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A pass over all formatting, after big pile of coqdoc changes
author Adam Chlipala <adam@chlipala.net>
date Fri, 27 Jul 2012 16:47:28 -0400
parents 5d5e44f905c7
children 0d66f1a710b8
rev   line source
adam@377 1 (* Copyright (c) 2009-2012, Adam Chlipala
adamc@227 2 *
adamc@227 3 * This work is licensed under a
adamc@227 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@227 5 * Unported License.
adamc@227 6 * The license text is available at:
adamc@227 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@227 8 *)
adamc@227 9
adamc@227 10 (* begin hide *)
adam@377 11 Require Import List.
adam@377 12
adam@314 13 Require Import DepList CpdtTactics.
adamc@227 14
adamc@227 15 Set Implicit Arguments.
adamc@227 16 (* end hide *)
adamc@227 17
adam@398 18 (** printing $ %({}*% #(<a/>*# *)
adam@398 19 (** printing ^ %*{})% #*<a/>)# *)
adam@398 20
adam@398 21
adamc@227 22
adamc@227 23 (** %\chapter{Universes and Axioms}% *)
adamc@227 24
adam@343 25 (** Many traditional theorems can be proved in Coq without special knowledge of CIC, the logic behind the prover. A development just seems to be using a particular ASCII notation for standard formulas based on %\index{set theory}%set theory. Nonetheless, as we saw in Chapter 4, CIC differs from set theory in starting from fewer orthogonal primitives. It is possible to define the usual logical connectives as derived notions. The foundation of it all is a dependently typed functional programming language, based on dependent function types and inductive type families. By using the facilities of this language directly, we can accomplish some things much more easily than in mainstream math.
adamc@227 26
adam@343 27 %\index{Gallina}%Gallina, which adds features to the more theoretical CIC%~\cite{CIC}%, is the logic implemented in Coq. It has a relatively simple foundation that can be defined rigorously in a page or two of formal proof rules. Still, there are some important subtleties that have practical ramifications. This chapter focuses on those subtleties, avoiding formal metatheory in favor of example code. *)
adamc@227 28
adamc@227 29
adamc@227 30 (** * The [Type] Hierarchy *)
adamc@227 31
adam@343 32 (** %\index{type hierarchy}%Every object in Gallina has a type. *)
adamc@227 33
adamc@227 34 Check 0.
adamc@227 35 (** %\vspace{-.15in}% [[
adamc@227 36 0
adamc@227 37 : nat
adamc@227 38
adamc@227 39 ]]
adamc@227 40
adamc@227 41 It is natural enough that zero be considered as a natural number. *)
adamc@227 42
adamc@227 43 Check nat.
adamc@227 44 (** %\vspace{-.15in}% [[
adamc@227 45 nat
adamc@227 46 : Set
adamc@227 47
adamc@227 48 ]]
adamc@227 49
adam@429 50 From a set theory perspective, it is unsurprising to consider the natural numbers as a "set." *)
adamc@227 51
adamc@227 52 Check Set.
adamc@227 53 (** %\vspace{-.15in}% [[
adamc@227 54 Set
adamc@227 55 : Type
adamc@227 56
adamc@227 57 ]]
adamc@227 58
adam@409 59 The type [Set] may be considered as the set of all sets, a concept that set theory handles in terms of%\index{class (in set theory)}% _classes_. In Coq, this more general notion is [Type]. *)
adamc@227 60
adamc@227 61 Check Type.
adamc@227 62 (** %\vspace{-.15in}% [[
adamc@227 63 Type
adamc@227 64 : Type
adamc@227 65
adamc@227 66 ]]
adamc@227 67
adam@429 68 Strangely enough, [Type] appears to be its own type. It is known that polymorphic languages with this property are inconsistent, via %\index{Girard's paradox}%Girard's paradox%~\cite{GirardsParadox}%. That is, using such a language to encode proofs is unwise, because it is possible to "prove" any proposition. What is really going on here?
adamc@227 69
adam@343 70 Let us repeat some of our queries after toggling a flag related to Coq's printing behavior.%\index{Vernacular commands!Set Printing Universes}% *)
adamc@227 71
adamc@227 72 Set Printing Universes.
adamc@227 73
adamc@227 74 Check nat.
adamc@227 75 (** %\vspace{-.15in}% [[
adamc@227 76 nat
adamc@227 77 : Set
adam@302 78 ]]
adam@398 79 *)
adamc@227 80
adamc@227 81 Check Set.
adamc@227 82 (** %\vspace{-.15in}% [[
adamc@227 83 Set
adamc@227 84 : Type $ (0)+1 ^
adamc@227 85
adam@302 86 ]]
adam@302 87 *)
adamc@227 88
adamc@227 89 Check Type.
adamc@227 90 (** %\vspace{-.15in}% [[
adamc@227 91 Type $ Top.3 ^
adamc@227 92 : Type $ (Top.3)+1 ^
adamc@227 93
adamc@227 94 ]]
adamc@227 95
adam@429 96 Occurrences of [Type] are annotated with some additional information, inside comments. These annotations have to do with the secret behind [Type]: it really stands for an infinite hierarchy of types. The type of [Set] is [Type(0)], the type of [Type(0)] is [Type(1)], the type of [Type(1)] is [Type(2)], and so on. This is how we avoid the "[Type : Type]" paradox. As a convenience, the universe hierarchy drives Coq's one variety of subtyping. Any term whose type is [Type] at level [i] is automatically also described by [Type] at level [j] when [j > i].
adamc@227 97
adam@398 98 In the outputs of our first [Check] query, we see that the type level of [Set]'s type is [(0)+1]. Here [0] stands for the level of [Set], and we increment it to arrive at the level that _classifies_ [Set].
adamc@227 99
adam@409 100 In the second query's output, we see that the occurrence of [Type] that we check is assigned a fresh%\index{universe variable}% _universe variable_ [Top.3]. The output type increments [Top.3] to move up a level in the universe hierarchy. As we write code that uses definitions whose types mention universe variables, unification may refine the values of those variables. Luckily, the user rarely has to worry about the details.
adamc@227 101
adam@409 102 Another crucial concept in CIC is%\index{predicativity}% _predicativity_. Consider these queries. *)
adamc@227 103
adamc@227 104 Check forall T : nat, fin T.
adamc@227 105 (** %\vspace{-.15in}% [[
adamc@227 106 forall T : nat, fin T
adamc@227 107 : Set
adam@302 108 ]]
adam@302 109 *)
adamc@227 110
adamc@227 111 Check forall T : Set, T.
adamc@227 112 (** %\vspace{-.15in}% [[
adamc@227 113 forall T : Set, T
adamc@227 114 : Type $ max(0, (0)+1) ^
adam@302 115 ]]
adam@302 116 *)
adamc@227 117
adamc@227 118 Check forall T : Type, T.
adamc@227 119 (** %\vspace{-.15in}% [[
adamc@227 120 forall T : Type $ Top.9 ^ , T
adamc@227 121 : Type $ max(Top.9, (Top.9)+1) ^
adamc@227 122
adamc@227 123 ]]
adamc@227 124
adamc@227 125 These outputs demonstrate the rule for determining which universe a [forall] type lives in. In particular, for a type [forall x : T1, T2], we take the maximum of the universes of [T1] and [T2]. In the first example query, both [T1] ([nat]) and [T2] ([fin T]) are in [Set], so the [forall] type is in [Set], too. In the second query, [T1] is [Set], which is at level [(0)+1]; and [T2] is [T], which is at level [0]. Thus, the [forall] exists at the maximum of these two levels. The third example illustrates the same outcome, where we replace [Set] with an occurrence of [Type] that is assigned universe variable [Top.9]. This universe variable appears in the places where [0] appeared in the previous query.
adamc@227 126
adam@287 127 The behind-the-scenes manipulation of universe variables gives us predicativity. Consider this simple definition of a polymorphic identity function, where the first argument [T] will automatically be marked as implicit, since it can be inferred from the type of the second argument [x]. *)
adamc@227 128
adamc@227 129 Definition id (T : Set) (x : T) : T := x.
adamc@227 130
adamc@227 131 Check id 0.
adamc@227 132 (** %\vspace{-.15in}% [[
adamc@227 133 id 0
adamc@227 134 : nat
adamc@227 135
adamc@227 136 Check id Set.
adam@343 137 ]]
adamc@227 138
adam@343 139 <<
adamc@227 140 Error: Illegal application (Type Error):
adamc@227 141 ...
adam@343 142 The 1st term has type "Type (* (Top.15)+1 *)" which should be coercible to "Set".
adam@343 143 >>
adamc@227 144
adam@343 145 The parameter [T] of [id] must be instantiated with a [Set]. The type [nat] is a [Set], but [Set] is not. We can try fixing the problem by generalizing our definition of [id]. *)
adamc@227 146
adamc@227 147 Reset id.
adamc@227 148 Definition id (T : Type) (x : T) : T := x.
adamc@227 149 Check id 0.
adamc@227 150 (** %\vspace{-.15in}% [[
adamc@227 151 id 0
adamc@227 152 : nat
adam@302 153 ]]
adam@302 154 *)
adamc@227 155
adamc@227 156 Check id Set.
adamc@227 157 (** %\vspace{-.15in}% [[
adamc@227 158 id Set
adamc@227 159 : Type $ Top.17 ^
adam@302 160 ]]
adam@302 161 *)
adamc@227 162
adamc@227 163 Check id Type.
adamc@227 164 (** %\vspace{-.15in}% [[
adamc@227 165 id Type $ Top.18 ^
adamc@227 166 : Type $ Top.19 ^
adam@302 167 ]]
adam@302 168 *)
adamc@227 169
adamc@227 170 (** So far so good. As we apply [id] to different [T] values, the inferred index for [T]'s [Type] occurrence automatically moves higher up the type hierarchy.
adamc@227 171 [[
adamc@227 172 Check id id.
adam@343 173 ]]
adamc@227 174
adam@343 175 <<
adamc@227 176 Error: Universe inconsistency (cannot enforce Top.16 < Top.16).
adam@343 177 >>
adamc@227 178
adam@429 179 %\index{universe inconsistency}%This error message reminds us that the universe variable for [T] still exists, even though it is usually hidden. To apply [id] to itself, that variable would need to be less than itself in the type hierarchy. Universe inconsistency error messages announce cases like this one where a term could only type-check by violating an implied constraint over universe variables. Such errors demonstrate that [Type] is _predicative_, where this word has a CIC meaning closely related to its usual mathematical meaning. A predicative system enforces the constraint that, for any object of quantified type, none of those quantifiers may ever be instantiated with the object itself. %\index{impredicativity}%Impredicativity is associated with popular paradoxes in set theory, involving inconsistent constructions like "the set of all sets that do not contain themselves" (%\index{Russell's paradox}%Russell's paradox). Similar paradoxes would result from uncontrolled impredicativity in Coq. *)
adamc@227 180
adamc@227 181
adamc@227 182 (** ** Inductive Definitions *)
adamc@227 183
adamc@227 184 (** Predicativity restrictions also apply to inductive definitions. As an example, let us consider a type of expression trees that allows injection of any native Coq value. The idea is that an [exp T] stands for a reflected expression of type [T].
adamc@227 185
adamc@227 186 [[
adamc@227 187 Inductive exp : Set -> Set :=
adamc@227 188 | Const : forall T : Set, T -> exp T
adamc@227 189 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 190 | Eq : forall T, exp T -> exp T -> exp bool.
adam@343 191 ]]
adamc@227 192
adam@343 193 <<
adamc@227 194 Error: Large non-propositional inductive types must be in Type.
adam@343 195 >>
adamc@227 196
adam@409 197 This definition is%\index{large inductive types}% _large_ in the sense that at least one of its constructors takes an argument whose type has type [Type]. Coq would be inconsistent if we allowed definitions like this one in their full generality. Instead, we must change [exp] to live in [Type]. We will go even further and move [exp]'s index to [Type] as well. *)
adamc@227 198
adamc@227 199 Inductive exp : Type -> Type :=
adamc@227 200 | Const : forall T, T -> exp T
adamc@227 201 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 202 | Eq : forall T, exp T -> exp T -> exp bool.
adamc@227 203
adamc@228 204 (** Note that before we had to include an annotation [: Set] for the variable [T] in [Const]'s type, but we need no annotation now. When the type of a variable is not known, and when that variable is used in a context where only types are allowed, Coq infers that the variable is of type [Type]. That is the right behavior here, but it was wrong for the [Set] version of [exp].
adamc@228 205
adamc@228 206 Our new definition is accepted. We can build some sample expressions. *)
adamc@227 207
adamc@227 208 Check Const 0.
adamc@227 209 (** %\vspace{-.15in}% [[
adamc@227 210 Const 0
adamc@227 211 : exp nat
adam@302 212 ]]
adam@302 213 *)
adamc@227 214
adamc@227 215 Check Pair (Const 0) (Const tt).
adamc@227 216 (** %\vspace{-.15in}% [[
adamc@227 217 Pair (Const 0) (Const tt)
adamc@227 218 : exp (nat * unit)
adam@302 219 ]]
adam@302 220 *)
adamc@227 221
adamc@227 222 Check Eq (Const Set) (Const Type).
adamc@227 223 (** %\vspace{-.15in}% [[
adamc@228 224 Eq (Const Set) (Const Type $ Top.59 ^ )
adamc@227 225 : exp bool
adamc@227 226
adamc@227 227 ]]
adamc@227 228
adamc@227 229 We can check many expressions, including fancy expressions that include types. However, it is not hard to hit a type-checking wall.
adamc@227 230
adamc@227 231 [[
adamc@227 232 Check Const (Const O).
adam@343 233 ]]
adamc@227 234
adam@343 235 <<
adamc@227 236 Error: Universe inconsistency (cannot enforce Top.42 < Top.42).
adam@343 237 >>
adamc@227 238
adamc@227 239 We are unable to instantiate the parameter [T] of [Const] with an [exp] type. To see why, it is helpful to print the annotated version of [exp]'s inductive definition. *)
adamc@227 240
adam@417 241 (** [[
adamc@227 242 Print exp.
adam@417 243 ]]
adam@417 244
adam@417 245 [[
adamc@227 246 Inductive exp
adamc@227 247 : Type $ Top.8 ^ ->
adamc@227 248 Type
adamc@227 249 $ max(0, (Top.11)+1, (Top.14)+1, (Top.15)+1, (Top.19)+1) ^ :=
adamc@227 250 Const : forall T : Type $ Top.11 ^ , T -> exp T
adamc@227 251 | Pair : forall (T1 : Type $ Top.14 ^ ) (T2 : Type $ Top.15 ^ ),
adamc@227 252 exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 253 | Eq : forall T : Type $ Top.19 ^ , exp T -> exp T -> exp bool
adamc@227 254
adamc@227 255 ]]
adamc@227 256
adam@398 257 We see that the index type of [exp] has been assigned to universe level [Top.8]. In addition, each of the four occurrences of [Type] in the types of the constructors gets its own universe variable. Each of these variables appears explicitly in the type of [exp]. In particular, any type [exp T] lives at a universe level found by incrementing by one the maximum of the four argument variables. A consequence of this is that [exp] _must_ live at a higher universe level than any type which may be passed to one of its constructors. This consequence led to the universe inconsistency.
adamc@227 258
adam@429 259 Strangely, the universe variable [Top.8] only appears in one place. Is there no restriction imposed on which types are valid arguments to [exp]? In fact, there is a restriction, but it only appears in a global set of universe constraints that are maintained "off to the side," not appearing explicitly in types. We can print the current database.%\index{Vernacular commands!Print Universes}% *)
adamc@227 260
adamc@227 261 Print Universes.
adamc@227 262 (** %\vspace{-.15in}% [[
adamc@227 263 Top.19 < Top.9 <= Top.8
adamc@227 264 Top.15 < Top.9 <= Top.8 <= Coq.Init.Datatypes.38
adamc@227 265 Top.14 < Top.9 <= Top.8 <= Coq.Init.Datatypes.37
adamc@227 266 Top.11 < Top.9 <= Top.8
adamc@227 267
adamc@227 268 ]]
adamc@227 269
adam@343 270 The command outputs many more constraints, but we have collected only those that mention [Top] variables. We see one constraint for each universe variable associated with a constructor argument from [exp]'s definition. Universe variable [Top.19] is the type argument to [Eq]. The constraint for [Top.19] effectively says that [Top.19] must be less than [Top.8], the universe of [exp]'s indices; an intermediate variable [Top.9] appears as an artifact of the way the constraint was generated.
adamc@227 271
adamc@227 272 The next constraint, for [Top.15], is more complicated. This is the universe of the second argument to the [Pair] constructor. Not only must [Top.15] be less than [Top.8], but it also comes out that [Top.8] must be less than [Coq.Init.Datatypes.38]. What is this new universe variable? It is from the definition of the [prod] inductive family, to which types of the form [A * B] are desugared. *)
adamc@227 273
adam@417 274 (* begin hide *)
adam@437 275 (* begin thide *)
adam@417 276 Inductive prod := pair.
adam@417 277 Reset prod.
adam@437 278 (* end thide *)
adam@417 279 (* end hide *)
adam@417 280
adam@417 281 (** [[
adamc@227 282 Print prod.
adam@417 283 ]]
adam@417 284
adam@417 285 [[
adamc@227 286 Inductive prod (A : Type $ Coq.Init.Datatypes.37 ^ )
adamc@227 287 (B : Type $ Coq.Init.Datatypes.38 ^ )
adamc@227 288 : Type $ max(Coq.Init.Datatypes.37, Coq.Init.Datatypes.38) ^ :=
adamc@227 289 pair : A -> B -> A * B
adamc@227 290
adamc@227 291 ]]
adamc@227 292
adamc@227 293 We see that the constraint is enforcing that indices to [exp] must not live in a higher universe level than [B]-indices to [prod]. The next constraint above establishes a symmetric condition for [A].
adamc@227 294
adamc@227 295 Thus it is apparent that Coq maintains a tortuous set of universe variable inequalities behind the scenes. It may look like some functions are polymorphic in the universe levels of their arguments, but what is really happening is imperative updating of a system of constraints, such that all uses of a function are consistent with a global set of universe levels. When the constraint system may not be evolved soundly, we get a universe inconsistency error.
adamc@227 296
adamc@227 297 %\medskip%
adamc@227 298
adam@398 299 Something interesting is revealed in the annotated definition of [prod]. A type [prod A B] lives at a universe that is the maximum of the universes of [A] and [B]. From our earlier experiments, we might expect that [prod]'s universe would in fact need to be _one higher_ than the maximum. The critical difference is that, in the definition of [prod], [A] and [B] are defined as _parameters_; that is, they appear named to the left of the main colon, rather than appearing (possibly unnamed) to the right.
adamc@227 300
adamc@231 301 Parameters are not as flexible as normal inductive type arguments. The range types of all of the constructors of a parameterized type must share the same parameters. Nonetheless, when it is possible to define a polymorphic type in this way, we gain the ability to use the new type family in more ways, without triggering universe inconsistencies. For instance, nested pairs of types are perfectly legal. *)
adamc@227 302
adamc@227 303 Check (nat, (Type, Set)).
adamc@227 304 (** %\vspace{-.15in}% [[
adamc@227 305 (nat, (Type $ Top.44 ^ , Set))
adamc@227 306 : Set * (Type $ Top.45 ^ * Type $ Top.46 ^ )
adamc@227 307 ]]
adamc@227 308
adamc@227 309 The same cannot be done with a counterpart to [prod] that does not use parameters. *)
adamc@227 310
adamc@227 311 Inductive prod' : Type -> Type -> Type :=
adamc@227 312 | pair' : forall A B : Type, A -> B -> prod' A B.
adamc@227 313 (** [[
adamc@227 314 Check (pair' nat (pair' Type Set)).
adam@343 315 ]]
adamc@227 316
adam@343 317 <<
adamc@227 318 Error: Universe inconsistency (cannot enforce Top.51 < Top.51).
adam@343 319 >>
adamc@227 320
adamc@233 321 The key benefit parameters bring us is the ability to avoid quantifying over types in the types of constructors. Such quantification induces less-than constraints, while parameters only introduce less-than-or-equal-to constraints.
adamc@233 322
adam@343 323 Coq includes one more (potentially confusing) feature related to parameters. While Gallina does not support real %\index{universe polymorphism}%universe polymorphism, there is a convenience facility that mimics universe polymorphism in some cases. We can illustrate what this means with a simple example. *)
adamc@233 324
adamc@233 325 Inductive foo (A : Type) : Type :=
adamc@233 326 | Foo : A -> foo A.
adamc@229 327
adamc@229 328 (* begin hide *)
adamc@229 329 Unset Printing Universes.
adamc@229 330 (* end hide *)
adamc@229 331
adamc@233 332 Check foo nat.
adamc@233 333 (** %\vspace{-.15in}% [[
adamc@233 334 foo nat
adamc@233 335 : Set
adam@302 336 ]]
adam@302 337 *)
adamc@233 338
adamc@233 339 Check foo Set.
adamc@233 340 (** %\vspace{-.15in}% [[
adamc@233 341 foo Set
adamc@233 342 : Type
adam@302 343 ]]
adam@302 344 *)
adamc@233 345
adamc@233 346 Check foo True.
adamc@233 347 (** %\vspace{-.15in}% [[
adamc@233 348 foo True
adamc@233 349 : Prop
adamc@233 350
adamc@233 351 ]]
adamc@233 352
adam@429 353 The basic pattern here is that Coq is willing to automatically build a "copied-and-pasted" version of an inductive definition, where some occurrences of [Type] have been replaced by [Set] or [Prop]. In each context, the type-checker tries to find the valid replacements that are lowest in the type hierarchy. Automatic cloning of definitions can be much more convenient than manual cloning. We have already taken advantage of the fact that we may re-use the same families of tuple and list types to form values in [Set] and [Type].
adamc@233 354
adamc@233 355 Imitation polymorphism can be confusing in some contexts. For instance, it is what is responsible for this weird behavior. *)
adamc@233 356
adamc@233 357 Inductive bar : Type := Bar : bar.
adamc@233 358
adamc@233 359 Check bar.
adamc@233 360 (** %\vspace{-.15in}% [[
adamc@233 361 bar
adamc@233 362 : Prop
adamc@233 363 ]]
adamc@233 364
adamc@233 365 The type that Coq comes up with may be used in strictly more contexts than the type one might have expected. *)
adamc@233 366
adamc@229 367
adam@388 368 (** ** Deciphering Baffling Messages About Inability to Unify *)
adam@388 369
adam@388 370 (** One of the most confusing sorts of Coq error messages arises from an interplay between universes, syntax notations, and %\index{implicit arguments}%implicit arguments. Consider the following innocuous lemma, which is symmetry of equality for the special case of types. *)
adam@388 371
adam@388 372 Theorem symmetry : forall A B : Type,
adam@388 373 A = B
adam@388 374 -> B = A.
adam@388 375 intros ? ? H; rewrite H; reflexivity.
adam@388 376 Qed.
adam@388 377
adam@388 378 (** Let us attempt an admittedly silly proof of the following theorem. *)
adam@388 379
adam@388 380 Theorem illustrative_but_silly_detour : unit = unit.
adam@388 381 (** [[
adam@388 382 apply symmetry.
adam@388 383 ]]
adam@388 384 <<
adam@388 385 Error: Impossible to unify "?35 = ?34" with "unit = unit".
adam@388 386 >>
adam@388 387
adam@398 388 Coq tells us that we cannot, in fact, apply our lemma [symmetry] here, but the error message seems defective. In particular, one might think that [apply] should unify [?35] and [?34] with [unit] to ensure that the unification goes through. In fact, the problem is in a part of the unification problem that is _not_ shown to us in this error message!
adam@388 389
adam@388 390 The following command is the secret to getting better error messages in such cases: *)
adam@388 391
adam@388 392 Set Printing All.
adam@388 393 (** [[
adam@388 394 apply symmetry.
adam@388 395 ]]
adam@388 396 <<
adam@388 397 Error: Impossible to unify "@eq Type ?46 ?45" with "@eq Set unit unit".
adam@388 398 >>
adam@388 399
adam@398 400 Now we can see the problem: it is the first, _implicit_ argument to the underlying equality function [eq] that disagrees across the two terms. The universe [Set] may be both an element and a subtype of [Type], but the two are not definitionally equal. *)
adam@388 401
adam@388 402 Abort.
adam@388 403
adam@388 404 (** A variety of changes to the theorem statement would lead to use of [Type] as the implicit argument of [eq]. Here is one such change. *)
adam@388 405
adam@388 406 Theorem illustrative_but_silly_detour : (unit : Type) = unit.
adam@388 407 apply symmetry; reflexivity.
adam@388 408 Qed.
adam@388 409
adam@388 410 (** There are many related issues that can come up with error messages, where one or both of notations and implicit arguments hide important details. The [Set Printing All] command turns off all such features and exposes underlying CIC terms.
adam@388 411
adam@388 412 For completeness, we mention one other class of confusing error message about inability to unify two terms that look obviously unifiable. Each unification variable has a scope; a unification variable instantiation may not mention variables that were not already defined within that scope, at the point in proof search where the unification variable was introduced. Consider this illustrative example: *)
adam@388 413
adam@388 414 Unset Printing All.
adam@388 415
adam@388 416 Theorem ex_symmetry : (exists x, x = 0) -> (exists x, 0 = x).
adam@435 417 eexists.
adam@388 418 (** %\vspace{-.15in}%[[
adam@388 419 H : exists x : nat, x = 0
adam@388 420 ============================
adam@388 421 0 = ?98
adam@388 422 ]]
adam@388 423 *)
adam@388 424
adam@388 425 destruct H.
adam@388 426 (** %\vspace{-.15in}%[[
adam@388 427 x : nat
adam@388 428 H : x = 0
adam@388 429 ============================
adam@388 430 0 = ?99
adam@388 431 ]]
adam@388 432 *)
adam@388 433
adam@388 434 (** [[
adam@388 435 symmetry; exact H.
adam@388 436 ]]
adam@388 437
adam@388 438 <<
adam@388 439 Error: In environment
adam@388 440 x : nat
adam@388 441 H : x = 0
adam@388 442 The term "H" has type "x = 0" while it is expected to have type
adam@388 443 "?99 = 0".
adam@388 444 >>
adam@388 445
adam@398 446 The problem here is that variable [x] was introduced by [destruct] _after_ we introduced [?99] with [eexists], so the instantiation of [?99] may not mention [x]. A simple reordering of the proof solves the problem. *)
adam@388 447
adam@388 448 Restart.
adam@388 449 destruct 1 as [x]; apply ex_intro with x; symmetry; assumption.
adam@388 450 Qed.
adam@388 451
adam@429 452 (** This restriction for unification variables may seem counterintuitive, but it follows from the fact that CIC contains no concept of unification variable. Rather, to construct the final proof term, at the point in a proof where the unification variable is introduced, we replace it with the instantiation we eventually find for it. It is simply syntactically illegal to refer there to variables that are not in scope. Without such a restriction, we could trivially "prove" such non-theorems as [exists n : nat, forall m : nat, n = m] by [econstructor; intro; reflexivity]. *)
adam@388 453
adam@388 454
adamc@229 455 (** * The [Prop] Universe *)
adamc@229 456
adam@429 457 (** In Chapter 4, we saw parallel versions of useful datatypes for "programs" and "proofs." The convention was that programs live in [Set], and proofs live in [Prop]. We gave little explanation for why it is useful to maintain this distinction. There is certainly documentation value from separating programs from proofs; in practice, different concerns apply to building the two types of objects. It turns out, however, that these concerns motivate formal differences between the two universes in Coq.
adamc@229 458
adamc@229 459 Recall the types [sig] and [ex], which are the program and proof versions of existential quantification. Their definitions differ only in one place, where [sig] uses [Type] and [ex] uses [Prop]. *)
adamc@229 460
adamc@229 461 Print sig.
adamc@229 462 (** %\vspace{-.15in}% [[
adamc@229 463 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@229 464 exist : forall x : A, P x -> sig P
adam@302 465 ]]
adam@302 466 *)
adamc@229 467
adamc@229 468 Print ex.
adamc@229 469 (** %\vspace{-.15in}% [[
adamc@229 470 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@229 471 ex_intro : forall x : A, P x -> ex P
adamc@229 472 ]]
adamc@229 473
adamc@229 474 It is natural to want a function to extract the first components of data structures like these. Doing so is easy enough for [sig]. *)
adamc@229 475
adamc@229 476 Definition projS A (P : A -> Prop) (x : sig P) : A :=
adamc@229 477 match x with
adamc@229 478 | exist v _ => v
adamc@229 479 end.
adamc@229 480
adam@429 481 (* begin hide *)
adam@437 482 (* begin thide *)
adam@429 483 Definition projE := O.
adam@437 484 (* end thide *)
adam@429 485 (* end hide *)
adam@429 486
adamc@229 487 (** We run into trouble with a version that has been changed to work with [ex].
adamc@229 488 [[
adamc@229 489 Definition projE A (P : A -> Prop) (x : ex P) : A :=
adamc@229 490 match x with
adamc@229 491 | ex_intro v _ => v
adamc@229 492 end.
adam@343 493 ]]
adamc@229 494
adam@343 495 <<
adamc@229 496 Error:
adamc@229 497 Incorrect elimination of "x" in the inductive type "ex":
adamc@229 498 the return type has sort "Type" while it should be "Prop".
adamc@229 499 Elimination of an inductive object of sort Prop
adamc@229 500 is not allowed on a predicate in sort Type
adamc@229 501 because proofs can be eliminated only to build proofs.
adam@343 502 >>
adamc@229 503
adam@429 504 In formal Coq parlance, %\index{elimination}%"elimination" means "pattern-matching." The typing rules of Gallina forbid us from pattern-matching on a discriminee whose type belongs to [Prop], whenever the result type of the [match] has a type besides [Prop]. This is a sort of "information flow" policy, where the type system ensures that the details of proofs can never have any effect on parts of a development that are not also marked as proofs.
adamc@229 505
adamc@229 506 This restriction matches informal practice. We think of programs and proofs as clearly separated, and, outside of constructive logic, the idea of computing with proofs is ill-formed. The distinction also has practical importance in Coq, where it affects the behavior of extraction.
adamc@229 507
adam@398 508 Recall that %\index{program extraction}%extraction is Coq's facility for translating Coq developments into programs in general-purpose programming languages like OCaml. Extraction _erases_ proofs and leaves programs intact. A simple example with [sig] and [ex] demonstrates the distinction. *)
adamc@229 509
adamc@229 510 Definition sym_sig (x : sig (fun n => n = 0)) : sig (fun n => 0 = n) :=
adamc@229 511 match x with
adamc@229 512 | exist n pf => exist _ n (sym_eq pf)
adamc@229 513 end.
adamc@229 514
adamc@229 515 Extraction sym_sig.
adamc@229 516 (** <<
adamc@229 517 (** val sym_sig : nat -> nat **)
adamc@229 518
adamc@229 519 let sym_sig x = x
adamc@229 520 >>
adamc@229 521
adamc@229 522 Since extraction erases proofs, the second components of [sig] values are elided, making [sig] a simple identity type family. The [sym_sig] operation is thus an identity function. *)
adamc@229 523
adamc@229 524 Definition sym_ex (x : ex (fun n => n = 0)) : ex (fun n => 0 = n) :=
adamc@229 525 match x with
adamc@229 526 | ex_intro n pf => ex_intro _ n (sym_eq pf)
adamc@229 527 end.
adamc@229 528
adamc@229 529 Extraction sym_ex.
adamc@229 530 (** <<
adamc@229 531 (** val sym_ex : __ **)
adamc@229 532
adamc@229 533 let sym_ex = __
adamc@229 534 >>
adamc@229 535
adam@435 536 In this example, the [ex] type itself is in [Prop], so whole [ex] packages are erased. Coq extracts every proposition as the (Coq-specific) type <<__>>, whose single constructor is <<__>>. Not only are proofs replaced by [__], but proof arguments to functions are also removed completely, as we see here.
adamc@229 537
adam@419 538 Extraction is very helpful as an optimization over programs that contain proofs. In languages like Haskell, advanced features make it possible to program with proofs, as a way of convincing the type checker to accept particular definitions. Unfortunately, when proofs are encoded as values in GADTs%~\cite{GADT}%, these proofs exist at runtime and consume resources. In contrast, with Coq, as long as all proofs are kept within [Prop], extraction is guaranteed to erase them.
adamc@229 539
adam@398 540 Many fans of the %\index{Curry-Howard correspondence}%Curry-Howard correspondence support the idea of _extracting programs from proofs_. In reality, few users of Coq and related tools do any such thing. Instead, extraction is better thought of as an optimization that reduces the runtime costs of expressive typing.
adamc@229 541
adamc@229 542 %\medskip%
adamc@229 543
adam@409 544 We have seen two of the differences between proofs and programs: proofs are subject to an elimination restriction and are elided by extraction. The remaining difference is that [Prop] is%\index{impredicativity}% _impredicative_, as this example shows. *)
adamc@229 545
adamc@229 546 Check forall P Q : Prop, P \/ Q -> Q \/ P.
adamc@229 547 (** %\vspace{-.15in}% [[
adamc@229 548 forall P Q : Prop, P \/ Q -> Q \/ P
adamc@229 549 : Prop
adamc@229 550
adamc@229 551 ]]
adamc@229 552
adamc@230 553 We see that it is possible to define a [Prop] that quantifies over other [Prop]s. This is fortunate, as we start wanting that ability even for such basic purposes as stating propositional tautologies. In the next section of this chapter, we will see some reasons why unrestricted impredicativity is undesirable. The impredicativity of [Prop] interacts crucially with the elimination restriction to avoid those pitfalls.
adamc@230 554
adamc@230 555 Impredicativity also allows us to implement a version of our earlier [exp] type that does not suffer from the weakness that we found. *)
adamc@230 556
adamc@230 557 Inductive expP : Type -> Prop :=
adamc@230 558 | ConstP : forall T, T -> expP T
adamc@230 559 | PairP : forall T1 T2, expP T1 -> expP T2 -> expP (T1 * T2)
adamc@230 560 | EqP : forall T, expP T -> expP T -> expP bool.
adamc@230 561
adamc@230 562 Check ConstP 0.
adamc@230 563 (** %\vspace{-.15in}% [[
adamc@230 564 ConstP 0
adamc@230 565 : expP nat
adam@302 566 ]]
adam@302 567 *)
adamc@230 568
adamc@230 569 Check PairP (ConstP 0) (ConstP tt).
adamc@230 570 (** %\vspace{-.15in}% [[
adamc@230 571 PairP (ConstP 0) (ConstP tt)
adamc@230 572 : expP (nat * unit)
adam@302 573 ]]
adam@302 574 *)
adamc@230 575
adamc@230 576 Check EqP (ConstP Set) (ConstP Type).
adamc@230 577 (** %\vspace{-.15in}% [[
adamc@230 578 EqP (ConstP Set) (ConstP Type)
adamc@230 579 : expP bool
adam@302 580 ]]
adam@302 581 *)
adamc@230 582
adamc@230 583 Check ConstP (ConstP O).
adamc@230 584 (** %\vspace{-.15in}% [[
adamc@230 585 ConstP (ConstP 0)
adamc@230 586 : expP (expP nat)
adamc@230 587
adamc@230 588 ]]
adamc@230 589
adam@287 590 In this case, our victory is really a shallow one. As we have marked [expP] as a family of proofs, we cannot deconstruct our expressions in the usual programmatic ways, which makes them almost useless for the usual purposes. Impredicative quantification is much more useful in defining inductive families that we really think of as judgments. For instance, this code defines a notion of equality that is strictly more permissive than the base equality [=]. *)
adamc@230 591
adamc@230 592 Inductive eqPlus : forall T, T -> T -> Prop :=
adamc@230 593 | Base : forall T (x : T), eqPlus x x
adamc@230 594 | Func : forall dom ran (f1 f2 : dom -> ran),
adamc@230 595 (forall x : dom, eqPlus (f1 x) (f2 x))
adamc@230 596 -> eqPlus f1 f2.
adamc@230 597
adamc@230 598 Check (Base 0).
adamc@230 599 (** %\vspace{-.15in}% [[
adamc@230 600 Base 0
adamc@230 601 : eqPlus 0 0
adam@302 602 ]]
adam@302 603 *)
adamc@230 604
adamc@230 605 Check (Func (fun n => n) (fun n => 0 + n) (fun n => Base n)).
adamc@230 606 (** %\vspace{-.15in}% [[
adamc@230 607 Func (fun n : nat => n) (fun n : nat => 0 + n) (fun n : nat => Base n)
adamc@230 608 : eqPlus (fun n : nat => n) (fun n : nat => 0 + n)
adam@302 609 ]]
adam@302 610 *)
adamc@230 611
adamc@230 612 Check (Base (Base 1)).
adamc@230 613 (** %\vspace{-.15in}% [[
adamc@230 614 Base (Base 1)
adamc@230 615 : eqPlus (Base 1) (Base 1)
adam@302 616 ]]
adam@302 617 *)
adamc@230 618
adam@343 619 (** Stating equality facts about proofs may seem baroque, but we have already seen its utility in the chapter on reasoning about equality proofs. *)
adam@343 620
adamc@230 621
adamc@230 622 (** * Axioms *)
adamc@230 623
adam@409 624 (** While the specific logic Gallina is hardcoded into Coq's implementation, it is possible to add certain logical rules in a controlled way. In other words, Coq may be used to reason about many different refinements of Gallina where strictly more theorems are provable. We achieve this by asserting%\index{axioms}% _axioms_ without proof.
adamc@230 625
adamc@230 626 We will motivate the idea by touring through some standard axioms, as enumerated in Coq's online FAQ. I will add additional commentary as appropriate. *)
adamc@230 627
adamc@230 628 (** ** The Basics *)
adamc@230 629
adam@343 630 (** One simple example of a useful axiom is the %\index{law of the excluded middle}%law of the excluded middle. *)
adamc@230 631
adamc@230 632 Require Import Classical_Prop.
adamc@230 633 Print classic.
adamc@230 634 (** %\vspace{-.15in}% [[
adamc@230 635 *** [ classic : forall P : Prop, P \/ ~ P ]
adamc@230 636 ]]
adamc@230 637
adam@343 638 In the implementation of module [Classical_Prop], this axiom was defined with the command%\index{Vernacular commands!Axiom}% *)
adamc@230 639
adamc@230 640 Axiom classic : forall P : Prop, P \/ ~ P.
adamc@230 641
adam@343 642 (** An [Axiom] may be declared with any type, in any of the universes. There is a synonym %\index{Vernacular commands!Parameter}%[Parameter] for [Axiom], and that synonym is often clearer for assertions not of type [Prop]. For instance, we can assert the existence of objects with certain properties. *)
adamc@230 643
adamc@230 644 Parameter n : nat.
adamc@230 645 Axiom positive : n > 0.
adamc@230 646 Reset n.
adamc@230 647
adam@429 648 (** This kind of "axiomatic presentation" of a theory is very common outside of higher-order logic. However, in Coq, it is almost always preferable to stick to defining your objects, functions, and predicates via inductive definitions and functional programming.
adamc@230 649
adam@409 650 In general, there is a significant burden associated with any use of axioms. It is easy to assert a set of axioms that together is%\index{inconsistent axioms}% _inconsistent_. That is, a set of axioms may imply [False], which allows any theorem to be proved, which defeats the purpose of a proof assistant. For example, we could assert the following axiom, which is consistent by itself but inconsistent when combined with [classic]. *)
adamc@230 651
adam@287 652 Axiom not_classic : ~ forall P : Prop, P \/ ~ P.
adamc@230 653
adamc@230 654 Theorem uhoh : False.
adam@287 655 generalize classic not_classic; tauto.
adamc@230 656 Qed.
adamc@230 657
adamc@230 658 Theorem uhoh_again : 1 + 1 = 3.
adamc@230 659 destruct uhoh.
adamc@230 660 Qed.
adamc@230 661
adamc@230 662 Reset not_classic.
adamc@230 663
adam@429 664 (** On the subject of the law of the excluded middle itself, this axiom is usually quite harmless, and many practical Coq developments assume it. It has been proved metatheoretically to be consistent with CIC. Here, "proved metatheoretically" means that someone proved on paper that excluded middle holds in a _model_ of CIC in set theory%~\cite{SetsInTypes}%. All of the other axioms that we will survey in this section hold in the same model, so they are all consistent together.
adamc@230 665
adam@409 666 Recall that Coq implements%\index{constructive logic}% _constructive_ logic by default, where excluded middle is not provable. Proofs in constructive logic can be thought of as programs. A [forall] quantifier denotes a dependent function type, and a disjunction denotes a variant type. In such a setting, excluded middle could be interpreted as a decision procedure for arbitrary propositions, which computability theory tells us cannot exist. Thus, constructive logic with excluded middle can no longer be associated with our usual notion of programming.
adamc@230 667
adam@398 668 Given all this, why is it all right to assert excluded middle as an axiom? The intuitive justification is that the elimination restriction for [Prop] prevents us from treating proofs as programs. An excluded middle axiom that quantified over [Set] instead of [Prop] _would_ be problematic. If a development used that axiom, we would not be able to extract the code to OCaml (soundly) without implementing a genuine universal decision procedure. In contrast, values whose types belong to [Prop] are always erased by extraction, so we sidestep the axiom's algorithmic consequences.
adamc@230 669
adam@343 670 Because the proper use of axioms is so precarious, there are helpful commands for determining which axioms a theorem relies on.%\index{Vernacular commands!Print Assumptions}% *)
adamc@230 671
adamc@230 672 Theorem t1 : forall P : Prop, P -> ~ ~ P.
adamc@230 673 tauto.
adamc@230 674 Qed.
adamc@230 675
adamc@230 676 Print Assumptions t1.
adam@343 677 (** <<
adamc@230 678 Closed under the global context
adam@343 679 >>
adam@302 680 *)
adamc@230 681
adamc@230 682 Theorem t2 : forall P : Prop, ~ ~ P -> P.
adamc@230 683 (** [[
adamc@230 684 tauto.
adam@343 685 ]]
adam@343 686 <<
adamc@230 687 Error: tauto failed.
adam@343 688 >>
adam@302 689 *)
adamc@230 690 intro P; destruct (classic P); tauto.
adamc@230 691 Qed.
adamc@230 692
adamc@230 693 Print Assumptions t2.
adamc@230 694 (** %\vspace{-.15in}% [[
adamc@230 695 Axioms:
adamc@230 696 classic : forall P : Prop, P \/ ~ P
adamc@230 697 ]]
adamc@230 698
adam@398 699 It is possible to avoid this dependence in some specific cases, where excluded middle _is_ provable, for decidable families of propositions. *)
adamc@230 700
adam@287 701 Theorem nat_eq_dec : forall n m : nat, n = m \/ n <> m.
adamc@230 702 induction n; destruct m; intuition; generalize (IHn m); intuition.
adamc@230 703 Qed.
adamc@230 704
adamc@230 705 Theorem t2' : forall n m : nat, ~ ~ (n = m) -> n = m.
adam@287 706 intros n m; destruct (nat_eq_dec n m); tauto.
adamc@230 707 Qed.
adamc@230 708
adamc@230 709 Print Assumptions t2'.
adam@343 710 (** <<
adamc@230 711 Closed under the global context
adam@343 712 >>
adamc@230 713
adamc@230 714 %\bigskip%
adamc@230 715
adam@409 716 Mainstream mathematical practice assumes excluded middle, so it can be useful to have it available in Coq developments, though it is also nice to know that a theorem is proved in a simpler formal system than classical logic. There is a similar story for%\index{proof irrelevance}% _proof irrelevance_, which simplifies proof issues that would not even arise in mainstream math. *)
adamc@230 717
adamc@230 718 Require Import ProofIrrelevance.
adamc@230 719 Print proof_irrelevance.
adamc@230 720 (** %\vspace{-.15in}% [[
adamc@230 721 *** [ proof_irrelevance : forall (P : Prop) (p1 p2 : P), p1 = p2 ]
adamc@230 722 ]]
adamc@230 723
adam@353 724 This axiom asserts that any two proofs of the same proposition are equal. If we replaced [p1 = p2] by [p1 <-> p2], then the statement would be provable. However, equality is a stronger notion than logical equivalence. Recall this example function from Chapter 6. *)
adamc@230 725
adamc@230 726 (* begin hide *)
adamc@230 727 Lemma zgtz : 0 > 0 -> False.
adamc@230 728 crush.
adamc@230 729 Qed.
adamc@230 730 (* end hide *)
adamc@230 731
adamc@230 732 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@230 733 match n with
adamc@230 734 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@230 735 | S n' => fun _ => n'
adamc@230 736 end.
adamc@230 737
adam@343 738 (** We might want to prove that different proofs of [n > 0] do not lead to different results from our richly typed predecessor function. *)
adamc@230 739
adamc@230 740 Theorem pred_strong1_irrel : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 741 destruct n; crush.
adamc@230 742 Qed.
adamc@230 743
adamc@230 744 (** The proof script is simple, but it involved peeking into the definition of [pred_strong1]. For more complicated function definitions, it can be considerably more work to prove that they do not discriminate on details of proof arguments. This can seem like a shame, since the [Prop] elimination restriction makes it impossible to write any function that does otherwise. Unfortunately, this fact is only true metatheoretically, unless we assert an axiom like [proof_irrelevance]. With that axiom, we can prove our theorem without consulting the definition of [pred_strong1]. *)
adamc@230 745
adamc@230 746 Theorem pred_strong1_irrel' : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 747 intros; f_equal; apply proof_irrelevance.
adamc@230 748 Qed.
adamc@230 749
adamc@230 750
adamc@230 751 (** %\bigskip%
adamc@230 752
adamc@230 753 In the chapter on equality, we already discussed some axioms that are related to proof irrelevance. In particular, Coq's standard library includes this axiom: *)
adamc@230 754
adamc@230 755 Require Import Eqdep.
adamc@230 756 Import Eq_rect_eq.
adamc@230 757 Print eq_rect_eq.
adamc@230 758 (** %\vspace{-.15in}% [[
adamc@230 759 *** [ eq_rect_eq :
adamc@230 760 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adamc@230 761 x = eq_rect p Q x p h ]
adamc@230 762 ]]
adamc@230 763
adam@429 764 This axiom says that it is permissible to simplify pattern matches over proofs of equalities like [e = e]. The axiom is logically equivalent to some simpler corollaries. In the theorem names, "UIP" stands for %\index{unicity of identity proofs}%"unicity of identity proofs", where "identity" is a synonym for "equality." *)
adamc@230 765
adam@426 766 Corollary UIP_refl : forall A (x : A) (pf : x = x), pf = eq_refl x.
adam@426 767 intros; replace pf with (eq_rect x (eq x) (eq_refl x) x pf); [
adamc@230 768 symmetry; apply eq_rect_eq
adamc@230 769 | exact (match pf as pf' return match pf' in _ = y return x = y with
adam@426 770 | eq_refl => eq_refl x
adamc@230 771 end = pf' with
adam@426 772 | eq_refl => eq_refl _
adamc@230 773 end) ].
adamc@230 774 Qed.
adamc@230 775
adamc@230 776 Corollary UIP : forall A (x y : A) (pf1 pf2 : x = y), pf1 = pf2.
adamc@230 777 intros; generalize pf1 pf2; subst; intros;
adamc@230 778 match goal with
adamc@230 779 | [ |- ?pf1 = ?pf2 ] => rewrite (UIP_refl pf1); rewrite (UIP_refl pf2); reflexivity
adamc@230 780 end.
adamc@230 781 Qed.
adamc@230 782
adam@436 783 (* begin hide *)
adam@437 784 (* begin thide *)
adam@436 785 Require Eqdep_dec.
adam@437 786 (* end thide *)
adam@436 787 (* end hide *)
adam@436 788
adamc@231 789 (** These corollaries are special cases of proof irrelevance. In developments that only need proof irrelevance for equality, there is no need to assert full irrelevance.
adamc@230 790
adamc@230 791 Another facet of proof irrelevance is that, like excluded middle, it is often provable for specific propositions. For instance, [UIP] is provable whenever the type [A] has a decidable equality operation. The module [Eqdep_dec] of the standard library contains a proof. A similar phenomenon applies to other notable cases, including less-than proofs. Thus, it is often possible to use proof irrelevance without asserting axioms.
adamc@230 792
adamc@230 793 %\bigskip%
adamc@230 794
adamc@230 795 There are two more basic axioms that are often assumed, to avoid complications that do not arise in set theory. *)
adamc@230 796
adamc@230 797 Require Import FunctionalExtensionality.
adamc@230 798 Print functional_extensionality_dep.
adamc@230 799 (** %\vspace{-.15in}% [[
adamc@230 800 *** [ functional_extensionality_dep :
adamc@230 801 forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
adamc@230 802 (forall x : A, f x = g x) -> f = g ]
adamc@230 803
adamc@230 804 ]]
adamc@230 805
adamc@230 806 This axiom says that two functions are equal if they map equal inputs to equal outputs. Such facts are not provable in general in CIC, but it is consistent to assume that they are.
adamc@230 807
adam@343 808 A simple corollary shows that the same property applies to predicates. *)
adamc@230 809
adamc@230 810 Corollary predicate_extensionality : forall (A : Type) (B : A -> Prop) (f g : forall x : A, B x),
adamc@230 811 (forall x : A, f x = g x) -> f = g.
adamc@230 812 intros; apply functional_extensionality_dep; assumption.
adamc@230 813 Qed.
adamc@230 814
adam@343 815 (** In some cases, one might prefer to assert this corollary as the axiom, to restrict the consequences to proofs and not programs. *)
adam@343 816
adamc@230 817
adamc@230 818 (** ** Axioms of Choice *)
adamc@230 819
adam@343 820 (** Some Coq axioms are also points of contention in mainstream math. The most prominent example is the %\index{axiom of choice}%axiom of choice. In fact, there are multiple versions that we might consider, and, considered in isolation, none of these versions means quite what it means in classical set theory.
adamc@230 821
adam@398 822 First, it is possible to implement a choice operator _without_ axioms in some potentially surprising cases. *)
adamc@230 823
adamc@230 824 Require Import ConstructiveEpsilon.
adamc@230 825 Check constructive_definite_description.
adamc@230 826 (** %\vspace{-.15in}% [[
adamc@230 827 constructive_definite_description
adamc@230 828 : forall (A : Set) (f : A -> nat) (g : nat -> A),
adamc@230 829 (forall x : A, g (f x) = x) ->
adamc@230 830 forall P : A -> Prop,
adamc@230 831 (forall x : A, {P x} + {~ P x}) ->
adamc@230 832 (exists! x : A, P x) -> {x : A | P x}
adam@302 833 ]]
adam@302 834 *)
adamc@230 835
adamc@230 836 Print Assumptions constructive_definite_description.
adam@343 837 (** <<
adamc@230 838 Closed under the global context
adam@343 839 >>
adamc@230 840
adam@398 841 This function transforms a decidable predicate [P] into a function that produces an element satisfying [P] from a proof that such an element exists. The functions [f] and [g], in conjunction with an associated injectivity property, are used to express the idea that the set [A] is countable. Under these conditions, a simple brute force algorithm gets the job done: we just enumerate all elements of [A], stopping when we find one satisfying [P]. The existence proof, specified in terms of _unique_ existence [exists!], guarantees termination. The definition of this operator in Coq uses some interesting techniques, as seen in the implementation of the [ConstructiveEpsilon] module.
adamc@230 842
adamc@230 843 Countable choice is provable in set theory without appealing to the general axiom of choice. To support the more general principle in Coq, we must also add an axiom. Here is a functional version of the axiom of unique choice. *)
adamc@230 844
adamc@230 845 Require Import ClassicalUniqueChoice.
adamc@230 846 Check dependent_unique_choice.
adamc@230 847 (** %\vspace{-.15in}% [[
adamc@230 848 dependent_unique_choice
adamc@230 849 : forall (A : Type) (B : A -> Type) (R : forall x : A, B x -> Prop),
adamc@230 850 (forall x : A, exists! y : B x, R x y) ->
adam@343 851 exists f : forall x : A, B x,
adam@343 852 forall x : A, R x (f x)
adamc@230 853 ]]
adamc@230 854
adamc@230 855 This axiom lets us convert a relational specification [R] into a function implementing that specification. We need only prove that [R] is truly a function. An alternate, stronger formulation applies to cases where [R] maps each input to one or more outputs. We also simplify the statement of the theorem by considering only non-dependent function types. *)
adamc@230 856
adam@436 857 (* begin hide *)
adam@437 858 (* begin thide *)
adam@436 859 Require RelationalChoice.
adam@437 860 (* end thide *)
adam@436 861 (* end hide *)
adam@436 862
adamc@230 863 Require Import ClassicalChoice.
adamc@230 864 Check choice.
adamc@230 865 (** %\vspace{-.15in}% [[
adamc@230 866 choice
adamc@230 867 : forall (A B : Type) (R : A -> B -> Prop),
adamc@230 868 (forall x : A, exists y : B, R x y) ->
adamc@230 869 exists f : A -> B, forall x : A, R x (f x)
adamc@230 870
adamc@230 871 ]]
adamc@230 872
adamc@230 873 This principle is proved as a theorem, based on the unique choice axiom and an additional axiom of relational choice from the [RelationalChoice] module.
adamc@230 874
adamc@230 875 In set theory, the axiom of choice is a fundamental philosophical commitment one makes about the universe of sets. In Coq, the choice axioms say something weaker. For instance, consider the simple restatement of the [choice] axiom where we replace existential quantification by its Curry-Howard analogue, subset types. *)
adamc@230 876
adamc@230 877 Definition choice_Set (A B : Type) (R : A -> B -> Prop) (H : forall x : A, {y : B | R x y})
adamc@230 878 : {f : A -> B | forall x : A, R x (f x)} :=
adamc@230 879 exist (fun f => forall x : A, R x (f x))
adamc@230 880 (fun x => proj1_sig (H x)) (fun x => proj2_sig (H x)).
adamc@230 881
adam@429 882 (** Via the Curry-Howard correspondence, this "axiom" can be taken to have the same meaning as the original. It is implemented trivially as a transformation not much deeper than uncurrying. Thus, we see that the utility of the axioms that we mentioned earlier comes in their usage to build programs from proofs. Normal set theory has no explicit proofs, so the meaning of the usual axiom of choice is subtlely different. In Gallina, the axioms implement a controlled relaxation of the restrictions on information flow from proofs to programs.
adamc@230 883
adam@429 884 However, when we combine an axiom of choice with the law of the excluded middle, the idea of "choice" becomes more interesting. Excluded middle gives us a highly non-computational way of constructing proofs, but it does not change the computational nature of programs. Thus, the axiom of choice is still giving us a way of translating between two different sorts of "programs," but the input programs (which are proofs) may be written in a rich language that goes beyond normal computability. This truly is more than repackaging a function with a different type.
adamc@230 885
adamc@230 886 %\bigskip%
adamc@230 887
adam@429 888 The Coq tools support a command-line flag %\index{impredicative Set}%<<-impredicative-set>>, which modifies Gallina in a more fundamental way by making [Set] impredicative. A term like [forall T : Set, T] has type [Set], and inductive definitions in [Set] may have constructors that quantify over arguments of any types. To maintain consistency, an elimination restriction must be imposed, similarly to the restriction for [Prop]. The restriction only applies to large inductive types, where some constructor quantifies over a type of type [Type]. In such cases, a value in this inductive type may only be pattern-matched over to yield a result type whose type is [Set] or [Prop]. This contrasts with [Prop], where the restriction applies even to non-large inductive types, and where the result type may only have type [Prop].
adamc@230 889
adamc@230 890 In old versions of Coq, [Set] was impredicative by default. Later versions make [Set] predicative to avoid inconsistency with some classical axioms. In particular, one should watch out when using impredicative [Set] with axioms of choice. In combination with excluded middle or predicate extensionality, this can lead to inconsistency. Impredicative [Set] can be useful for modeling inherently impredicative mathematical concepts, but almost all Coq developments get by fine without it. *)
adamc@230 891
adamc@230 892 (** ** Axioms and Computation *)
adamc@230 893
adam@398 894 (** One additional axiom-related wrinkle arises from an aspect of Gallina that is very different from set theory: a notion of _computational equivalence_ is central to the definition of the formal system. Axioms tend not to play well with computation. Consider this example. We start by implementing a function that uses a type equality proof to perform a safe type-cast. *)
adamc@230 895
adamc@230 896 Definition cast (x y : Set) (pf : x = y) (v : x) : y :=
adamc@230 897 match pf with
adam@426 898 | eq_refl => v
adamc@230 899 end.
adamc@230 900
adamc@230 901 (** Computation over programs that use [cast] can proceed smoothly. *)
adamc@230 902
adam@426 903 Eval compute in (cast (eq_refl (nat -> nat)) (fun n => S n)) 12.
adam@343 904 (** %\vspace{-.15in}%[[
adamc@230 905 = 13
adamc@230 906 : nat
adam@302 907 ]]
adam@302 908 *)
adamc@230 909
adamc@230 910 (** Things do not go as smoothly when we use [cast] with proofs that rely on axioms. *)
adamc@230 911
adamc@230 912 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 913 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 914 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 915 Qed.
adamc@230 916
adamc@230 917 Eval compute in (cast t3 (fun _ => First)) 12.
adamc@230 918 (** [[
adamc@230 919 = match t3 in (_ = P) return P with
adam@426 920 | eq_refl => fun n : nat => First
adamc@230 921 end 12
adamc@230 922 : fin (12 + 1)
adamc@230 923 ]]
adamc@230 924
adamc@230 925 Computation gets stuck in a pattern-match on the proof [t3]. The structure of [t3] is not known, so the match cannot proceed. It turns out a more basic problem leads to this particular situation. We ended the proof of [t3] with [Qed], so the definition of [t3] is not available to computation. That is easily fixed. *)
adamc@230 926
adamc@230 927 Reset t3.
adamc@230 928
adamc@230 929 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 930 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 931 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 932 Defined.
adamc@230 933
adamc@230 934 Eval compute in (cast t3 (fun _ => First)) 12.
adamc@230 935 (** [[
adamc@230 936 = match
adamc@230 937 match
adamc@230 938 match
adamc@230 939 functional_extensionality
adamc@230 940 ....
adamc@230 941 ]]
adamc@230 942
adam@398 943 We elide most of the details. A very unwieldy tree of nested matches on equality proofs appears. This time evaluation really _is_ stuck on a use of an axiom.
adamc@230 944
adamc@230 945 If we are careful in using tactics to prove an equality, we can still compute with casts over the proof. *)
adamc@230 946
adamc@230 947 Lemma plus1 : forall n, S n = n + 1.
adamc@230 948 induction n; simpl; intuition.
adamc@230 949 Defined.
adamc@230 950
adamc@230 951 Theorem t4 : forall n, fin (S n) = fin (n + 1).
adamc@230 952 intro; f_equal; apply plus1.
adamc@230 953 Defined.
adamc@230 954
adamc@230 955 Eval compute in cast (t4 13) First.
adamc@230 956 (** %\vspace{-.15in}% [[
adamc@230 957 = First
adamc@230 958 : fin (13 + 1)
adam@302 959 ]]
adam@343 960
adam@426 961 This simple computational reduction hides the use of a recursive function to produce a suitable [eq_refl] proof term. The recursion originates in our use of [induction] in [t4]'s proof. *)
adam@343 962
adam@344 963
adam@344 964 (** ** Methods for Avoiding Axioms *)
adam@344 965
adam@409 966 (** The last section demonstrated one reason to avoid axioms: they interfere with computational behavior of terms. A further reason is to reduce the philosophical commitment of a theorem. The more axioms one assumes, the harder it becomes to convince oneself that the formal system corresponds appropriately to one's intuitions. A refinement of this last point, in applications like %\index{proof-carrying code}%proof-carrying code%~\cite{PCC}% in computer security, has to do with minimizing the size of a%\index{trusted code base}% _trusted code base_. To convince ourselves that a theorem is true, we must convince ourselves of the correctness of the program that checks the theorem. Axioms effectively become new source code for the checking program, increasing the effort required to perform a correctness audit.
adam@344 967
adam@429 968 An earlier section gave one example of avoiding an axiom. We proved that [pred_strong1] is agnostic to details of the proofs passed to it as arguments, by unfolding the definition of the function. A "simpler" proof keeps the function definition opaque and instead applies a proof irrelevance axiom. By accepting a more complex proof, we reduce our philosophical commitment and trusted base. (By the way, the less-than relation that the proofs in question here prove turns out to admit proof irrelevance as a theorem provable within normal Gallina!)
adam@344 969
adam@344 970 One dark secret of the [dep_destruct] tactic that we have used several times is reliance on an axiom. Consider this simple case analysis principle for [fin] values: *)
adam@344 971
adam@344 972 Theorem fin_cases : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 973 intros; dep_destruct f; eauto.
adam@344 974 Qed.
adam@344 975
adam@429 976 (* begin hide *)
adam@429 977 Require Import JMeq.
adam@437 978 (* begin thide *)
adam@429 979 Definition jme := (JMeq, JMeq_eq).
adam@437 980 (* end thide *)
adam@429 981 (* end hide *)
adam@429 982
adam@344 983 Print Assumptions fin_cases.
adam@344 984 (** %\vspace{-.15in}%[[
adam@344 985 Axioms:
adam@429 986 JMeq_eq : forall (A : Type) (x y : A), JMeq x y -> x = y
adam@344 987 ]]
adam@344 988
adam@344 989 The proof depends on the [JMeq_eq] axiom that we met in the chapter on equality proofs. However, a smarter tactic could have avoided an axiom dependence. Here is an alternate proof via a slightly strange looking lemma. *)
adam@344 990
adam@344 991 (* begin thide *)
adam@344 992 Lemma fin_cases_again' : forall n (f : fin n),
adam@344 993 match n return fin n -> Prop with
adam@344 994 | O => fun _ => False
adam@344 995 | S n' => fun f => f = First \/ exists f', f = Next f'
adam@344 996 end f.
adam@344 997 destruct f; eauto.
adam@344 998 Qed.
adam@344 999
adam@344 1000 (** We apply a variant of the %\index{convoy pattern}%convoy pattern, which we are used to seeing in function implementations. Here, the pattern helps us state a lemma in a form where the argument to [fin] is a variable. Recall that, thanks to basic typing rules for pattern-matching, [destruct] will only work effectively on types whose non-parameter arguments are variables. The %\index{tactics!exact}%[exact] tactic, which takes as argument a literal proof term, now gives us an easy way of proving the original theorem. *)
adam@344 1001
adam@344 1002 Theorem fin_cases_again : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 1003 intros; exact (fin_cases_again' f).
adam@344 1004 Qed.
adam@344 1005 (* end thide *)
adam@344 1006
adam@344 1007 Print Assumptions fin_cases_again.
adam@344 1008 (** %\vspace{-.15in}%
adam@344 1009 <<
adam@344 1010 Closed under the global context
adam@344 1011 >>
adam@344 1012
adam@345 1013 *)
adam@345 1014
adam@345 1015 (* begin thide *)
adam@345 1016 (** As the Curry-Howard correspondence might lead us to expect, the same pattern may be applied in programming as in proving. Axioms are relevant in programming, too, because, while Coq includes useful extensions like [Program] that make dependently typed programming more straightforward, in general these extensions generate code that relies on axioms about equality. We can use clever pattern matching to write our code axiom-free.
adam@345 1017
adam@429 1018 As an example, consider a [Set] version of [fin_cases]. We use [Set] types instead of [Prop] types, so that return values have computational content and may be used to guide the behavior of algorithms. Beside that, we are essentially writing the same "proof" in a more explicit way. *)
adam@345 1019
adam@345 1020 Definition finOut n (f : fin n) : match n return fin n -> Type with
adam@345 1021 | O => fun _ => Empty_set
adam@345 1022 | _ => fun f => {f' : _ | f = Next f'} + {f = First}
adam@345 1023 end f :=
adam@345 1024 match f with
adam@426 1025 | First _ => inright _ (eq_refl _)
adam@426 1026 | Next _ f' => inleft _ (exist _ f' (eq_refl _))
adam@345 1027 end.
adam@345 1028 (* end thide *)
adam@345 1029
adam@345 1030 (** As another example, consider the following type of formulas in first-order logic. The intent of the type definition will not be important in what follows, but we give a quick intuition for the curious reader. Our formulas may include [forall] quantification over arbitrary [Type]s, and we index formulas by environments telling which variables are in scope and what their types are; such an environment is a [list Type]. A constructor [Inject] lets us include any Coq [Prop] as a formula, and [VarEq] and [Lift] can be used for variable references, in what is essentially the de Bruijn index convention. (Again, the detail in this paragraph is not important to understand the discussion that follows!) *)
adam@344 1031
adam@344 1032 Inductive formula : list Type -> Type :=
adam@344 1033 | Inject : forall Ts, Prop -> formula Ts
adam@344 1034 | VarEq : forall T Ts, T -> formula (T :: Ts)
adam@344 1035 | Lift : forall T Ts, formula Ts -> formula (T :: Ts)
adam@344 1036 | Forall : forall T Ts, formula (T :: Ts) -> formula Ts
adam@344 1037 | And : forall Ts, formula Ts -> formula Ts -> formula Ts.
adam@344 1038
adam@344 1039 (** This example is based on my own experiences implementing variants of a program logic called XCAP%~\cite{XCAP}%, which also includes an inductive predicate for characterizing which formulas are provable. Here I include a pared-down version of such a predicate, with only two constructors, which is sufficient to illustrate certain tricky issues. *)
adam@344 1040
adam@344 1041 Inductive proof : formula nil -> Prop :=
adam@344 1042 | PInject : forall (P : Prop), P -> proof (Inject nil P)
adam@344 1043 | PAnd : forall p q, proof p -> proof q -> proof (And p q).
adam@344 1044
adam@429 1045 (** Let us prove a lemma showing that a "[P /\ Q -> P]" rule is derivable within the rules of [proof]. *)
adam@344 1046
adam@344 1047 Theorem proj1 : forall p q, proof (And p q) -> proof p.
adam@344 1048 destruct 1.
adam@344 1049 (** %\vspace{-.15in}%[[
adam@344 1050 p : formula nil
adam@344 1051 q : formula nil
adam@344 1052 P : Prop
adam@344 1053 H : P
adam@344 1054 ============================
adam@344 1055 proof p
adam@344 1056 ]]
adam@344 1057 *)
adam@344 1058
adam@344 1059 (** We are reminded that [induction] and [destruct] do not work effectively on types with non-variable arguments. The first subgoal, shown above, is clearly unprovable. (Consider the case where [p = Inject nil False].)
adam@344 1060
adam@344 1061 An application of the %\index{tactics!dependent destruction}%[dependent destruction] tactic (the basis for [dep_destruct]) solves the problem handily. We use a shorthand with the %\index{tactics!intros}%[intros] tactic that lets us use question marks for variable names that do not matter. *)
adam@344 1062
adam@344 1063 Restart.
adam@344 1064 Require Import Program.
adam@344 1065 intros ? ? H; dependent destruction H; auto.
adam@344 1066 Qed.
adam@344 1067
adam@344 1068 Print Assumptions proj1.
adam@344 1069 (** %\vspace{-.15in}%[[
adam@344 1070 Axioms:
adam@344 1071 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1072 x = eq_rect p Q x p h
adam@344 1073 ]]
adam@344 1074
adam@344 1075 Unfortunately, that built-in tactic appeals to an axiom. It is still possible to avoid axioms by giving the proof via another odd-looking lemma. Here is a first attempt that fails at remaining axiom-free, using a common equality-based trick for supporting induction on non-variable arguments to type families. The trick works fine without axioms for datatypes more traditional than [formula], but we run into trouble with our current type. *)
adam@344 1076
adam@344 1077 Lemma proj1_again' : forall r, proof r
adam@344 1078 -> forall p q, r = And p q -> proof p.
adam@344 1079 destruct 1; crush.
adam@344 1080 (** %\vspace{-.15in}%[[
adam@344 1081 H0 : Inject [] P = And p q
adam@344 1082 ============================
adam@344 1083 proof p
adam@344 1084 ]]
adam@344 1085
adam@344 1086 The first goal looks reasonable. Hypothesis [H0] is clearly contradictory, as [discriminate] can show. *)
adam@344 1087
adam@344 1088 discriminate.
adam@344 1089 (** %\vspace{-.15in}%[[
adam@344 1090 H : proof p
adam@344 1091 H1 : And p q = And p0 q0
adam@344 1092 ============================
adam@344 1093 proof p0
adam@344 1094 ]]
adam@344 1095
adam@344 1096 It looks like we are almost done. Hypothesis [H1] gives [p = p0] by injectivity of constructors, and then [H] finishes the case. *)
adam@344 1097
adam@344 1098 injection H1; intros.
adam@344 1099
adam@429 1100 (* begin hide *)
adam@437 1101 (* begin thide *)
adam@429 1102 Definition existT' := existT.
adam@437 1103 (* end thide *)
adam@429 1104 (* end hide *)
adam@429 1105
adam@429 1106 (** Unfortunately, the "equality" that we expected between [p] and [p0] comes in a strange form:
adam@344 1107
adam@344 1108 [[
adam@344 1109 H3 : existT (fun Ts : list Type => formula Ts) []%list p =
adam@344 1110 existT (fun Ts : list Type => formula Ts) []%list p0
adam@344 1111 ============================
adam@344 1112 proof p0
adam@344 1113 ]]
adam@344 1114
adam@345 1115 It may take a bit of tinkering, but, reviewing Chapter 3's discussion of writing injection principles manually, it makes sense that an [existT] type is the most direct way to express the output of [injection] on a dependently typed constructor. The constructor [And] is dependently typed, since it takes a parameter [Ts] upon which the types of [p] and [q] depend. Let us not dwell further here on why this goal appears; the reader may like to attempt the (impossible) exercise of building a better injection lemma for [And], without using axioms.
adam@344 1116
adam@344 1117 How exactly does an axiom come into the picture here? Let us ask [crush] to finish the proof. *)
adam@344 1118
adam@344 1119 crush.
adam@344 1120 Qed.
adam@344 1121
adam@344 1122 Print Assumptions proj1_again'.
adam@344 1123 (** %\vspace{-.15in}%[[
adam@344 1124 Axioms:
adam@344 1125 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1126 x = eq_rect p Q x p h
adam@344 1127 ]]
adam@344 1128
adam@344 1129 It turns out that this familiar axiom about equality (or some other axiom) is required to deduce [p = p0] from the hypothesis [H3] above. The soundness of that proof step is neither provable nor disprovable in Gallina.
adam@344 1130
adam@344 1131 Hope is not lost, however. We can produce an even stranger looking lemma, which gives us the theorem without axioms. *)
adam@344 1132
adam@344 1133 Lemma proj1_again'' : forall r, proof r
adam@344 1134 -> match r with
adam@344 1135 | And Ps p _ => match Ps return formula Ps -> Prop with
adam@344 1136 | nil => fun p => proof p
adam@344 1137 | _ => fun _ => True
adam@344 1138 end p
adam@344 1139 | _ => True
adam@344 1140 end.
adam@344 1141 destruct 1; auto.
adam@344 1142 Qed.
adam@344 1143
adam@344 1144 Theorem proj1_again : forall p q, proof (And p q) -> proof p.
adam@344 1145 intros ? ? H; exact (proj1_again'' H).
adam@344 1146 Qed.
adam@344 1147
adam@344 1148 Print Assumptions proj1_again.
adam@344 1149 (** <<
adam@344 1150 Closed under the global context
adam@344 1151 >>
adam@344 1152
adam@377 1153 This example illustrates again how some of the same design patterns we learned for dependently typed programming can be used fruitfully in theorem statements.
adam@377 1154
adam@377 1155 %\medskip%
adam@377 1156
adam@398 1157 To close the chapter, we consider one final way to avoid dependence on axioms. Often this task is equivalent to writing definitions such that they _compute_. That is, we want Coq's normal reduction to be able to run certain programs to completion. Here is a simple example where such computation can get stuck. In proving properties of such functions, we would need to apply axioms like %\index{axiom K}%K manually to make progress.
adam@377 1158
adam@377 1159 Imagine we are working with %\index{deep embedding}%deeply embedded syntax of some programming language, where each term is considered to be in the scope of a number of free variables that hold normal Coq values. To enforce proper typing, we will need to model a Coq typing environment somehow. One natural choice is as a list of types, where variable number [i] will be treated as a reference to the [i]th element of the list. *)
adam@377 1160
adam@377 1161 Section withTypes.
adam@377 1162 Variable types : list Set.
adam@377 1163
adam@377 1164 (** To give the semantics of terms, we will need to represent value environments, which assign each variable a term of the proper type. *)
adam@377 1165
adam@377 1166 Variable values : hlist (fun x : Set => x) types.
adam@377 1167
adam@377 1168 (** Now imagine that we are writing some procedure that operates on a distinguished variable of type [nat]. A hypothesis formalizes this assumption, using the standard library function [nth_error] for looking up list elements by position. *)
adam@377 1169
adam@377 1170 Variable natIndex : nat.
adam@377 1171 Variable natIndex_ok : nth_error types natIndex = Some nat.
adam@377 1172
adam@377 1173 (** It is not hard to use this hypothesis to write a function for extracting the [nat] value in position [natIndex] of [values], starting with two helpful lemmas, each of which we finish with [Defined] to mark the lemma as transparent, so that its definition may be expanded during evaluation. *)
adam@377 1174
adam@377 1175 Lemma nth_error_nil : forall A n x,
adam@377 1176 nth_error (@nil A) n = Some x
adam@377 1177 -> False.
adam@377 1178 destruct n; simpl; unfold error; congruence.
adam@377 1179 Defined.
adam@377 1180
adam@377 1181 Implicit Arguments nth_error_nil [A n x].
adam@377 1182
adam@377 1183 Lemma Some_inj : forall A (x y : A),
adam@377 1184 Some x = Some y
adam@377 1185 -> x = y.
adam@377 1186 congruence.
adam@377 1187 Defined.
adam@377 1188
adam@377 1189 Fixpoint getNat (types' : list Set) (values' : hlist (fun x : Set => x) types')
adam@377 1190 (natIndex : nat) : (nth_error types' natIndex = Some nat) -> nat :=
adam@377 1191 match values' with
adam@377 1192 | HNil => fun pf => match nth_error_nil pf with end
adam@377 1193 | HCons t ts x values'' =>
adam@377 1194 match natIndex return nth_error (t :: ts) natIndex = Some nat -> nat with
adam@377 1195 | O => fun pf =>
adam@377 1196 match Some_inj pf in _ = T return T with
adam@426 1197 | eq_refl => x
adam@377 1198 end
adam@377 1199 | S natIndex' => getNat values'' natIndex'
adam@377 1200 end
adam@377 1201 end.
adam@377 1202 End withTypes.
adam@377 1203
adam@377 1204 (** The problem becomes apparent when we experiment with running [getNat] on a concrete [types] list. *)
adam@377 1205
adam@377 1206 Definition myTypes := unit :: nat :: bool :: nil.
adam@377 1207 Definition myValues : hlist (fun x : Set => x) myTypes :=
adam@377 1208 tt ::: 3 ::: false ::: HNil.
adam@377 1209
adam@377 1210 Definition myNatIndex := 1.
adam@377 1211
adam@377 1212 Theorem myNatIndex_ok : nth_error myTypes myNatIndex = Some nat.
adam@377 1213 reflexivity.
adam@377 1214 Defined.
adam@377 1215
adam@377 1216 Eval compute in getNat myValues myNatIndex myNatIndex_ok.
adam@377 1217 (** %\vspace{-.15in}%[[
adam@377 1218 = 3
adam@377 1219 ]]
adam@377 1220
adam@398 1221 We have not hit the problem yet, since we proceeded with a concrete equality proof for [myNatIndex_ok]. However, consider a case where we want to reason about the behavior of [getNat] _independently_ of a specific proof. *)
adam@377 1222
adam@377 1223 Theorem getNat_is_reasonable : forall pf, getNat myValues myNatIndex pf = 3.
adam@377 1224 intro; compute.
adam@377 1225 (**
adam@377 1226 <<
adam@377 1227 1 subgoal
adam@377 1228 >>
adam@377 1229 %\vspace{-.3in}%[[
adam@377 1230 pf : nth_error myTypes myNatIndex = Some nat
adam@377 1231 ============================
adam@377 1232 match
adam@377 1233 match
adam@377 1234 pf in (_ = y)
adam@377 1235 return (nat = match y with
adam@377 1236 | Some H => H
adam@377 1237 | None => nat
adam@377 1238 end)
adam@377 1239 with
adam@377 1240 | eq_refl => eq_refl
adam@377 1241 end in (_ = T) return T
adam@377 1242 with
adam@377 1243 | eq_refl => 3
adam@377 1244 end = 3
adam@377 1245 ]]
adam@377 1246
adam@377 1247 Since the details of the equality proof [pf] are not known, computation can proceed no further. A rewrite with axiom K would allow us to make progress, but we can rethink the definitions a bit to avoid depending on axioms. *)
adam@377 1248
adam@377 1249 Abort.
adam@377 1250
adam@377 1251 (** Here is a definition of a function that turns out to be useful, though no doubt its purpose will be mysterious for now. A call [update ls n x] overwrites the [n]th position of the list [ls] with the value [x], padding the end of the list with extra [x] values as needed to ensure sufficient length. *)
adam@377 1252
adam@377 1253 Fixpoint copies A (x : A) (n : nat) : list A :=
adam@377 1254 match n with
adam@377 1255 | O => nil
adam@377 1256 | S n' => x :: copies x n'
adam@377 1257 end.
adam@377 1258
adam@377 1259 Fixpoint update A (ls : list A) (n : nat) (x : A) : list A :=
adam@377 1260 match ls with
adam@377 1261 | nil => copies x n ++ x :: nil
adam@377 1262 | y :: ls' => match n with
adam@377 1263 | O => x :: ls'
adam@377 1264 | S n' => y :: update ls' n' x
adam@377 1265 end
adam@377 1266 end.
adam@377 1267
adam@377 1268 (** Now let us revisit the definition of [getNat]. *)
adam@377 1269
adam@377 1270 Section withTypes'.
adam@377 1271 Variable types : list Set.
adam@377 1272 Variable natIndex : nat.
adam@377 1273
adam@429 1274 (** Here is the trick: instead of asserting properties about the list [types], we build a "new" list that is _guaranteed by construction_ to have those properties. *)
adam@377 1275
adam@377 1276 Definition types' := update types natIndex nat.
adam@377 1277
adam@377 1278 Variable values : hlist (fun x : Set => x) types'.
adam@377 1279
adam@377 1280 (** Now a bit of dependent pattern matching helps us rewrite [getNat] in a way that avoids any use of equality proofs. *)
adam@377 1281
adam@378 1282 Fixpoint skipCopies (n : nat)
adam@378 1283 : hlist (fun x : Set => x) (copies nat n ++ nat :: nil) -> nat :=
adam@378 1284 match n with
adam@378 1285 | O => fun vs => hhd vs
adam@378 1286 | S n' => fun vs => skipCopies n' (htl vs)
adam@378 1287 end.
adam@378 1288
adam@377 1289 Fixpoint getNat' (types'' : list Set) (natIndex : nat)
adam@377 1290 : hlist (fun x : Set => x) (update types'' natIndex nat) -> nat :=
adam@377 1291 match types'' with
adam@378 1292 | nil => skipCopies natIndex
adam@377 1293 | t :: types0 =>
adam@377 1294 match natIndex return hlist (fun x : Set => x)
adam@377 1295 (update (t :: types0) natIndex nat) -> nat with
adam@377 1296 | O => fun vs => hhd vs
adam@377 1297 | S natIndex' => fun vs => getNat' types0 natIndex' (htl vs)
adam@377 1298 end
adam@377 1299 end.
adam@377 1300 End withTypes'.
adam@377 1301
adam@398 1302 (** Now the surprise comes in how easy it is to _use_ [getNat']. While typing works by modification of a types list, we can choose parameters so that the modification has no effect. *)
adam@377 1303
adam@377 1304 Theorem getNat_is_reasonable : getNat' myTypes myNatIndex myValues = 3.
adam@377 1305 reflexivity.
adam@377 1306 Qed.
adam@377 1307
adam@377 1308 (** The same parameters as before work without alteration, and we avoid use of axioms. *)