fun x => (fun x' => (x + x'))) N) N)"
adam@381 434 ]]
adam@381 435 *)
adam@381 436
adam@497 437 (** However, it is not necessary to convert to first-order form to support many common operations on terms. For instance, we can implement substitution of terms for variables. The key insight here is to _tag variables with terms_, so that, on encountering a variable, we can simply replace it by the term in its tag. We will call this function initially on a term with exactly one free variable, tagged with the appropriate substitute. During recursion, new variables are added, but they are only tagged with their own term equivalents. Note that this function [squash] is parameterized over a specific [var] choice. *)
adam@381 438
adam@381 439 Fixpoint squash var t (e : term (term var) t) : term var t :=
adam@381 440 match e with
adam@381 441 | Var _ e1 => e1
adam@381 442
adam@381 443 | Const n => Const n
adam@381 444 | Plus e1 e2 => Plus (squash e1) (squash e2)
adam@381 445
adam@381 446 | Abs _ _ e1 => Abs (fun x => squash (e1 (Var x)))
adam@381 447 | App _ _ e1 e2 => App (squash e1) (squash e2)
adam@381 448
adam@381 449 | Let _ _ e1 e2 => Let (squash e1) (fun x => squash (e2 (Var x)))
adam@381 450 end.
adam@381 451
adam@381 452 (** To define the final substitution function over terms with single free variables, we define [Term1], an analogue to [Term] that we defined before for closed terms. *)
adam@381 453
adam@381 454 Definition Term1 (t1 t2 : type) := forall var, var t1 -> term var t2.
adam@381 455
adam@381 456 (** Substitution is defined by (1) instantiating a [Term1] to tag variables with terms and (2) applying the result to a specific term to be substituted. Note how the parameter [var] of [squash] is instantiated: the body of [Subst] is itself a polymorphic quantification over [var], standing for a variable tag choice in the output term; and we use that input to compute a tag choice for the input term. *)
adam@381 457
adam@381 458 Definition Subst (t1 t2 : type) (E : Term1 t1 t2) (E' : Term t1) : Term t2 :=
adam@381 459 fun var => squash (E (term var) (E' var)).
adam@381 460
adam@381 461 Eval compute in Subst (fun _ x => Plus (Var x) (Const 3)) three_the_hard_way.
adam@381 462 (** %\vspace{-.15in}%[[
adam@381 463 = fun var : type -> Type =>
adam@381 464 Plus
adam@381 465 (App
adam@381 466 (App
adam@381 467 (Abs
adam@381 468 (fun x : var Nat =>
adam@381 469 Abs (fun y : var Nat => Plus (Var x) (Var y))))
adam@381 470 (Const 1)) (Const 2)) (Const 3)
adam@381 471 ]]
adam@381 472
adam@398 473 One further development, which may seem surprising at first, is that we can also implement a usual term denotation function, when we _tag variables with their denotations_. *)
adam@381 474
adam@381 475 Fixpoint termDenote t (e : term typeDenote t) : typeDenote t :=
adam@381 476 match e with
adam@381 477 | Var _ v => v
adam@381 478
adam@381 479 | Const n => n
adam@381 480 | Plus e1 e2 => termDenote e1 + termDenote e2
adam@381 481
adam@381 482 | Abs _ _ e1 => fun x => termDenote (e1 x)
adam@381 483 | App _ _ e1 e2 => (termDenote e1) (termDenote e2)
adam@381 484
adam@381 485 | Let _ _ e1 e2 => termDenote (e2 (termDenote e1))
adam@381 486 end.
adam@381 487
adam@381 488 Definition TermDenote t (E : Term t) : typeDenote t :=
adam@381 489 termDenote (E typeDenote).
adam@381 490
adam@381 491 Eval compute in TermDenote three_the_hard_way.
adam@381 492 (** %\vspace{-.15in}%[[
adam@381 493 = 3
adam@381 494 ]]
adam@381 495
adam@381 496 To summarize, the PHOAS representation has all the expressive power of more standard first-order encodings, and a variety of translations are actually much more pleasant to implement than usual, thanks to the novel ability to tag variables with data. *)
adam@381 497
adam@381 498
adam@381 499 (** ** Verifying Program Transformations *)
adam@381 500
adam@381 501 (** Let us now revisit the three example program transformations from the last section. Each is easy to implement with PHOAS, and the last is substantially easier than with first-order representations.
adam@381 502
adam@381 503 First, we have the recursive identity function, following the same pattern as in the previous subsection, with a helper function, polymorphic in a tag choice; and a final function that instantiates the choice appropriately. *)
adam@381 504
adam@381 505 Fixpoint ident var t (e : term var t) : term var t :=
adam@381 506 match e with
adam@381 507 | Var _ x => Var x
adam@381 508
adam@381 509 | Const n => Const n
adam@381 510 | Plus e1 e2 => Plus (ident e1) (ident e2)
adam@381 511
adam@381 512 | Abs _ _ e1 => Abs (fun x => ident (e1 x))
adam@381 513 | App _ _ e1 e2 => App (ident e1) (ident e2)
adam@381 514
adam@381 515 | Let _ _ e1 e2 => Let (ident e1) (fun x => ident (e2 x))
adam@381 516 end.
adam@381 517
adam@381 518 Definition Ident t (E : Term t) : Term t := fun var =>
adam@381 519 ident (E var).
adam@381 520
adam@381 521 (** Proving correctness is both easier and harder than in the last section, easier because we do not need to manipulate substitutions, and harder because we do the induction in an extra lemma about [ident], to establish the correctness theorem for [Ident]. *)
adam@381 522
adam@381 523 Lemma identSound : forall t (e : term typeDenote t),
adam@381 524 termDenote (ident e) = termDenote e.
adam@497 525 induction e; pl.
adam@381 526 Qed.
adam@381 527
adam@381 528 Theorem IdentSound : forall t (E : Term t),
adam@381 529 TermDenote (Ident E) = TermDenote E.
adam@381 530 intros; apply identSound.
adam@381 531 Qed.
adam@381 532
adam@381 533 (** The translation of the constant-folding function and its proof work more or less the same way. *)
adam@381 534
adam@381 535 Fixpoint cfold var t (e : term var t) : term var t :=
adam@381 536 match e with
adam@381 537 | Plus e1 e2 =>
adam@381 538 let e1' := cfold e1 in
adam@381 539 let e2' := cfold e2 in
adam@381 540 match e1', e2' with
adam@381 541 | Const n1, Const n2 => Const (n1 + n2)
adam@381 542 | _, _ => Plus e1' e2'
adam@381 543 end
adam@381 544
adam@381 545 | Abs _ _ e1 => Abs (fun x => cfold (e1 x))
adam@381 546 | App _ _ e1 e2 => App (cfold e1) (cfold e2)
adam@381 547
adam@381 548 | Let _ _ e1 e2 => Let (cfold e1) (fun x => cfold (e2 x))
adam@381 549
adam@381 550 | e => e
adam@381 551 end.
adam@381 552
adam@381 553 Definition Cfold t (E : Term t) : Term t := fun var =>
adam@381 554 cfold (E var).
adam@381 555
adam@381 556 Lemma cfoldSound : forall t (e : term typeDenote t),
adam@381 557 termDenote (cfold e) = termDenote e.
adam@497 558 induction e; pl;
adam@381 559 repeat (match goal with
adam@414 560 | [ |- context[match ?E with Var _ _ => _ | _ => _ end] ] =>
adam@414 561 dep_destruct E
adam@497 562 end; pl).
adam@381 563 Qed.
adam@381 564
adam@381 565 Theorem CfoldSound : forall t (E : Term t),
adam@381 566 TermDenote (Cfold E) = TermDenote E.
adam@381 567 intros; apply cfoldSound.
adam@381 568 Qed.
adam@381 569
adam@381 570 (** Things get more interesting in the [Let]-removal optimization. Our recursive helper function adapts the key idea from our earlier definitions of [squash] and [Subst]: tag variables with terms. We have a straightforward generalization of [squash], where only the [Let] case has changed, to tag the new variable with the term it is bound to, rather than just tagging the variable with itself as a term. *)
adam@381 571
adam@381 572 Fixpoint unlet var t (e : term (term var) t) : term var t :=
adam@381 573 match e with
adam@381 574 | Var _ e1 => e1
adam@381 575
adam@381 576 | Const n => Const n
adam@381 577 | Plus e1 e2 => Plus (unlet e1) (unlet e2)
adam@381 578
adam@381 579 | Abs _ _ e1 => Abs (fun x => unlet (e1 (Var x)))
adam@381 580 | App _ _ e1 e2 => App (unlet e1) (unlet e2)
adam@381 581
adam@381 582 | Let _ _ e1 e2 => unlet (e2 (unlet e1))
adam@381 583 end.
adam@381 584
adam@381 585 Definition Unlet t (E : Term t) : Term t := fun var =>
adam@381 586 unlet (E (term var)).
adam@381 587
adam@381 588 (** We can test [Unlet] first on an uninteresting example, [three_the_hard_way], which does not use [Let]. *)
adam@381 589
adam@381 590 Eval compute in Unlet three_the_hard_way.
adam@381 591 (** %\vspace{-.15in}%[[
adam@381 592 = fun var : type -> Type =>
adam@381 593 App
adam@381 594 (App
adam@381 595 (Abs
adam@381 596 (fun x : var Nat =>
adam@381 597 Abs (fun x0 : var Nat => Plus (Var x) (Var x0))))
adam@381 598 (Const 1)) (Const 2)
adam@381 599 ]]
adam@381 600
adam@381 601 Next, we try a more interesting example, with some extra [Let]s introduced in [three_the_hard_way]. *)
adam@381 602
adam@381 603 Definition three_a_harder_way : Term Nat := fun _ =>
adam@381 604 Let (Const 1) (fun x => Let (Const 2) (fun y => App (App (add _) (Var x)) (Var y))).
adam@381 605
adam@381 606 Eval compute in Unlet three_a_harder_way.
adam@381 607 (** %\vspace{-.15in}%[[
adam@381 608 = fun var : type -> Type =>
adam@381 609 App
adam@381 610 (App
adam@381 611 (Abs
adam@381 612 (fun x : var Nat =>
adam@381 613 Abs (fun x0 : var Nat => Plus (Var x) (Var x0))))
adam@381 614 (Const 1)) (Const 2)
adam@381 615 ]]
adam@381 616
adam@381 617 The output is the same as in the previous test, confirming that [Unlet] operates properly here.
adam@381 618
adam@381 619 Now we need to state a correctness theorem for [Unlet], based on an inductively proved lemma about [unlet]. It is not at all obvious how to arrive at a proper induction principle for the lemma. The problem is that we want to relate two instantiations of the same [Term], in a way where we know they share the same structure. Note that, while [Unlet] is defined to consider all possible [var] choices in the output term, the correctness proof conveniently only depends on the case of [var := typeDenote]. Thus, one parallel instantiation will set [var := typeDenote], to take the denotation of the original term. The other parallel instantiation will set [var := term typeDenote], to perform the [unlet] transformation in the original term.
adam@381 620
adam@381 621 Here is a relation formalizing the idea that two terms are structurally the same, differing only by replacing the variable data of one with another isomorphic set of variable data in some possibly different type family. *)
adam@381 622
adam@381 623 Section wf.
adam@381 624 Variables var1 var2 : type -> Type.
adam@381 625
adam@381 626 (** To formalize the tag isomorphism, we will use lists of values with the following record type. Each entry has an object language type and an appropriate tag for that type, in each of the two tag families [var1] and [var2]. *)
adam@381 627
adam@381 628 Record varEntry := {
adam@381 629 Ty : type;
adam@381 630 First : var1 Ty;
adam@381 631 Second : var2 Ty
adam@381 632 }.
adam@381 633
adam@381 634 (** Here is the inductive relation definition. An instance [wf G e1 e2] asserts that terms [e1] and [e2] are equivalent up to the variable tag isomorphism [G]. Note how the [Var] rule looks up an entry in [G], and the [Abs] and [Let] rules include recursive [wf] invocations inside the scopes of quantifiers to introduce parallel tag values to be considered as isomorphic. *)
adam@381 635
adam@381 636 Inductive wf : list varEntry -> forall t, term var1 t -> term var2 t -> Prop :=
adam@381 637 | WfVar : forall G t x x', In {| Ty := t; First := x; Second := x' |} G
adam@381 638 -> wf G (Var x) (Var x')
adam@381 639
adam@381 640 | WfConst : forall G n, wf G (Const n) (Const n)
adam@381 641
adam@381 642 | WfPlus : forall G e1 e2 e1' e2', wf G e1 e1'
adam@381 643 -> wf G e2 e2'
adam@381 644 -> wf G (Plus e1 e2) (Plus e1' e2')
adam@381 645
adam@381 646 | WfAbs : forall G dom ran (e1 : _ dom -> term _ ran) e1',
adam@381 647 (forall x1 x2, wf ({| First := x1; Second := x2 |} :: G) (e1 x1) (e1' x2))
adam@381 648 -> wf G (Abs e1) (Abs e1')
adam@381 649
adam@381 650 | WfApp : forall G dom ran (e1 : term _ (Func dom ran)) (e2 : term _ dom) e1' e2',
adam@381 651 wf G e1 e1'
adam@381 652 -> wf G e2 e2'
adam@381 653 -> wf G (App e1 e2) (App e1' e2')
adam@381 654
adam@381 655 | WfLet : forall G t1 t2 e1 e1' (e2 : _ t1 -> term _ t2) e2', wf G e1 e1'
adam@381 656 -> (forall x1 x2, wf ({| First := x1; Second := x2 |} :: G) (e2 x1) (e2' x2))
adam@381 657 -> wf G (Let e1 e2) (Let e1' e2').
adam@381 658 End wf.
adam@381 659
adam@381 660 (** We can state a well-formedness condition for closed terms: for any two choices of tag type families, the parallel instantiations belong to the [wf] relation, starting from an empty variable isomorphism. *)
adam@381 661
adam@381 662 Definition Wf t (E : Term t) := forall var1 var2, wf nil (E var1) (E var2).
adam@381 663
adam@465 664 (** After digesting the syntactic details of [Wf], it is probably not hard to see that reasonable term encodings will satisfy it. For example: *)
adam@381 665
adam@381 666 Theorem three_the_hard_way_Wf : Wf three_the_hard_way.
adam@381 667 red; intros; repeat match goal with
adam@381 668 | [ |- wf _ _ _ ] => constructor; intros
adam@381 669 end; intuition.
adam@381 670 Qed.
adam@381 671
adam@497 672 (** Now we are ready to give a nice simple proof of correctness for [unlet]. First, we add one hint to apply a small variant of a standard library theorem connecting [Forall], a higher-order predicate asserting that every element of a list satisfies some property; and [In], the list membership predicate. *)
adam@381 673
adam@381 674 Hint Extern 1 => match goal with
adam@381 675 | [ H1 : Forall _ _, H2 : In _ _ |- _ ] => apply (Forall_In H1 _ H2)
adam@381 676 end.
adam@381 677
adam@381 678 (** The rest of the proof is about as automated as we could hope for. *)
adam@381 679
adam@381 680 Lemma unletSound : forall G t (e1 : term _ t) e2,
adam@381 681 wf G e1 e2
adam@381 682 -> Forall (fun ve => termDenote (First ve) = Second ve) G
adam@381 683 -> termDenote (unlet e1) = termDenote e2.
adam@497 684 induction 1; pl.
adam@381 685 Qed.
adam@381 686
adam@381 687 Theorem UnletSound : forall t (E : Term t), Wf E
adam@381 688 -> TermDenote (Unlet E) = TermDenote E.
adam@381 689 intros; eapply unletSound; eauto.
adam@381 690 Qed.
adam@381 691
adam@381 692 (** With this example, it is not obvious that the PHOAS encoding is more tractable than dependent de Bruijn. Where the de Bruijn version had [lift] and its helper functions, here we have [Wf] and its auxiliary definitions. In practice, [Wf] is defined once per object language, while such operations as [lift] often need to operate differently for different examples, forcing new implementations for new transformations.
adam@381 693
adam@381 694 The reader may also have come up with another objection: via Curry-Howard, [wf] proofs may be thought of as first-order encodings of term syntax! For instance, the [In] hypothesis of rule [WfVar] is equivalent to a [member] value. There is some merit to this objection. However, as the proofs above show, we are able to reason about transformations using first-order representation only for their inputs, not their outputs. Furthermore, explicit numbering of variables remains absent from the proofs.
adam@381 695
adam@381 696 Have we really avoided first-order reasoning about the output terms of translations? The answer depends on some subtle issues, which deserve a subsection of their own. *)
adam@381 697
adam@381 698
adam@381 699 (** ** Establishing Term Well-Formedness *)
adam@381 700
adam@434 701 (** Can there be values of type [Term t] that are not well-formed according to [Wf]? We expect that Gallina satisfies key%\index{parametricity}% _parametricity_ %\cite{parametricity}% properties, which indicate how polymorphic types may only be inhabited by specific values. We omit details of parametricity theorems here, but [forall t (E : Term t), Wf E] follows the flavor of such theorems. One option would be to assert that fact as an axiom, "proving" that any output of any of our translations is well-formed. We could even prove the soundness of the theorem on paper meta-theoretically, say by considering some particular model of CIC.
adam@381 702
adam@381 703 To be more cautious, we could prove [Wf] for every term that interests us, threading such proofs through all transformations. Here is an example exercise of that kind, for [Unlet].
adam@381 704
adam@398 705 First, we prove that [wf] is _monotone_, in that a given instance continues to hold as we add new variable pairs to the variable isomorphism. *)
adam@381 706
adam@381 707 Hint Constructors wf.
adam@381 708 Hint Extern 1 (In _ _) => simpl; tauto.
adam@381 709 Hint Extern 1 (Forall _ _) => eapply Forall_weaken; [ eassumption | simpl ].
adam@381 710
adam@381 711 Lemma wf_monotone : forall var1 var2 G t (e1 : term var1 t) (e2 : term var2 t),
adam@381 712 wf G e1 e2
adam@381 713 -> forall G', Forall (fun x => In x G') G
adam@381 714 -> wf G' e1 e2.
adam@497 715 induction 1; pl; auto 6.
adam@381 716 Qed.
adam@381 717
adam@381 718 Hint Resolve wf_monotone Forall_In'.
adam@381 719
adam@381 720 (** Now we are ready to prove that [unlet] preserves any [wf] instance. The key invariant has to do with the parallel execution of [unlet] on two different [var] instantiations of a particular term. Since [unlet] uses [term] as the type of variable data, our variable isomorphism context [G] contains pairs of terms, which, conveniently enough, allows us to state the invariant that any pair of terms in the context is also related by [wf]. *)
adam@381 721
adam@381 722 Hint Extern 1 (wf _ _ _) => progress simpl.
adam@381 723
adam@381 724 Lemma unletWf : forall var1 var2 G t (e1 : term (term var1) t) (e2 : term (term var2) t),
adam@381 725 wf G e1 e2
adam@381 726 -> forall G', Forall (fun ve => wf G' (First ve) (Second ve)) G
adam@381 727 -> wf G' (unlet e1) (unlet e2).
adam@497 728 induction 1; pl; eauto 9.
adam@381 729 Qed.
adam@381 730
adam@381 731 (** Repackaging [unletWf] into a theorem about [Wf] and [Unlet] is straightforward. *)
adam@381 732
adam@381 733 Theorem UnletWf : forall t (E : Term t), Wf E
adam@381 734 -> Wf (Unlet E).
adam@381 735 red; intros; apply unletWf with nil; auto.
adam@381 736 Qed.
adam@381 737
adam@381 738 (** This example demonstrates how we may need to use reasoning reminiscent of that associated with first-order representations, though the bookkeeping details are generally easier to manage, and bookkeeping theorems may generally be proved separately from the independently interesting theorems about program transformations. *)
adam@381 739
adam@381 740
adam@381 741 (** ** A Few More Remarks *)
adam@381 742
adam@381 743 (** Higher-order encodings derive their strength from reuse of the meta language's binding constructs. As a result, we can write encoded terms so that they look very similar to their informal counterparts, without variable numbering schemes like for de Bruijn indices. The example encodings above have demonstrated this fact, but modulo the clunkiness of explicit use of the constructors of [term]. After defining a few new Coq syntax notations, we can work with terms in an even more standard form. *)
adam@381 744
adam@381 745 Infix "-->" := Func (right associativity, at level 52).
adam@381 746
adam@381 747 Notation "^" := Var.
adam@381 748 Notation "#" := Const.
adam@381 749 Infix "@" := App (left associativity, at level 50).
adam@381 750 Infix "@+" := Plus (left associativity, at level 50).
adam@381 751 Notation "\ x : t , e" := (Abs (dom := t) (fun x => e))
adam@381 752 (no associativity, at level 51, x at level 0).
adam@381 753 Notation "[ e ]" := (fun _ => e).
adam@381 754
adam@381 755 Example Add : Term (Nat --> Nat --> Nat) :=
adam@381 756 [\x : Nat, \y : Nat, ^x @+ ^y].
adam@381 757
adam@381 758 Example Three_the_hard_way : Term Nat :=
adam@381 759 [Add _ @ #1 @ #2].
adam@381 760
adam@381 761 Eval compute in TermDenote Three_the_hard_way.
adam@381 762 (** %\vspace{-.15in}%[[
adam@381 763 = 3
adam@381 764 ]]
adam@381 765 *)
adam@381 766
adam@381 767 End HigherOrder.
adam@381 768
adam@381 769 (** The PHOAS approach shines here because we are working with an object language that has an easy embedding into Coq. That is, there is a straightforward recursive function translating object terms into terms of Gallina. All Gallina programs terminate, so clearly we cannot hope to find such embeddings for Turing-complete languages; and non-Turing-complete languages may still require much more involved translations. I have some work%~\cite{CompilerPOPL10}% on modeling semantics of Turing-complete languages with PHOAS, but my impression is that there are many more advances left to be made in this field, possibly with completely new term representations that we have not yet been clever enough to think up. *)