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Finish complete proofreading pass
author Adam Chlipala <adam@chlipala.net>
date Wed, 29 Aug 2012 17:17:17 -0400
parents 92f386c33e94
children 4320c1a967c2
rev   line source
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@381 29 Ltac t := 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@398 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 familiy parametrized 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@381 144 induction e; t.
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@381 179 induction e; t;
adam@381 180 repeat (match goal with
adam@414 181 | [ |- context[match ?E with Var _ _ _ => _ | _ => _ end] ] =>
adam@414 182 dep_destruct E
adam@381 183 end; t).
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@381 269 induction m; destruct n; dep_destruct s; t.
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@381 278 induction e; t;
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@381 282 end; t.
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@381 299 induction e; t.
adam@381 300 Qed.
adam@381 301
adam@381 302 (** 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 do that 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@414 316 (** 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}\index{HOAS}% _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@414 335 An alternate higher-order encoding is%\index{parametric higher-order abstract syntax}\index{PHOAS}% _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 parametrize 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@398 370 (** 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! Even though these formal parameters appear as underscores, they _are_ mentioned in the function bodies that type inference calculates. *)
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@381 378
adam@381 379 (** ** Functional Programming with PHOAS *)
adam@381 380
adam@398 381 (** 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@381 382
adam@381 383 Fixpoint countVars t (e : term (fun _ => unit) t) : nat :=
adam@381 384 match e with
adam@381 385 | Var _ _ => 1
adam@381 386
adam@381 387 | Const _ => 0
adam@381 388 | Plus e1 e2 => countVars e1 + countVars e2
adam@381 389
adam@381 390 | Abs _ _ e1 => countVars (e1 tt)
adam@381 391 | App _ _ e1 e2 => countVars e1 + countVars e2
adam@381 392
adam@381 393 | Let _ _ e1 e2 => countVars e1 + countVars (e2 tt)
adam@381 394 end.
adam@381 395
adam@381 396 (** 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@381 397
adam@381 398 Definition CountVars t (E : Term t) := countVars (E (fun _ => unit)).
adam@381 399
adam@381 400 (** It is easy to test that [CountVars] operates properly. *)
adam@381 401
adam@381 402 Eval compute in CountVars three_the_hard_way.
adam@381 403 (** %\vspace{-.15in}%[[
adam@381 404 = 2
adam@381 405 ]]
adam@381 406 *)
adam@381 407
adam@381 408 (** 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@381 409
adam@381 410 Require Import String.
adam@381 411 Open Scope string_scope.
adam@381 412
adam@381 413 Fixpoint pretty t (e : term (fun _ => string) t) (x : string) : string :=
adam@381 414 match e with
adam@381 415 | Var _ s => s
adam@381 416
adam@381 417 | Const _ => "N"
adam@381 418 | Plus e1 e2 => "(" ++ pretty e1 x ++ " + " ++ pretty e2 x ++ ")"
adam@381 419
adam@381 420 | Abs _ _ e1 => "(fun " ++ x ++ " => " ++ pretty (e1 x) (x ++ "'") ++ ")"
adam@381 421 | App _ _ e1 e2 => "(" ++ pretty e1 x ++ " " ++ pretty e2 x ++ ")"
adam@381 422
adam@381 423 | Let _ _ e1 e2 => "(let " ++ x ++ " = " ++ pretty e1 x ++ " in "
adam@381 424 ++ pretty (e2 x) (x ++ "'") ++ ")"
adam@381 425 end.
adam@381 426
adam@381 427 Definition Pretty t (E : Term t) := pretty (E (fun _ => string)) "x".
adam@381 428
adam@381 429 Eval compute in Pretty three_the_hard_way.
adam@381 430 (** %\vspace{-.15in}%[[
adam@381 431 = "(((fun x => (fun x' => (x + x'))) N) N)"
adam@381 432 ]]
adam@381 433 *)
adam@381 434
adam@398 435 (** 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 one term in another. 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 parametrized over a specific [var] choice. *)
adam@381 436
adam@381 437 Fixpoint squash var t (e : term (term var) t) : term var t :=
adam@381 438 match e with
adam@381 439 | Var _ e1 => e1
adam@381 440
adam@381 441 | Const n => Const n
adam@381 442 | Plus e1 e2 => Plus (squash e1) (squash e2)
adam@381 443
adam@381 444 | Abs _ _ e1 => Abs (fun x => squash (e1 (Var x)))
adam@381 445 | App _ _ e1 e2 => App (squash e1) (squash e2)
adam@381 446
adam@381 447 | Let _ _ e1 e2 => Let (squash e1) (fun x => squash (e2 (Var x)))
adam@381 448 end.
adam@381 449
adam@381 450 (** 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@381 451
adam@381 452 Definition Term1 (t1 t2 : type) := forall var, var t1 -> term var t2.
adam@381 453
adam@381 454 (** 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@381 455
adam@381 456 Definition Subst (t1 t2 : type) (E : Term1 t1 t2) (E' : Term t1) : Term t2 :=
adam@381 457 fun var => squash (E (term var) (E' var)).
adam@381 458
adam@381 459 Eval compute in Subst (fun _ x => Plus (Var x) (Const 3)) three_the_hard_way.
adam@381 460 (** %\vspace{-.15in}%[[
adam@381 461 = fun var : type -> Type =>
adam@381 462 Plus
adam@381 463 (App
adam@381 464 (App
adam@381 465 (Abs
adam@381 466 (fun x : var Nat =>
adam@381 467 Abs (fun y : var Nat => Plus (Var x) (Var y))))
adam@381 468 (Const 1)) (Const 2)) (Const 3)
adam@381 469 ]]
adam@381 470
adam@398 471 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@381 472
adam@381 473 Fixpoint termDenote t (e : term typeDenote t) : typeDenote t :=
adam@381 474 match e with
adam@381 475 | Var _ v => v
adam@381 476
adam@381 477 | Const n => n
adam@381 478 | Plus e1 e2 => termDenote e1 + termDenote e2
adam@381 479
adam@381 480 | Abs _ _ e1 => fun x => termDenote (e1 x)
adam@381 481 | App _ _ e1 e2 => (termDenote e1) (termDenote e2)
adam@381 482
adam@381 483 | Let _ _ e1 e2 => termDenote (e2 (termDenote e1))
adam@381 484 end.
adam@381 485
adam@381 486 Definition TermDenote t (E : Term t) : typeDenote t :=
adam@381 487 termDenote (E typeDenote).
adam@381 488
adam@381 489 Eval compute in TermDenote three_the_hard_way.
adam@381 490 (** %\vspace{-.15in}%[[
adam@381 491 = 3
adam@381 492 ]]
adam@381 493
adam@381 494 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@381 495
adam@381 496
adam@381 497 (** ** Verifying Program Transformations *)
adam@381 498
adam@381 499 (** 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@381 500
adam@381 501 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@381 502
adam@381 503 Fixpoint ident var t (e : term var t) : term var t :=
adam@381 504 match e with
adam@381 505 | Var _ x => Var x
adam@381 506
adam@381 507 | Const n => Const n
adam@381 508 | Plus e1 e2 => Plus (ident e1) (ident e2)
adam@381 509
adam@381 510 | Abs _ _ e1 => Abs (fun x => ident (e1 x))
adam@381 511 | App _ _ e1 e2 => App (ident e1) (ident e2)
adam@381 512
adam@381 513 | Let _ _ e1 e2 => Let (ident e1) (fun x => ident (e2 x))
adam@381 514 end.
adam@381 515
adam@381 516 Definition Ident t (E : Term t) : Term t := fun var =>
adam@381 517 ident (E var).
adam@381 518
adam@381 519 (** 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@381 520
adam@381 521 Lemma identSound : forall t (e : term typeDenote t),
adam@381 522 termDenote (ident e) = termDenote e.
adam@381 523 induction e; t.
adam@381 524 Qed.
adam@381 525
adam@381 526 Theorem IdentSound : forall t (E : Term t),
adam@381 527 TermDenote (Ident E) = TermDenote E.
adam@381 528 intros; apply identSound.
adam@381 529 Qed.
adam@381 530
adam@381 531 (** The translation of the constant-folding function and its proof work more or less the same way. *)
adam@381 532
adam@381 533 Fixpoint cfold var t (e : term var t) : term var t :=
adam@381 534 match e with
adam@381 535 | Plus e1 e2 =>
adam@381 536 let e1' := cfold e1 in
adam@381 537 let e2' := cfold e2 in
adam@381 538 match e1', e2' with
adam@381 539 | Const n1, Const n2 => Const (n1 + n2)
adam@381 540 | _, _ => Plus e1' e2'
adam@381 541 end
adam@381 542
adam@381 543 | Abs _ _ e1 => Abs (fun x => cfold (e1 x))
adam@381 544 | App _ _ e1 e2 => App (cfold e1) (cfold e2)
adam@381 545
adam@381 546 | Let _ _ e1 e2 => Let (cfold e1) (fun x => cfold (e2 x))
adam@381 547
adam@381 548 | e => e
adam@381 549 end.
adam@381 550
adam@381 551 Definition Cfold t (E : Term t) : Term t := fun var =>
adam@381 552 cfold (E var).
adam@381 553
adam@381 554 Lemma cfoldSound : forall t (e : term typeDenote t),
adam@381 555 termDenote (cfold e) = termDenote e.
adam@381 556 induction e; t;
adam@381 557 repeat (match goal with
adam@414 558 | [ |- context[match ?E with Var _ _ => _ | _ => _ end] ] =>
adam@414 559 dep_destruct E
adam@381 560 end; t).
adam@381 561 Qed.
adam@381 562
adam@381 563 Theorem CfoldSound : forall t (E : Term t),
adam@381 564 TermDenote (Cfold E) = TermDenote E.
adam@381 565 intros; apply cfoldSound.
adam@381 566 Qed.
adam@381 567
adam@381 568 (** 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@381 569
adam@381 570 Fixpoint unlet var t (e : term (term var) t) : term var t :=
adam@381 571 match e with
adam@381 572 | Var _ e1 => e1
adam@381 573
adam@381 574 | Const n => Const n
adam@381 575 | Plus e1 e2 => Plus (unlet e1) (unlet e2)
adam@381 576
adam@381 577 | Abs _ _ e1 => Abs (fun x => unlet (e1 (Var x)))
adam@381 578 | App _ _ e1 e2 => App (unlet e1) (unlet e2)
adam@381 579
adam@381 580 | Let _ _ e1 e2 => unlet (e2 (unlet e1))
adam@381 581 end.
adam@381 582
adam@381 583 Definition Unlet t (E : Term t) : Term t := fun var =>
adam@381 584 unlet (E (term var)).
adam@381 585
adam@381 586 (** We can test [Unlet] first on an uninteresting example, [three_the_hard_way], which does not use [Let]. *)
adam@381 587
adam@381 588 Eval compute in Unlet three_the_hard_way.
adam@381 589 (** %\vspace{-.15in}%[[
adam@381 590 = fun var : type -> Type =>
adam@381 591 App
adam@381 592 (App
adam@381 593 (Abs
adam@381 594 (fun x : var Nat =>
adam@381 595 Abs (fun x0 : var Nat => Plus (Var x) (Var x0))))
adam@381 596 (Const 1)) (Const 2)
adam@381 597 ]]
adam@381 598
adam@381 599 Next, we try a more interesting example, with some extra [Let]s introduced in [three_the_hard_way]. *)
adam@381 600
adam@381 601 Definition three_a_harder_way : Term Nat := fun _ =>
adam@381 602 Let (Const 1) (fun x => Let (Const 2) (fun y => App (App (add _) (Var x)) (Var y))).
adam@381 603
adam@381 604 Eval compute in Unlet three_a_harder_way.
adam@381 605 (** %\vspace{-.15in}%[[
adam@381 606 = fun var : type -> Type =>
adam@381 607 App
adam@381 608 (App
adam@381 609 (Abs
adam@381 610 (fun x : var Nat =>
adam@381 611 Abs (fun x0 : var Nat => Plus (Var x) (Var x0))))
adam@381 612 (Const 1)) (Const 2)
adam@381 613 ]]
adam@381 614
adam@381 615 The output is the same as in the previous test, confirming that [Unlet] operates properly here.
adam@381 616
adam@381 617 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@381 618
adam@381 619 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@381 620
adam@381 621 Section wf.
adam@381 622 Variables var1 var2 : type -> Type.
adam@381 623
adam@381 624 (** 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@381 625
adam@381 626 Record varEntry := {
adam@381 627 Ty : type;
adam@381 628 First : var1 Ty;
adam@381 629 Second : var2 Ty
adam@381 630 }.
adam@381 631
adam@381 632 (** 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@381 633
adam@381 634 Inductive wf : list varEntry -> forall t, term var1 t -> term var2 t -> Prop :=
adam@381 635 | WfVar : forall G t x x', In {| Ty := t; First := x; Second := x' |} G
adam@381 636 -> wf G (Var x) (Var x')
adam@381 637
adam@381 638 | WfConst : forall G n, wf G (Const n) (Const n)
adam@381 639
adam@381 640 | WfPlus : forall G e1 e2 e1' e2', wf G e1 e1'
adam@381 641 -> wf G e2 e2'
adam@381 642 -> wf G (Plus e1 e2) (Plus e1' e2')
adam@381 643
adam@381 644 | WfAbs : forall G dom ran (e1 : _ dom -> term _ ran) e1',
adam@381 645 (forall x1 x2, wf ({| First := x1; Second := x2 |} :: G) (e1 x1) (e1' x2))
adam@381 646 -> wf G (Abs e1) (Abs e1')
adam@381 647
adam@381 648 | WfApp : forall G dom ran (e1 : term _ (Func dom ran)) (e2 : term _ dom) e1' e2',
adam@381 649 wf G e1 e1'
adam@381 650 -> wf G e2 e2'
adam@381 651 -> wf G (App e1 e2) (App e1' e2')
adam@381 652
adam@381 653 | WfLet : forall G t1 t2 e1 e1' (e2 : _ t1 -> term _ t2) e2', wf G e1 e1'
adam@381 654 -> (forall x1 x2, wf ({| First := x1; Second := x2 |} :: G) (e2 x1) (e2' x2))
adam@381 655 -> wf G (Let e1 e2) (Let e1' e2').
adam@381 656 End wf.
adam@381 657
adam@381 658 (** 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@381 659
adam@381 660 Definition Wf t (E : Term t) := forall var1 var2, wf nil (E var1) (E var2).
adam@381 661
adam@381 662 (** After digesting the syntactic details of [Wf], it is probably not hard to see that reasonable term encodings will satsify it. For example: *)
adam@381 663
adam@381 664 Theorem three_the_hard_way_Wf : Wf three_the_hard_way.
adam@381 665 red; intros; repeat match goal with
adam@381 666 | [ |- wf _ _ _ ] => constructor; intros
adam@381 667 end; intuition.
adam@381 668 Qed.
adam@381 669
adam@381 670 (** Now we are ready to give a nice simple proof of correctness for [unlet]. First, we add one hint to apply 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@381 671
adam@381 672 Hint Extern 1 => match goal with
adam@381 673 | [ H1 : Forall _ _, H2 : In _ _ |- _ ] => apply (Forall_In H1 _ H2)
adam@381 674 end.
adam@381 675
adam@381 676 (** The rest of the proof is about as automated as we could hope for. *)
adam@381 677
adam@381 678 Lemma unletSound : forall G t (e1 : term _ t) e2,
adam@381 679 wf G e1 e2
adam@381 680 -> Forall (fun ve => termDenote (First ve) = Second ve) G
adam@381 681 -> termDenote (unlet e1) = termDenote e2.
adam@381 682 induction 1; t.
adam@381 683 Qed.
adam@381 684
adam@381 685 Theorem UnletSound : forall t (E : Term t), Wf E
adam@381 686 -> TermDenote (Unlet E) = TermDenote E.
adam@381 687 intros; eapply unletSound; eauto.
adam@381 688 Qed.
adam@381 689
adam@381 690 (** 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@381 691
adam@381 692 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@381 693
adam@381 694 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@381 695
adam@381 696
adam@381 697 (** ** Establishing Term Well-Formedness *)
adam@381 698
adam@434 699 (** 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@381 700
adam@381 701 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@381 702
adam@398 703 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@381 704
adam@381 705 Hint Constructors wf.
adam@381 706 Hint Extern 1 (In _ _) => simpl; tauto.
adam@381 707 Hint Extern 1 (Forall _ _) => eapply Forall_weaken; [ eassumption | simpl ].
adam@381 708
adam@381 709 Lemma wf_monotone : forall var1 var2 G t (e1 : term var1 t) (e2 : term var2 t),
adam@381 710 wf G e1 e2
adam@381 711 -> forall G', Forall (fun x => In x G') G
adam@381 712 -> wf G' e1 e2.
adam@381 713 induction 1; t; auto 6.
adam@381 714 Qed.
adam@381 715
adam@381 716 Hint Resolve wf_monotone Forall_In'.
adam@381 717
adam@381 718 (** 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@381 719
adam@381 720 Hint Extern 1 (wf _ _ _) => progress simpl.
adam@381 721
adam@381 722 Lemma unletWf : forall var1 var2 G t (e1 : term (term var1) t) (e2 : term (term var2) t),
adam@381 723 wf G e1 e2
adam@381 724 -> forall G', Forall (fun ve => wf G' (First ve) (Second ve)) G
adam@381 725 -> wf G' (unlet e1) (unlet e2).
adam@381 726 induction 1; t; eauto 9.
adam@381 727 Qed.
adam@381 728
adam@381 729 (** Repackaging [unletWf] into a theorem about [Wf] and [Unlet] is straightforward. *)
adam@381 730
adam@381 731 Theorem UnletWf : forall t (E : Term t), Wf E
adam@381 732 -> Wf (Unlet E).
adam@381 733 red; intros; apply unletWf with nil; auto.
adam@381 734 Qed.
adam@381 735
adam@381 736 (** 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@381 737
adam@381 738
adam@381 739 (** ** A Few More Remarks *)
adam@381 740
adam@381 741 (** 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@381 742
adam@381 743 Infix "-->" := Func (right associativity, at level 52).
adam@381 744
adam@381 745 Notation "^" := Var.
adam@381 746 Notation "#" := Const.
adam@381 747 Infix "@" := App (left associativity, at level 50).
adam@381 748 Infix "@+" := Plus (left associativity, at level 50).
adam@381 749 Notation "\ x : t , e" := (Abs (dom := t) (fun x => e))
adam@381 750 (no associativity, at level 51, x at level 0).
adam@381 751 Notation "[ e ]" := (fun _ => e).
adam@381 752
adam@381 753 Example Add : Term (Nat --> Nat --> Nat) :=
adam@381 754 [\x : Nat, \y : Nat, ^x @+ ^y].
adam@381 755
adam@381 756 Example Three_the_hard_way : Term Nat :=
adam@381 757 [Add _ @ #1 @ #2].
adam@381 758
adam@381 759 Eval compute in TermDenote Three_the_hard_way.
adam@381 760 (** %\vspace{-.15in}%[[
adam@381 761 = 3
adam@381 762 ]]
adam@381 763 *)
adam@381 764
adam@381 765 End HigherOrder.
adam@381 766
adam@381 767 (** 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. *)