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Get it working in Coq 8.4beta1; use nice coqdoc notation for italics
author Adam Chlipala <adam@chlipala.net>
date Wed, 06 Jun 2012 11:25:13 -0400
parents d5112c099fbf
children 73f8165a3c1d
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@381 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{%#<a href="http://www.cis.upenn.edu/~bcpierce/sf/"><i>#Software Foundations#</i></a>#%}\footnote{\url{http://www.cis.upenn.edu/~bcpierce/sf/}}% 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@398 149 Axiom admit : forall T, T.
adam@381 150
adam@381 151 Fixpoint cfold G t (e : term G t) : term G t :=
adam@381 152 match e with
adam@381 153 | Plus G e1 e2 =>
adam@381 154 let e1' := cfold e1 in
adam@381 155 let e2' := cfold e2 in
adam@398 156 let maybeOpt := match e1' return _ with
adam@398 157 | Const _ n1 =>
adam@398 158 match e2' return _ with
adam@398 159 | Const _ n2 => Some (Const (n1 + n2))
adam@398 160 | _ => None
adam@398 161 end
adam@398 162 | _ => None
adam@398 163 end in
adam@398 164 match maybeOpt with
adam@398 165 | None => Plus e1' e2'
adam@398 166 | Some e' => e'
adam@398 167 end
adam@381 168
adam@381 169 | Abs _ _ _ e1 => Abs (cfold e1)
adam@381 170 | App _ _ _ e1 e2 => App (cfold e1) (cfold e2)
adam@381 171
adam@381 172 | Let _ _ _ e1 e2 => Let (cfold e1) (cfold e2)
adam@381 173
adam@381 174 | e => e
adam@381 175 end.
adam@381 176
adam@381 177 (** The correctness proof is more complex, but only slightly so. *)
adam@381 178
adam@381 179 Theorem cfoldSound : forall G t (e : term G t) s,
adam@381 180 termDenote (cfold e) s = termDenote e s.
adam@381 181 induction e; t;
adam@381 182 repeat (match goal with
adam@381 183 | [ |- context[match ?E with
adam@381 184 | Var _ _ _ => _ | Const _ _ => _ | Plus _ _ _ => _
adam@381 185 | Abs _ _ _ _ => _ | App _ _ _ _ _ => _
adam@381 186 | Let _ _ _ _ _ => _
adam@381 187 end] ] => dep_destruct E
adam@381 188 end; t).
adam@381 189 Qed.
adam@381 190
adam@398 191 (** 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@381 192
adam@398 193 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@381 194
adam@381 195 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@381 196
adam@381 197 Fixpoint insertAt (t : type) (G : list type) (n : nat) {struct n} : list type :=
adam@381 198 match n with
adam@381 199 | O => t :: G
adam@381 200 | S n' => match G with
adam@381 201 | nil => t :: G
adam@381 202 | t' :: G' => t' :: insertAt t G' n'
adam@381 203 end
adam@381 204 end.
adam@381 205
adam@381 206 (** Another function lifts bound variable instances, which we represent with [member] values. *)
adam@381 207
adam@381 208 Fixpoint liftVar t G (x : member t G) t' n : member t (insertAt t' G n) :=
adam@381 209 match x with
adam@381 210 | HFirst G' => match n return member t (insertAt t' (t :: G') n) with
adam@381 211 | O => HNext HFirst
adam@381 212 | _ => HFirst
adam@381 213 end
adam@381 214 | HNext t'' G' x' => match n return member t (insertAt t' (t'' :: G') n) with
adam@381 215 | O => HNext (HNext x')
adam@381 216 | S n' => HNext (liftVar x' t' n')
adam@381 217 end
adam@381 218 end.
adam@381 219
adam@381 220 (** The final helper function for lifting allows us to insert a new variable anywhere in a typing context. *)
adam@381 221
adam@381 222 Fixpoint lift' G t' n t (e : term G t) : term (insertAt t' G n) t :=
adam@381 223 match e with
adam@381 224 | Var _ _ x => Var (liftVar x t' n)
adam@381 225
adam@381 226 | Const _ n => Const n
adam@381 227 | Plus _ e1 e2 => Plus (lift' t' n e1) (lift' t' n e2)
adam@381 228
adam@381 229 | Abs _ _ _ e1 => Abs (lift' t' (S n) e1)
adam@381 230 | App _ _ _ e1 e2 => App (lift' t' n e1) (lift' t' n e2)
adam@381 231
adam@381 232 | Let _ _ _ e1 e2 => Let (lift' t' n e1) (lift' t' (S n) e2)
adam@381 233 end.
adam@381 234
adam@398 235 (** 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@381 236
adam@381 237 Definition lift G t' t (e : term G t) : term (t' :: G) t :=
adam@381 238 lift' t' O e.
adam@381 239
adam@381 240 (** 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@381 241
adam@381 242 Fixpoint unlet G t (e : term G t) G' : hlist (term G') G -> term G' t :=
adam@381 243 match e with
adam@381 244 | Var _ _ x => fun s => hget s x
adam@381 245
adam@381 246 | Const _ n => fun _ => Const n
adam@381 247 | Plus _ e1 e2 => fun s => Plus (unlet e1 s) (unlet e2 s)
adam@381 248
adam@381 249 | Abs _ _ _ e1 => fun s => Abs (unlet e1 (Var HFirst ::: hmap (lift _) s))
adam@381 250 | App _ _ _ e1 e2 => fun s => App (unlet e1 s) (unlet e2 s)
adam@381 251
adam@381 252 | Let _ t1 _ e1 e2 => fun s => unlet e2 (unlet e1 s ::: s)
adam@381 253 end.
adam@381 254
adam@381 255 (** 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@381 256
adam@381 257 Fixpoint insertAtS (t : type) (x : typeDenote t) (G : list type) (n : nat) {struct n}
adam@381 258 : hlist typeDenote G -> hlist typeDenote (insertAt t G n) :=
adam@381 259 match n with
adam@381 260 | O => fun s => x ::: s
adam@381 261 | S n' => match G return hlist typeDenote G
adam@381 262 -> hlist typeDenote (insertAt t G (S n')) with
adam@381 263 | nil => fun s => x ::: s
adam@381 264 | t' :: G' => fun s => hhd s ::: insertAtS t x n' (htl s)
adam@381 265 end
adam@381 266 end.
adam@381 267
adam@381 268 Implicit Arguments insertAtS [t G].
adam@381 269
adam@381 270 (** 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@381 271
adam@381 272 Lemma liftVarSound : forall t' (x : typeDenote t') t G (m : member t G) s n,
adam@381 273 hget s m = hget (insertAtS x n s) (liftVar m t' n).
adam@381 274 induction m; destruct n; dep_destruct s; t.
adam@381 275 Qed.
adam@381 276
adam@381 277 Hint Resolve liftVarSound.
adam@381 278
adam@381 279 (** An analogous lemma establishes correctness of [lift']. *)
adam@381 280
adam@381 281 Lemma lift'Sound : forall G t' (x : typeDenote t') t (e : term G t) n s,
adam@381 282 termDenote e s = termDenote (lift' t' n e) (insertAtS x n s).
adam@381 283 induction e; t;
adam@381 284 repeat match goal with
adam@381 285 | [ IH : forall n s, _ = termDenote (lift' _ n ?E) _
adam@381 286 |- context[lift' _ (S ?N) ?E] ] => specialize (IH (S N))
adam@381 287 end; t.
adam@381 288 Qed.
adam@381 289
adam@381 290 (** Correctness of [lift] itself is an easy corollary. *)
adam@381 291
adam@381 292 Lemma liftSound : forall G t' (x : typeDenote t') t (e : term G t) s,
adam@381 293 termDenote (lift t' e) (x ::: s) = termDenote e s.
adam@381 294 unfold lift; intros; rewrite (lift'Sound _ x e O); trivial.
adam@381 295 Qed.
adam@381 296
adam@381 297 Hint Rewrite hget_hmap hmap_hmap liftSound.
adam@381 298
adam@381 299 (** Finally, we can prove correctness of [unletSound] for terms in arbitrary typing environments. *)
adam@381 300
adam@381 301 Lemma unletSound' : forall G t (e : term G t) G' (s : hlist (term G') G) s1,
adam@381 302 termDenote (unlet e s) s1
adam@381 303 = termDenote e (hmap (fun t' (e' : term G' t') => termDenote e' s1) s).
adam@381 304 induction e; t.
adam@381 305 Qed.
adam@381 306
adam@381 307 (** 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@381 308
adam@381 309 Theorem unletSound : forall t (e : term nil t),
adam@381 310 termDenote (unlet e HNil) HNil = termDenote e HNil.
adam@381 311 intros; apply unletSound'.
adam@381 312 Qed.
adam@381 313
adam@381 314 End FirstOrder.
adam@381 315
adam@381 316 (** 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@381 317
adam@381 318
adam@381 319 (** * Parametric Higher-Order Abstract Syntax *)
adam@381 320
adam@398 321 (** 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@381 322
adam@381 323 Module HigherOrder.
adam@381 324
adam@398 325 (** 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@381 326
adam@381 327 (** %\vspace{-.15in}%[[
adam@381 328 Inductive term : type -> Type :=
adam@381 329 | Const : nat -> term Nat
adam@381 330 | Plus : term Nat -> term Nat -> term Nat
adam@381 331
adam@381 332 | Abs : forall dom ran, (term dom -> term ran) -> term (Func dom ran)
adam@381 333 | App : forall dom ran, term (Func dom ran) -> term dom -> term ran
adam@381 334
adam@381 335 | Let : forall t1 t2, term t1 -> (term t1 -> term t2) -> term t2.
adam@381 336 ]]
adam@381 337
adam@381 338 However, Coq rejects this definition for failing to meet the %\index{strict positivity restriction}%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@381 339
adam@398 340 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@381 341
adam@381 342 Section var.
adam@381 343 Variable var : type -> Type.
adam@381 344
adam@381 345 Inductive term : type -> Type :=
adam@381 346 | Var : forall t, var t -> term t
adam@381 347
adam@381 348 | Const : nat -> term Nat
adam@381 349 | Plus : term Nat -> term Nat -> term Nat
adam@381 350
adam@381 351 | Abs : forall dom ran, (var dom -> term ran) -> term (Func dom ran)
adam@381 352 | App : forall dom ran, term (Func dom ran) -> term dom -> term ran
adam@381 353
adam@381 354 | Let : forall t1 t2, term t1 -> (var t1 -> term t2) -> term t2.
adam@381 355 End var.
adam@381 356
adam@381 357 Implicit Arguments Var [var t].
adam@381 358 Implicit Arguments Const [var].
adam@381 359 Implicit Arguments Abs [var dom ran].
adam@381 360
adam@398 361 (** 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@381 362
adam@381 363 We write the final type of a closed term using polymorphic quantification over all possible choices of [var] type family. *)
adam@381 364
adam@381 365 Definition Term t := forall var, term var t.
adam@381 366
adam@398 367 (** 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@381 368
adam@381 369 Example add : Term (Func Nat (Func Nat Nat)) := fun var =>
adam@381 370 Abs (fun x => Abs (fun y => Plus (Var x) (Var y))).
adam@381 371
adam@381 372 Example three_the_hard_way : Term Nat := fun var =>
adam@381 373 App (App (add var) (Const 1)) (Const 2).
adam@381 374
adam@398 375 (** 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@381 376
adam@381 377 Example add' : Term (Func Nat (Func Nat Nat)) := fun _ =>
adam@381 378 Abs (fun x => Abs (fun y => Plus (Var x) (Var y))).
adam@381 379
adam@381 380 Example three_the_hard_way' : Term Nat := fun _ =>
adam@381 381 App (App (add' _) (Const 1)) (Const 2).
adam@381 382
adam@381 383
adam@381 384 (** ** Functional Programming with PHOAS *)
adam@381 385
adam@398 386 (** 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@381 387
adam@381 388 Fixpoint countVars t (e : term (fun _ => unit) t) : nat :=
adam@381 389 match e with
adam@381 390 | Var _ _ => 1
adam@381 391
adam@381 392 | Const _ => 0
adam@381 393 | Plus e1 e2 => countVars e1 + countVars e2
adam@381 394
adam@381 395 | Abs _ _ e1 => countVars (e1 tt)
adam@381 396 | App _ _ e1 e2 => countVars e1 + countVars e2
adam@381 397
adam@381 398 | Let _ _ e1 e2 => countVars e1 + countVars (e2 tt)
adam@381 399 end.
adam@381 400
adam@381 401 (** 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@381 402
adam@381 403 Definition CountVars t (E : Term t) := countVars (E (fun _ => unit)).
adam@381 404
adam@381 405 (** It is easy to test that [CountVars] operates properly. *)
adam@381 406
adam@381 407 Eval compute in CountVars three_the_hard_way.
adam@381 408 (** %\vspace{-.15in}%[[
adam@381 409 = 2
adam@381 410 ]]
adam@381 411 *)
adam@381 412
adam@381 413 (** 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@381 414
adam@381 415 Require Import String.
adam@381 416 Open Scope string_scope.
adam@381 417
adam@381 418 Fixpoint pretty t (e : term (fun _ => string) t) (x : string) : string :=
adam@381 419 match e with
adam@381 420 | Var _ s => s
adam@381 421
adam@381 422 | Const _ => "N"
adam@381 423 | Plus e1 e2 => "(" ++ pretty e1 x ++ " + " ++ pretty e2 x ++ ")"
adam@381 424
adam@381 425 | Abs _ _ e1 => "(fun " ++ x ++ " => " ++ pretty (e1 x) (x ++ "'") ++ ")"
adam@381 426 | App _ _ e1 e2 => "(" ++ pretty e1 x ++ " " ++ pretty e2 x ++ ")"
adam@381 427
adam@381 428 | Let _ _ e1 e2 => "(let " ++ x ++ " = " ++ pretty e1 x ++ " in "
adam@381 429 ++ pretty (e2 x) (x ++ "'") ++ ")"
adam@381 430 end.
adam@381 431
adam@381 432 Definition Pretty t (E : Term t) := pretty (E (fun _ => string)) "x".
adam@381 433
adam@381 434 Eval compute in Pretty three_the_hard_way.
adam@381 435 (** %\vspace{-.15in}%[[
adam@381 436 = "(((fun x => (fun x' => (x + x'))) N) N)"
adam@381 437 ]]
adam@381 438 *)
adam@381 439
adam@398 440 (** 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@381 441
adam@381 442 Fixpoint squash var t (e : term (term var) t) : term var t :=
adam@381 443 match e with
adam@381 444 | Var _ e1 => e1
adam@381 445
adam@381 446 | Const n => Const n
adam@381 447 | Plus e1 e2 => Plus (squash e1) (squash e2)
adam@381 448
adam@381 449 | Abs _ _ e1 => Abs (fun x => squash (e1 (Var x)))
adam@381 450 | App _ _ e1 e2 => App (squash e1) (squash e2)
adam@381 451
adam@381 452 | Let _ _ e1 e2 => Let (squash e1) (fun x => squash (e2 (Var x)))
adam@381 453 end.
adam@381 454
adam@381 455 (** 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@381 456
adam@381 457 Definition Term1 (t1 t2 : type) := forall var, var t1 -> term var t2.
adam@381 458
adam@381 459 (** 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@381 460
adam@381 461 Definition Subst (t1 t2 : type) (E : Term1 t1 t2) (E' : Term t1) : Term t2 :=
adam@381 462 fun var => squash (E (term var) (E' var)).
adam@381 463
adam@381 464 Eval compute in Subst (fun _ x => Plus (Var x) (Const 3)) three_the_hard_way.
adam@381 465 (** %\vspace{-.15in}%[[
adam@381 466 = fun var : type -> Type =>
adam@381 467 Plus
adam@381 468 (App
adam@381 469 (App
adam@381 470 (Abs
adam@381 471 (fun x : var Nat =>
adam@381 472 Abs (fun y : var Nat => Plus (Var x) (Var y))))
adam@381 473 (Const 1)) (Const 2)) (Const 3)
adam@381 474 ]]
adam@381 475
adam@398 476 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@381 477
adam@381 478 Fixpoint termDenote t (e : term typeDenote t) : typeDenote t :=
adam@381 479 match e with
adam@381 480 | Var _ v => v
adam@381 481
adam@381 482 | Const n => n
adam@381 483 | Plus e1 e2 => termDenote e1 + termDenote e2
adam@381 484
adam@381 485 | Abs _ _ e1 => fun x => termDenote (e1 x)
adam@381 486 | App _ _ e1 e2 => (termDenote e1) (termDenote e2)
adam@381 487
adam@381 488 | Let _ _ e1 e2 => termDenote (e2 (termDenote e1))
adam@381 489 end.
adam@381 490
adam@381 491 Definition TermDenote t (E : Term t) : typeDenote t :=
adam@381 492 termDenote (E typeDenote).
adam@381 493
adam@381 494 Eval compute in TermDenote three_the_hard_way.
adam@381 495 (** %\vspace{-.15in}%[[
adam@381 496 = 3
adam@381 497 ]]
adam@381 498
adam@381 499 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@381 500
adam@381 501
adam@381 502 (** ** Verifying Program Transformations *)
adam@381 503
adam@381 504 (** 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@381 505
adam@381 506 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@381 507
adam@381 508 Fixpoint ident var t (e : term var t) : term var t :=
adam@381 509 match e with
adam@381 510 | Var _ x => Var x
adam@381 511
adam@381 512 | Const n => Const n
adam@381 513 | Plus e1 e2 => Plus (ident e1) (ident e2)
adam@381 514
adam@381 515 | Abs _ _ e1 => Abs (fun x => ident (e1 x))
adam@381 516 | App _ _ e1 e2 => App (ident e1) (ident e2)
adam@381 517
adam@381 518 | Let _ _ e1 e2 => Let (ident e1) (fun x => ident (e2 x))
adam@381 519 end.
adam@381 520
adam@381 521 Definition Ident t (E : Term t) : Term t := fun var =>
adam@381 522 ident (E var).
adam@381 523
adam@381 524 (** 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@381 525
adam@381 526 Lemma identSound : forall t (e : term typeDenote t),
adam@381 527 termDenote (ident e) = termDenote e.
adam@381 528 induction e; t.
adam@381 529 Qed.
adam@381 530
adam@381 531 Theorem IdentSound : forall t (E : Term t),
adam@381 532 TermDenote (Ident E) = TermDenote E.
adam@381 533 intros; apply identSound.
adam@381 534 Qed.
adam@381 535
adam@381 536 (** The translation of the constant-folding function and its proof work more or less the same way. *)
adam@381 537
adam@381 538 Fixpoint cfold var t (e : term var t) : term var t :=
adam@381 539 match e with
adam@381 540 | Plus e1 e2 =>
adam@381 541 let e1' := cfold e1 in
adam@381 542 let e2' := cfold e2 in
adam@381 543 match e1', e2' with
adam@381 544 | Const n1, Const n2 => Const (n1 + n2)
adam@381 545 | _, _ => Plus e1' e2'
adam@381 546 end
adam@381 547
adam@381 548 | Abs _ _ e1 => Abs (fun x => cfold (e1 x))
adam@381 549 | App _ _ e1 e2 => App (cfold e1) (cfold e2)
adam@381 550
adam@381 551 | Let _ _ e1 e2 => Let (cfold e1) (fun x => cfold (e2 x))
adam@381 552
adam@381 553 | e => e
adam@381 554 end.
adam@381 555
adam@381 556 Definition Cfold t (E : Term t) : Term t := fun var =>
adam@381 557 cfold (E var).
adam@381 558
adam@381 559 Lemma cfoldSound : forall t (e : term typeDenote t),
adam@381 560 termDenote (cfold e) = termDenote e.
adam@381 561 induction e; t;
adam@381 562 repeat (match goal with
adam@381 563 | [ |- context[match ?E with
adam@381 564 | Var _ _ => _ | Const _ => _ | Plus _ _ => _
adam@381 565 | Abs _ _ _ => _ | App _ _ _ _ => _
adam@381 566 | Let _ _ _ _ => _
adam@381 567 end] ] => dep_destruct E
adam@381 568 end; t).
adam@381 569 Qed.
adam@381 570
adam@381 571 Theorem CfoldSound : forall t (E : Term t),
adam@381 572 TermDenote (Cfold E) = TermDenote E.
adam@381 573 intros; apply cfoldSound.
adam@381 574 Qed.
adam@381 575
adam@381 576 (** 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@381 577
adam@381 578 Fixpoint unlet var t (e : term (term var) t) : term var t :=
adam@381 579 match e with
adam@381 580 | Var _ e1 => e1
adam@381 581
adam@381 582 | Const n => Const n
adam@381 583 | Plus e1 e2 => Plus (unlet e1) (unlet e2)
adam@381 584
adam@381 585 | Abs _ _ e1 => Abs (fun x => unlet (e1 (Var x)))
adam@381 586 | App _ _ e1 e2 => App (unlet e1) (unlet e2)
adam@381 587
adam@381 588 | Let _ _ e1 e2 => unlet (e2 (unlet e1))
adam@381 589 end.
adam@381 590
adam@381 591 Definition Unlet t (E : Term t) : Term t := fun var =>
adam@381 592 unlet (E (term var)).
adam@381 593
adam@381 594 (** We can test [Unlet] first on an uninteresting example, [three_the_hard_way], which does not use [Let]. *)
adam@381 595
adam@381 596 Eval compute in Unlet three_the_hard_way.
adam@381 597 (** %\vspace{-.15in}%[[
adam@381 598 = fun var : type -> Type =>
adam@381 599 App
adam@381 600 (App
adam@381 601 (Abs
adam@381 602 (fun x : var Nat =>
adam@381 603 Abs (fun x0 : var Nat => Plus (Var x) (Var x0))))
adam@381 604 (Const 1)) (Const 2)
adam@381 605 ]]
adam@381 606
adam@381 607 Next, we try a more interesting example, with some extra [Let]s introduced in [three_the_hard_way]. *)
adam@381 608
adam@381 609 Definition three_a_harder_way : Term Nat := fun _ =>
adam@381 610 Let (Const 1) (fun x => Let (Const 2) (fun y => App (App (add _) (Var x)) (Var y))).
adam@381 611
adam@381 612 Eval compute in Unlet three_a_harder_way.
adam@381 613 (** %\vspace{-.15in}%[[
adam@381 614 = fun var : type -> Type =>
adam@381 615 App
adam@381 616 (App
adam@381 617 (Abs
adam@381 618 (fun x : var Nat =>
adam@381 619 Abs (fun x0 : var Nat => Plus (Var x) (Var x0))))
adam@381 620 (Const 1)) (Const 2)
adam@381 621 ]]
adam@381 622
adam@381 623 The output is the same as in the previous test, confirming that [Unlet] operates properly here.
adam@381 624
adam@381 625 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@381 626
adam@381 627 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@381 628
adam@381 629 Section wf.
adam@381 630 Variables var1 var2 : type -> Type.
adam@381 631
adam@381 632 (** 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@381 633
adam@381 634 Record varEntry := {
adam@381 635 Ty : type;
adam@381 636 First : var1 Ty;
adam@381 637 Second : var2 Ty
adam@381 638 }.
adam@381 639
adam@381 640 (** 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@381 641
adam@381 642 Inductive wf : list varEntry -> forall t, term var1 t -> term var2 t -> Prop :=
adam@381 643 | WfVar : forall G t x x', In {| Ty := t; First := x; Second := x' |} G
adam@381 644 -> wf G (Var x) (Var x')
adam@381 645
adam@381 646 | WfConst : forall G n, wf G (Const n) (Const n)
adam@381 647
adam@381 648 | WfPlus : forall G e1 e2 e1' e2', wf G e1 e1'
adam@381 649 -> wf G e2 e2'
adam@381 650 -> wf G (Plus e1 e2) (Plus e1' e2')
adam@381 651
adam@381 652 | WfAbs : forall G dom ran (e1 : _ dom -> term _ ran) e1',
adam@381 653 (forall x1 x2, wf ({| First := x1; Second := x2 |} :: G) (e1 x1) (e1' x2))
adam@381 654 -> wf G (Abs e1) (Abs e1')
adam@381 655
adam@381 656 | WfApp : forall G dom ran (e1 : term _ (Func dom ran)) (e2 : term _ dom) e1' e2',
adam@381 657 wf G e1 e1'
adam@381 658 -> wf G e2 e2'
adam@381 659 -> wf G (App e1 e2) (App e1' e2')
adam@381 660
adam@381 661 | WfLet : forall G t1 t2 e1 e1' (e2 : _ t1 -> term _ t2) e2', wf G e1 e1'
adam@381 662 -> (forall x1 x2, wf ({| First := x1; Second := x2 |} :: G) (e2 x1) (e2' x2))
adam@381 663 -> wf G (Let e1 e2) (Let e1' e2').
adam@381 664 End wf.
adam@381 665
adam@381 666 (** 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@381 667
adam@381 668 Definition Wf t (E : Term t) := forall var1 var2, wf nil (E var1) (E var2).
adam@381 669
adam@381 670 (** After digesting the syntactic details of [Wf], it is probably not hard to see that reasonable term encodings will satsify it. For example: *)
adam@381 671
adam@381 672 Theorem three_the_hard_way_Wf : Wf three_the_hard_way.
adam@381 673 red; intros; repeat match goal with
adam@381 674 | [ |- wf _ _ _ ] => constructor; intros
adam@381 675 end; intuition.
adam@381 676 Qed.
adam@381 677
adam@381 678 (** 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@381 679
adam@381 680 Hint Extern 1 => match goal with
adam@381 681 | [ H1 : Forall _ _, H2 : In _ _ |- _ ] => apply (Forall_In H1 _ H2)
adam@381 682 end.
adam@381 683
adam@381 684 (** The rest of the proof is about as automated as we could hope for. *)
adam@381 685
adam@381 686 Lemma unletSound : forall G t (e1 : term _ t) e2,
adam@381 687 wf G e1 e2
adam@381 688 -> Forall (fun ve => termDenote (First ve) = Second ve) G
adam@381 689 -> termDenote (unlet e1) = termDenote e2.
adam@381 690 induction 1; t.
adam@381 691 Qed.
adam@381 692
adam@381 693 Theorem UnletSound : forall t (E : Term t), Wf E
adam@381 694 -> TermDenote (Unlet E) = TermDenote E.
adam@381 695 intros; eapply unletSound; eauto.
adam@381 696 Qed.
adam@381 697
adam@381 698 (** 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@381 699
adam@381 700 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@381 701
adam@381 702 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@381 703
adam@381 704
adam@381 705 (** ** Establishing Term Well-Formedness *)
adam@381 706
adam@398 707 (** 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@381 708
adam@381 709 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@381 710
adam@398 711 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@381 712
adam@381 713 Hint Constructors wf.
adam@381 714 Hint Extern 1 (In _ _) => simpl; tauto.
adam@381 715 Hint Extern 1 (Forall _ _) => eapply Forall_weaken; [ eassumption | simpl ].
adam@381 716
adam@381 717 Lemma wf_monotone : forall var1 var2 G t (e1 : term var1 t) (e2 : term var2 t),
adam@381 718 wf G e1 e2
adam@381 719 -> forall G', Forall (fun x => In x G') G
adam@381 720 -> wf G' e1 e2.
adam@381 721 induction 1; t; auto 6.
adam@381 722 Qed.
adam@381 723
adam@381 724 Hint Resolve wf_monotone Forall_In'.
adam@381 725
adam@381 726 (** 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@381 727
adam@381 728 Hint Extern 1 (wf _ _ _) => progress simpl.
adam@381 729
adam@381 730 Lemma unletWf : forall var1 var2 G t (e1 : term (term var1) t) (e2 : term (term var2) t),
adam@381 731 wf G e1 e2
adam@381 732 -> forall G', Forall (fun ve => wf G' (First ve) (Second ve)) G
adam@381 733 -> wf G' (unlet e1) (unlet e2).
adam@381 734 induction 1; t; eauto 9.
adam@381 735 Qed.
adam@381 736
adam@381 737 (** Repackaging [unletWf] into a theorem about [Wf] and [Unlet] is straightforward. *)
adam@381 738
adam@381 739 Theorem UnletWf : forall t (E : Term t), Wf E
adam@381 740 -> Wf (Unlet E).
adam@381 741 red; intros; apply unletWf with nil; auto.
adam@381 742 Qed.
adam@381 743
adam@381 744 (** 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@381 745
adam@381 746
adam@381 747 (** ** A Few More Remarks *)
adam@381 748
adam@381 749 (** 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@381 750
adam@381 751 Infix "-->" := Func (right associativity, at level 52).
adam@381 752
adam@381 753 Notation "^" := Var.
adam@381 754 Notation "#" := Const.
adam@381 755 Infix "@" := App (left associativity, at level 50).
adam@381 756 Infix "@+" := Plus (left associativity, at level 50).
adam@381 757 Notation "\ x : t , e" := (Abs (dom := t) (fun x => e))
adam@381 758 (no associativity, at level 51, x at level 0).
adam@381 759 Notation "[ e ]" := (fun _ => e).
adam@381 760
adam@381 761 Example Add : Term (Nat --> Nat --> Nat) :=
adam@381 762 [\x : Nat, \y : Nat, ^x @+ ^y].
adam@381 763
adam@381 764 Example Three_the_hard_way : Term Nat :=
adam@381 765 [Add _ @ #1 @ #2].
adam@381 766
adam@381 767 Eval compute in TermDenote Three_the_hard_way.
adam@381 768 (** %\vspace{-.15in}%[[
adam@381 769 = 3
adam@381 770 ]]
adam@381 771 *)
adam@381 772
adam@381 773 End HigherOrder.
adam@381 774
adam@381 775 (** 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. *)