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MIT 6.822
author Adam Chlipala <adam@chlipala.net>
date Mon, 18 Dec 2017 17:05:53 -0500
parents 2c8c693ddaba
children af97676583f3
rev   line source
adam@534 1 (* Copyright (c) 2009-2012, 2015, 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@534 13 Require Import DepList Cpdt.CpdtTactics.
adamc@227 14
adamc@227 15 Set Implicit Arguments.
adam@534 16 Set Asymmetric Patterns.
adamc@227 17 (* end hide *)
adamc@227 18
adam@398 19 (** printing $ %({}*% #(<a/>*# *)
adam@398 20 (** printing ^ %*{})% #*<a/>)# *)
adam@398 21
adam@398 22
adamc@227 23
adamc@227 24 (** %\chapter{Universes and Axioms}% *)
adamc@227 25
adam@343 26 (** 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 27
adam@343 28 %\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 29
adamc@227 30
adamc@227 31 (** * The [Type] Hierarchy *)
adamc@227 32
adam@343 33 (** %\index{type hierarchy}%Every object in Gallina has a type. *)
adamc@227 34
adamc@227 35 Check 0.
adamc@227 36 (** %\vspace{-.15in}% [[
adamc@227 37 0
adamc@227 38 : nat
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
adam@429 49 From a set theory perspective, it is unsurprising to consider the natural numbers as a "set." *)
adamc@227 50
adamc@227 51 Check Set.
adamc@227 52 (** %\vspace{-.15in}% [[
adamc@227 53 Set
adamc@227 54 : Type
adamc@227 55 ]]
adamc@227 56
adam@409 57 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 58
adamc@227 59 Check Type.
adamc@227 60 (** %\vspace{-.15in}% [[
adamc@227 61 Type
adamc@227 62 : Type
adamc@227 63 ]]
adamc@227 64
adam@429 65 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 66
adam@343 67 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 68
adamc@227 69 Set Printing Universes.
adamc@227 70
adamc@227 71 Check nat.
adamc@227 72 (** %\vspace{-.15in}% [[
adamc@227 73 nat
adamc@227 74 : Set
adam@302 75 ]]
adam@398 76 *)
adamc@227 77
adamc@227 78 Check Set.
adamc@227 79 (** %\vspace{-.15in}% [[
adamc@227 80 Set
adamc@227 81 : Type $ (0)+1 ^
adam@302 82 ]]
adam@302 83 *)
adamc@227 84
adamc@227 85 Check Type.
adamc@227 86 (** %\vspace{-.15in}% [[
adamc@227 87 Type $ Top.3 ^
adamc@227 88 : Type $ (Top.3)+1 ^
adamc@227 89 ]]
adamc@227 90
adam@429 91 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 92
adam@398 93 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 94
adam@488 95 In the third 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 96
adam@409 97 Another crucial concept in CIC is%\index{predicativity}% _predicativity_. Consider these queries. *)
adamc@227 98
adamc@227 99 Check forall T : nat, fin T.
adamc@227 100 (** %\vspace{-.15in}% [[
adamc@227 101 forall T : nat, fin T
adamc@227 102 : Set
adam@302 103 ]]
adam@302 104 *)
adamc@227 105
adamc@227 106 Check forall T : Set, T.
adamc@227 107 (** %\vspace{-.15in}% [[
adamc@227 108 forall T : Set, T
adamc@227 109 : Type $ max(0, (0)+1) ^
adam@302 110 ]]
adam@302 111 *)
adamc@227 112
adamc@227 113 Check forall T : Type, T.
adamc@227 114 (** %\vspace{-.15in}% [[
adamc@227 115 forall T : Type $ Top.9 ^ , T
adamc@227 116 : Type $ max(Top.9, (Top.9)+1) ^
adamc@227 117 ]]
adamc@227 118
adamc@227 119 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@227 120
adam@287 121 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@227 122
adamc@227 123 Definition id (T : Set) (x : T) : T := x.
adamc@227 124
adamc@227 125 Check id 0.
adamc@227 126 (** %\vspace{-.15in}% [[
adamc@227 127 id 0
adamc@227 128 : nat
adamc@227 129
adamc@227 130 Check id Set.
adam@343 131 ]]
adamc@227 132
adam@343 133 <<
adamc@227 134 Error: Illegal application (Type Error):
adamc@227 135 ...
adam@479 136 The 1st term has type "Type (* (Top.15)+1 *)"
adam@479 137 which should be coercible to "Set".
adam@343 138 >>
adamc@227 139
adam@343 140 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@227 141
adamc@227 142 Reset id.
adamc@227 143 Definition id (T : Type) (x : T) : T := x.
adamc@227 144 Check id 0.
adamc@227 145 (** %\vspace{-.15in}% [[
adamc@227 146 id 0
adamc@227 147 : nat
adam@302 148 ]]
adam@302 149 *)
adamc@227 150
adamc@227 151 Check id Set.
adamc@227 152 (** %\vspace{-.15in}% [[
adamc@227 153 id Set
adamc@227 154 : Type $ Top.17 ^
adam@302 155 ]]
adam@302 156 *)
adamc@227 157
adamc@227 158 Check id Type.
adamc@227 159 (** %\vspace{-.15in}% [[
adamc@227 160 id Type $ Top.18 ^
adamc@227 161 : Type $ Top.19 ^
adam@302 162 ]]
adam@302 163 *)
adamc@227 164
adamc@227 165 (** 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@227 166 [[
adamc@227 167 Check id id.
adam@343 168 ]]
adamc@227 169
adam@343 170 <<
adamc@227 171 Error: Universe inconsistency (cannot enforce Top.16 < Top.16).
adam@343 172 >>
adamc@227 173
adam@479 174 %\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, when an object is defined using some sort of quantifier, none of the 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@227 175
adamc@227 176
adamc@227 177 (** ** Inductive Definitions *)
adamc@227 178
adam@505 179 (** 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 an encoded expression of type [T].
adamc@227 180 [[
adamc@227 181 Inductive exp : Set -> Set :=
adamc@227 182 | Const : forall T : Set, T -> exp T
adamc@227 183 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 184 | Eq : forall T, exp T -> exp T -> exp bool.
adam@343 185 ]]
adamc@227 186
adam@343 187 <<
adamc@227 188 Error: Large non-propositional inductive types must be in Type.
adam@343 189 >>
adamc@227 190
adam@409 191 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@227 192
adamc@227 193 Inductive exp : Type -> Type :=
adamc@227 194 | Const : forall T, T -> exp T
adamc@227 195 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 196 | Eq : forall T, exp T -> exp T -> exp bool.
adamc@227 197
adam@505 198 (** 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], the right behavior here, though it was wrong for the [Set] version of [exp].
adamc@228 199
adamc@228 200 Our new definition is accepted. We can build some sample expressions. *)
adamc@227 201
adamc@227 202 Check Const 0.
adamc@227 203 (** %\vspace{-.15in}% [[
adamc@227 204 Const 0
adamc@227 205 : exp nat
adam@302 206 ]]
adam@302 207 *)
adamc@227 208
adamc@227 209 Check Pair (Const 0) (Const tt).
adamc@227 210 (** %\vspace{-.15in}% [[
adamc@227 211 Pair (Const 0) (Const tt)
adamc@227 212 : exp (nat * unit)
adam@302 213 ]]
adam@302 214 *)
adamc@227 215
adamc@227 216 Check Eq (Const Set) (Const Type).
adamc@227 217 (** %\vspace{-.15in}% [[
adamc@228 218 Eq (Const Set) (Const Type $ Top.59 ^ )
adamc@227 219 : exp bool
adamc@227 220 ]]
adamc@227 221
adamc@227 222 We can check many expressions, including fancy expressions that include types. However, it is not hard to hit a type-checking wall.
adamc@227 223 [[
adamc@227 224 Check Const (Const O).
adam@343 225 ]]
adamc@227 226
adam@343 227 <<
adamc@227 228 Error: Universe inconsistency (cannot enforce Top.42 < Top.42).
adam@343 229 >>
adamc@227 230
adamc@227 231 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. *)
adam@417 232 (** [[
adamc@227 233 Print exp.
adam@417 234 ]]
adam@444 235 %\vspace{-.15in}%[[
adamc@227 236 Inductive exp
adamc@227 237 : Type $ Top.8 ^ ->
adamc@227 238 Type
adamc@227 239 $ max(0, (Top.11)+1, (Top.14)+1, (Top.15)+1, (Top.19)+1) ^ :=
adamc@227 240 Const : forall T : Type $ Top.11 ^ , T -> exp T
adamc@227 241 | Pair : forall (T1 : Type $ Top.14 ^ ) (T2 : Type $ Top.15 ^ ),
adamc@227 242 exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 243 | Eq : forall T : Type $ Top.19 ^ , exp T -> exp T -> exp bool
adamc@227 244 ]]
adamc@227 245
adam@505 246 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. Therefore, [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@227 247
adam@429 248 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@227 249
adamc@227 250 Print Universes.
adamc@227 251 (** %\vspace{-.15in}% [[
adamc@227 252 Top.19 < Top.9 <= Top.8
adamc@227 253 Top.15 < Top.9 <= Top.8 <= Coq.Init.Datatypes.38
adamc@227 254 Top.14 < Top.9 <= Top.8 <= Coq.Init.Datatypes.37
adamc@227 255 Top.11 < Top.9 <= Top.8
adamc@227 256 ]]
adamc@227 257
adam@343 258 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@227 259
adamc@227 260 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@227 261
adam@417 262 (* begin hide *)
adam@437 263 (* begin thide *)
adam@417 264 Inductive prod := pair.
adam@417 265 Reset prod.
adam@437 266 (* end thide *)
adam@417 267 (* end hide *)
adam@417 268
adam@444 269 (** %\vspace{-.3in}%[[
adamc@227 270 Print prod.
adam@417 271 ]]
adam@444 272 %\vspace{-.15in}%[[
adamc@227 273 Inductive prod (A : Type $ Coq.Init.Datatypes.37 ^ )
adamc@227 274 (B : Type $ Coq.Init.Datatypes.38 ^ )
adamc@227 275 : Type $ max(Coq.Init.Datatypes.37, Coq.Init.Datatypes.38) ^ :=
adamc@227 276 pair : A -> B -> A * B
adamc@227 277 ]]
adamc@227 278
adamc@227 279 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@227 280
adamc@227 281 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@227 282
adamc@227 283 %\medskip%
adamc@227 284
adam@505 285 The annotated definition of [prod] reveals something interesting. 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@227 286
adamc@231 287 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@227 288
adamc@227 289 Check (nat, (Type, Set)).
adamc@227 290 (** %\vspace{-.15in}% [[
adamc@227 291 (nat, (Type $ Top.44 ^ , Set))
adamc@227 292 : Set * (Type $ Top.45 ^ * Type $ Top.46 ^ )
adamc@227 293 ]]
adamc@227 294
adamc@227 295 The same cannot be done with a counterpart to [prod] that does not use parameters. *)
adamc@227 296
adamc@227 297 Inductive prod' : Type -> Type -> Type :=
adamc@227 298 | pair' : forall A B : Type, A -> B -> prod' A B.
adam@444 299 (** %\vspace{-.15in}%[[
adamc@227 300 Check (pair' nat (pair' Type Set)).
adam@343 301 ]]
adamc@227 302
adam@343 303 <<
adamc@227 304 Error: Universe inconsistency (cannot enforce Top.51 < Top.51).
adam@343 305 >>
adamc@227 306
adamc@233 307 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@233 308
adam@343 309 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@233 310
adamc@233 311 Inductive foo (A : Type) : Type :=
adamc@233 312 | Foo : A -> foo A.
adamc@229 313
adamc@229 314 (* begin hide *)
adamc@229 315 Unset Printing Universes.
adamc@229 316 (* end hide *)
adamc@229 317
adamc@233 318 Check foo nat.
adamc@233 319 (** %\vspace{-.15in}% [[
adamc@233 320 foo nat
adamc@233 321 : Set
adam@302 322 ]]
adam@302 323 *)
adamc@233 324
adamc@233 325 Check foo Set.
adamc@233 326 (** %\vspace{-.15in}% [[
adamc@233 327 foo Set
adamc@233 328 : Type
adam@302 329 ]]
adam@302 330 *)
adamc@233 331
adamc@233 332 Check foo True.
adamc@233 333 (** %\vspace{-.15in}% [[
adamc@233 334 foo True
adamc@233 335 : Prop
adamc@233 336 ]]
adamc@233 337
adam@429 338 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@233 339
adamc@233 340 Imitation polymorphism can be confusing in some contexts. For instance, it is what is responsible for this weird behavior. *)
adamc@233 341
adamc@233 342 Inductive bar : Type := Bar : bar.
adamc@233 343
adamc@233 344 Check bar.
adamc@233 345 (** %\vspace{-.15in}% [[
adamc@233 346 bar
adamc@233 347 : Prop
adamc@233 348 ]]
adamc@233 349
adamc@233 350 The type that Coq comes up with may be used in strictly more contexts than the type one might have expected. *)
adamc@233 351
adamc@229 352
adam@388 353 (** ** Deciphering Baffling Messages About Inability to Unify *)
adam@388 354
adam@388 355 (** 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@388 356
adam@388 357 Theorem symmetry : forall A B : Type,
adam@388 358 A = B
adam@388 359 -> B = A.
adam@388 360 intros ? ? H; rewrite H; reflexivity.
adam@388 361 Qed.
adam@388 362
adam@388 363 (** Let us attempt an admittedly silly proof of the following theorem. *)
adam@388 364
adam@388 365 Theorem illustrative_but_silly_detour : unit = unit.
adam@444 366 (** %\vspace{-.25in}%[[
adam@444 367 apply symmetry.
adam@388 368 ]]
adam@388 369 <<
adam@388 370 Error: Impossible to unify "?35 = ?34" with "unit = unit".
adam@388 371 >>
adam@388 372
adam@458 373 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 issue is in a part of the unification problem that is _not_ shown to us in this error message!
adam@388 374
adam@510 375 The following command is the secret to getting better error messages in such cases:%\index{Vernacular commands!Set Printing All}% *)
adam@388 376
adam@388 377 Set Printing All.
adam@444 378 (** %\vspace{-.15in}%[[
adam@444 379 apply symmetry.
adam@388 380 ]]
adam@388 381 <<
adam@388 382 Error: Impossible to unify "@eq Type ?46 ?45" with "@eq Set unit unit".
adam@388 383 >>
adam@388 384
adam@398 385 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@388 386
adam@388 387 Abort.
adam@388 388
adam@388 389 (** 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@388 390
adam@388 391 Theorem illustrative_but_silly_detour : (unit : Type) = unit.
adam@388 392 apply symmetry; reflexivity.
adam@388 393 Qed.
adam@388 394
adam@388 395 (** 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@388 396
adam@388 397 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@388 398
adam@388 399 Unset Printing All.
adam@388 400
adam@388 401 Theorem ex_symmetry : (exists x, x = 0) -> (exists x, 0 = x).
adam@435 402 eexists.
adam@388 403 (** %\vspace{-.15in}%[[
adam@388 404 H : exists x : nat, x = 0
adam@388 405 ============================
adam@388 406 0 = ?98
adam@388 407 ]]
adam@388 408 *)
adam@388 409
adam@388 410 destruct H.
adam@388 411 (** %\vspace{-.15in}%[[
adam@388 412 x : nat
adam@388 413 H : x = 0
adam@388 414 ============================
adam@388 415 0 = ?99
adam@388 416 ]]
adam@388 417 *)
adam@388 418
adam@444 419 (** %\vspace{-.2in}%[[
adam@444 420 symmetry; exact H.
adam@388 421 ]]
adam@388 422
adam@388 423 <<
adam@388 424 Error: In environment
adam@388 425 x : nat
adam@388 426 H : x = 0
adam@388 427 The term "H" has type "x = 0" while it is expected to have type
adam@388 428 "?99 = 0".
adam@388 429 >>
adam@388 430
adam@398 431 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@388 432
adam@388 433 Restart.
adam@388 434 destruct 1 as [x]; apply ex_intro with x; symmetry; assumption.
adam@388 435 Qed.
adam@388 436
adam@429 437 (** 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@388 438
adam@388 439
adamc@229 440 (** * The [Prop] Universe *)
adamc@229 441
adam@429 442 (** 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@229 443
adamc@229 444 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@229 445
adamc@229 446 Print sig.
adamc@229 447 (** %\vspace{-.15in}% [[
adamc@229 448 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@229 449 exist : forall x : A, P x -> sig P
adam@302 450 ]]
adam@302 451 *)
adamc@229 452
adamc@229 453 Print ex.
adamc@229 454 (** %\vspace{-.15in}% [[
adamc@229 455 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@229 456 ex_intro : forall x : A, P x -> ex P
adamc@229 457 ]]
adamc@229 458
adamc@229 459 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@229 460
adamc@229 461 Definition projS A (P : A -> Prop) (x : sig P) : A :=
adamc@229 462 match x with
adamc@229 463 | exist v _ => v
adamc@229 464 end.
adamc@229 465
adam@429 466 (* begin hide *)
adam@437 467 (* begin thide *)
adam@429 468 Definition projE := O.
adam@437 469 (* end thide *)
adam@429 470 (* end hide *)
adam@429 471
adamc@229 472 (** We run into trouble with a version that has been changed to work with [ex].
adamc@229 473 [[
adamc@229 474 Definition projE A (P : A -> Prop) (x : ex P) : A :=
adamc@229 475 match x with
adamc@229 476 | ex_intro v _ => v
adamc@229 477 end.
adam@343 478 ]]
adamc@229 479
adam@343 480 <<
adamc@229 481 Error:
adamc@229 482 Incorrect elimination of "x" in the inductive type "ex":
adamc@229 483 the return type has sort "Type" while it should be "Prop".
adamc@229 484 Elimination of an inductive object of sort Prop
adamc@229 485 is not allowed on a predicate in sort Type
adamc@229 486 because proofs can be eliminated only to build proofs.
adam@343 487 >>
adamc@229 488
adam@429 489 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@229 490
adamc@229 491 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@229 492
adam@398 493 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@229 494
adamc@229 495 Definition sym_sig (x : sig (fun n => n = 0)) : sig (fun n => 0 = n) :=
adamc@229 496 match x with
adamc@229 497 | exist n pf => exist _ n (sym_eq pf)
adamc@229 498 end.
adamc@229 499
adamc@229 500 Extraction sym_sig.
adamc@229 501 (** <<
adamc@229 502 (** val sym_sig : nat -> nat **)
adamc@229 503
adamc@229 504 let sym_sig x = x
adamc@229 505 >>
adamc@229 506
adamc@229 507 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@229 508
adamc@229 509 Definition sym_ex (x : ex (fun n => n = 0)) : ex (fun n => 0 = n) :=
adamc@229 510 match x with
adamc@229 511 | ex_intro n pf => ex_intro _ n (sym_eq pf)
adamc@229 512 end.
adamc@229 513
adamc@229 514 Extraction sym_ex.
adamc@229 515 (** <<
adamc@229 516 (** val sym_ex : __ **)
adamc@229 517
adamc@229 518 let sym_ex = __
adamc@229 519 >>
adamc@229 520
adam@435 521 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@229 522
adam@419 523 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@229 524
adam@398 525 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@229 526
adamc@229 527 %\medskip%
adamc@229 528
adam@409 529 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@229 530
adamc@229 531 Check forall P Q : Prop, P \/ Q -> Q \/ P.
adamc@229 532 (** %\vspace{-.15in}% [[
adamc@229 533 forall P Q : Prop, P \/ Q -> Q \/ P
adamc@229 534 : Prop
adamc@229 535 ]]
adamc@229 536
adamc@230 537 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@230 538
adamc@230 539 Impredicativity also allows us to implement a version of our earlier [exp] type that does not suffer from the weakness that we found. *)
adamc@230 540
adamc@230 541 Inductive expP : Type -> Prop :=
adamc@230 542 | ConstP : forall T, T -> expP T
adamc@230 543 | PairP : forall T1 T2, expP T1 -> expP T2 -> expP (T1 * T2)
adamc@230 544 | EqP : forall T, expP T -> expP T -> expP bool.
adamc@230 545
adamc@230 546 Check ConstP 0.
adamc@230 547 (** %\vspace{-.15in}% [[
adamc@230 548 ConstP 0
adamc@230 549 : expP nat
adam@302 550 ]]
adam@302 551 *)
adamc@230 552
adamc@230 553 Check PairP (ConstP 0) (ConstP tt).
adamc@230 554 (** %\vspace{-.15in}% [[
adamc@230 555 PairP (ConstP 0) (ConstP tt)
adamc@230 556 : expP (nat * unit)
adam@302 557 ]]
adam@302 558 *)
adamc@230 559
adamc@230 560 Check EqP (ConstP Set) (ConstP Type).
adamc@230 561 (** %\vspace{-.15in}% [[
adamc@230 562 EqP (ConstP Set) (ConstP Type)
adamc@230 563 : expP bool
adam@302 564 ]]
adam@302 565 *)
adamc@230 566
adamc@230 567 Check ConstP (ConstP O).
adamc@230 568 (** %\vspace{-.15in}% [[
adamc@230 569 ConstP (ConstP 0)
adamc@230 570 : expP (expP nat)
adamc@230 571 ]]
adamc@230 572
adam@287 573 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@230 574
adamc@230 575 Inductive eqPlus : forall T, T -> T -> Prop :=
adamc@230 576 | Base : forall T (x : T), eqPlus x x
adamc@230 577 | Func : forall dom ran (f1 f2 : dom -> ran),
adamc@230 578 (forall x : dom, eqPlus (f1 x) (f2 x))
adamc@230 579 -> eqPlus f1 f2.
adamc@230 580
adamc@230 581 Check (Base 0).
adamc@230 582 (** %\vspace{-.15in}% [[
adamc@230 583 Base 0
adamc@230 584 : eqPlus 0 0
adam@302 585 ]]
adam@302 586 *)
adamc@230 587
adamc@230 588 Check (Func (fun n => n) (fun n => 0 + n) (fun n => Base n)).
adamc@230 589 (** %\vspace{-.15in}% [[
adamc@230 590 Func (fun n : nat => n) (fun n : nat => 0 + n) (fun n : nat => Base n)
adamc@230 591 : eqPlus (fun n : nat => n) (fun n : nat => 0 + n)
adam@302 592 ]]
adam@302 593 *)
adamc@230 594
adamc@230 595 Check (Base (Base 1)).
adamc@230 596 (** %\vspace{-.15in}% [[
adamc@230 597 Base (Base 1)
adamc@230 598 : eqPlus (Base 1) (Base 1)
adam@302 599 ]]
adam@302 600 *)
adamc@230 601
adam@343 602 (** Stating equality facts about proofs may seem baroque, but we have already seen its utility in the chapter on reasoning about equality proofs. *)
adam@343 603
adamc@230 604
adamc@230 605 (** * Axioms *)
adamc@230 606
adam@409 607 (** 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@230 608
adamc@230 609 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@230 610
adamc@230 611 (** ** The Basics *)
adamc@230 612
adam@343 613 (** One simple example of a useful axiom is the %\index{law of the excluded middle}%law of the excluded middle. *)
adamc@230 614
adamc@230 615 Require Import Classical_Prop.
adamc@230 616 Print classic.
adamc@230 617 (** %\vspace{-.15in}% [[
adamc@230 618 *** [ classic : forall P : Prop, P \/ ~ P ]
adamc@230 619 ]]
adamc@230 620
adam@343 621 In the implementation of module [Classical_Prop], this axiom was defined with the command%\index{Vernacular commands!Axiom}% *)
adamc@230 622
adamc@230 623 Axiom classic : forall P : Prop, P \/ ~ P.
adamc@230 624
adam@343 625 (** 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@230 626
adam@458 627 Parameter num : nat.
adam@458 628 Axiom positive : num > 0.
adam@458 629 Reset num.
adamc@230 630
adam@429 631 (** 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@230 632
adam@409 633 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@230 634
adam@287 635 Axiom not_classic : ~ forall P : Prop, P \/ ~ P.
adamc@230 636
adamc@230 637 Theorem uhoh : False.
adam@287 638 generalize classic not_classic; tauto.
adamc@230 639 Qed.
adamc@230 640
adamc@230 641 Theorem uhoh_again : 1 + 1 = 3.
adamc@230 642 destruct uhoh.
adamc@230 643 Qed.
adamc@230 644
adamc@230 645 Reset not_classic.
adamc@230 646
adam@429 647 (** 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@230 648
adam@475 649 Recall that Coq implements%\index{constructive logic}% _constructive_ logic by default, where the law of the 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@230 650
adam@398 651 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@230 652
adam@343 653 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@230 654
adamc@230 655 Theorem t1 : forall P : Prop, P -> ~ ~ P.
adamc@230 656 tauto.
adamc@230 657 Qed.
adamc@230 658
adamc@230 659 Print Assumptions t1.
adam@343 660 (** <<
adamc@230 661 Closed under the global context
adam@343 662 >>
adam@302 663 *)
adamc@230 664
adamc@230 665 Theorem t2 : forall P : Prop, ~ ~ P -> P.
adam@444 666 (** %\vspace{-.25in}%[[
adamc@230 667 tauto.
adam@343 668 ]]
adam@343 669 <<
adamc@230 670 Error: tauto failed.
adam@343 671 >>
adam@302 672 *)
adamc@230 673 intro P; destruct (classic P); tauto.
adamc@230 674 Qed.
adamc@230 675
adamc@230 676 Print Assumptions t2.
adamc@230 677 (** %\vspace{-.15in}% [[
adamc@230 678 Axioms:
adamc@230 679 classic : forall P : Prop, P \/ ~ P
adamc@230 680 ]]
adamc@230 681
adam@398 682 It is possible to avoid this dependence in some specific cases, where excluded middle _is_ provable, for decidable families of propositions. *)
adamc@230 683
adam@287 684 Theorem nat_eq_dec : forall n m : nat, n = m \/ n <> m.
adamc@230 685 induction n; destruct m; intuition; generalize (IHn m); intuition.
adamc@230 686 Qed.
adamc@230 687
adamc@230 688 Theorem t2' : forall n m : nat, ~ ~ (n = m) -> n = m.
adam@287 689 intros n m; destruct (nat_eq_dec n m); tauto.
adamc@230 690 Qed.
adamc@230 691
adamc@230 692 Print Assumptions t2'.
adam@343 693 (** <<
adamc@230 694 Closed under the global context
adam@343 695 >>
adamc@230 696
adamc@230 697 %\bigskip%
adamc@230 698
adam@409 699 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@230 700
adamc@230 701 Require Import ProofIrrelevance.
adamc@230 702 Print proof_irrelevance.
adam@458 703
adamc@230 704 (** %\vspace{-.15in}% [[
adamc@230 705 *** [ proof_irrelevance : forall (P : Prop) (p1 p2 : P), p1 = p2 ]
adamc@230 706 ]]
adamc@230 707
adam@458 708 This axiom asserts that any two proofs of the same proposition are equal. Recall this example function from Chapter 6. *)
adamc@230 709
adamc@230 710 (* begin hide *)
adamc@230 711 Lemma zgtz : 0 > 0 -> False.
adamc@230 712 crush.
adamc@230 713 Qed.
adamc@230 714 (* end hide *)
adamc@230 715
adamc@230 716 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@230 717 match n with
adamc@230 718 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@230 719 | S n' => fun _ => n'
adamc@230 720 end.
adamc@230 721
adam@343 722 (** We might want to prove that different proofs of [n > 0] do not lead to different results from our richly typed predecessor function. *)
adamc@230 723
adamc@230 724 Theorem pred_strong1_irrel : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 725 destruct n; crush.
adamc@230 726 Qed.
adamc@230 727
adamc@230 728 (** 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@230 729
adamc@230 730 Theorem pred_strong1_irrel' : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 731 intros; f_equal; apply proof_irrelevance.
adamc@230 732 Qed.
adamc@230 733
adamc@230 734
adamc@230 735 (** %\bigskip%
adamc@230 736
adamc@230 737 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@230 738
adamc@230 739 Require Import Eqdep.
adamc@230 740 Import Eq_rect_eq.
adamc@230 741 Print eq_rect_eq.
adamc@230 742 (** %\vspace{-.15in}% [[
adamc@230 743 *** [ eq_rect_eq :
adamc@230 744 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adamc@230 745 x = eq_rect p Q x p h ]
adamc@230 746 ]]
adamc@230 747
adam@429 748 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@230 749
adam@426 750 Corollary UIP_refl : forall A (x : A) (pf : x = x), pf = eq_refl x.
adam@426 751 intros; replace pf with (eq_rect x (eq x) (eq_refl x) x pf); [
adamc@230 752 symmetry; apply eq_rect_eq
adamc@230 753 | exact (match pf as pf' return match pf' in _ = y return x = y with
adam@426 754 | eq_refl => eq_refl x
adamc@230 755 end = pf' with
adam@426 756 | eq_refl => eq_refl _
adamc@230 757 end) ].
adamc@230 758 Qed.
adamc@230 759
adamc@230 760 Corollary UIP : forall A (x y : A) (pf1 pf2 : x = y), pf1 = pf2.
adamc@230 761 intros; generalize pf1 pf2; subst; intros;
adamc@230 762 match goal with
adamc@230 763 | [ |- ?pf1 = ?pf2 ] => rewrite (UIP_refl pf1); rewrite (UIP_refl pf2); reflexivity
adamc@230 764 end.
adamc@230 765 Qed.
adamc@230 766
adam@436 767 (* begin hide *)
adam@437 768 (* begin thide *)
adam@436 769 Require Eqdep_dec.
adam@437 770 (* end thide *)
adam@436 771 (* end hide *)
adam@436 772
adamc@231 773 (** 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@230 774
adamc@230 775 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@230 776
adamc@230 777 %\bigskip%
adamc@230 778
adamc@230 779 There are two more basic axioms that are often assumed, to avoid complications that do not arise in set theory. *)
adamc@230 780
adamc@230 781 Require Import FunctionalExtensionality.
adamc@230 782 Print functional_extensionality_dep.
adamc@230 783 (** %\vspace{-.15in}% [[
adamc@230 784 *** [ functional_extensionality_dep :
adamc@230 785 forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
adamc@230 786 (forall x : A, f x = g x) -> f = g ]
adamc@230 787
adamc@230 788 ]]
adamc@230 789
adamc@230 790 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@230 791
adam@343 792 A simple corollary shows that the same property applies to predicates. *)
adamc@230 793
adamc@230 794 Corollary predicate_extensionality : forall (A : Type) (B : A -> Prop) (f g : forall x : A, B x),
adamc@230 795 (forall x : A, f x = g x) -> f = g.
adamc@230 796 intros; apply functional_extensionality_dep; assumption.
adamc@230 797 Qed.
adamc@230 798
adam@343 799 (** In some cases, one might prefer to assert this corollary as the axiom, to restrict the consequences to proofs and not programs. *)
adam@343 800
adamc@230 801
adamc@230 802 (** ** Axioms of Choice *)
adamc@230 803
adam@343 804 (** 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@230 805
adam@398 806 First, it is possible to implement a choice operator _without_ axioms in some potentially surprising cases. *)
adamc@230 807
adamc@230 808 Require Import ConstructiveEpsilon.
adamc@230 809 Check constructive_definite_description.
adamc@230 810 (** %\vspace{-.15in}% [[
adamc@230 811 constructive_definite_description
adamc@230 812 : forall (A : Set) (f : A -> nat) (g : nat -> A),
adamc@230 813 (forall x : A, g (f x) = x) ->
adamc@230 814 forall P : A -> Prop,
adam@505 815 (forall x : A, {P x} + { ~ P x}) ->
adamc@230 816 (exists! x : A, P x) -> {x : A | P x}
adam@302 817 ]]
adam@302 818 *)
adamc@230 819
adamc@230 820 Print Assumptions constructive_definite_description.
adam@343 821 (** <<
adamc@230 822 Closed under the global context
adam@343 823 >>
adamc@230 824
adam@398 825 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@230 826
adamc@230 827 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@230 828
adamc@230 829 Require Import ClassicalUniqueChoice.
adamc@230 830 Check dependent_unique_choice.
adamc@230 831 (** %\vspace{-.15in}% [[
adamc@230 832 dependent_unique_choice
adamc@230 833 : forall (A : Type) (B : A -> Type) (R : forall x : A, B x -> Prop),
adamc@230 834 (forall x : A, exists! y : B x, R x y) ->
adam@343 835 exists f : forall x : A, B x,
adam@343 836 forall x : A, R x (f x)
adamc@230 837 ]]
adamc@230 838
adamc@230 839 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@230 840
adam@436 841 (* begin hide *)
adam@437 842 (* begin thide *)
adam@436 843 Require RelationalChoice.
adam@437 844 (* end thide *)
adam@436 845 (* end hide *)
adam@436 846
adamc@230 847 Require Import ClassicalChoice.
adamc@230 848 Check choice.
adamc@230 849 (** %\vspace{-.15in}% [[
adamc@230 850 choice
adamc@230 851 : forall (A B : Type) (R : A -> B -> Prop),
adamc@230 852 (forall x : A, exists y : B, R x y) ->
adamc@230 853 exists f : A -> B, forall x : A, R x (f x)
adam@444 854 ]]
adamc@230 855
adamc@230 856 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@230 857
adamc@230 858 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@230 859
adamc@230 860 Definition choice_Set (A B : Type) (R : A -> B -> Prop) (H : forall x : A, {y : B | R x y})
adamc@230 861 : {f : A -> B | forall x : A, R x (f x)} :=
adamc@230 862 exist (fun f => forall x : A, R x (f x))
adamc@230 863 (fun x => proj1_sig (H x)) (fun x => proj2_sig (H x)).
adamc@230 864
adam@458 865 (** %\smallskip{}%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 subtly different. In Gallina, the axioms implement a controlled relaxation of the restrictions on information flow from proofs to programs.
adamc@230 866
adam@505 867 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 combination truly is more than repackaging a function with a different type.
adamc@230 868
adamc@230 869 %\bigskip%
adamc@230 870
adam@505 871 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 rule contrasts with the rule for [Prop], where the restriction applies even to non-large inductive types, and where the result type may only have type [Prop].
adamc@230 872
adam@505 873 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, inconsistency can result. Impredicative [Set] can be useful for modeling inherently impredicative mathematical concepts, but almost all Coq developments get by fine without it. *)
adamc@230 874
adamc@230 875 (** ** Axioms and Computation *)
adamc@230 876
adam@398 877 (** 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@230 878
adamc@230 879 Definition cast (x y : Set) (pf : x = y) (v : x) : y :=
adamc@230 880 match pf with
adam@426 881 | eq_refl => v
adamc@230 882 end.
adamc@230 883
adamc@230 884 (** Computation over programs that use [cast] can proceed smoothly. *)
adamc@230 885
adam@426 886 Eval compute in (cast (eq_refl (nat -> nat)) (fun n => S n)) 12.
adam@343 887 (** %\vspace{-.15in}%[[
adamc@230 888 = 13
adamc@230 889 : nat
adam@302 890 ]]
adam@302 891 *)
adamc@230 892
adamc@230 893 (** Things do not go as smoothly when we use [cast] with proofs that rely on axioms. *)
adamc@230 894
adamc@230 895 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 896 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 897 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 898 Qed.
adamc@230 899
adamc@230 900 Eval compute in (cast t3 (fun _ => First)) 12.
adam@444 901 (** %\vspace{-.15in}%[[
adamc@230 902 = match t3 in (_ = P) return P with
adam@426 903 | eq_refl => fun n : nat => First
adamc@230 904 end 12
adamc@230 905 : fin (12 + 1)
adamc@230 906 ]]
adamc@230 907
adam@458 908 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 mistake is easily fixed. *)
adamc@230 909
adamc@230 910 Reset t3.
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 Defined.
adamc@230 916
adamc@230 917 Eval compute in (cast t3 (fun _ => First)) 12.
adam@444 918 (** %\vspace{-.15in}%[[
adamc@230 919 = match
adamc@230 920 match
adamc@230 921 match
adamc@230 922 functional_extensionality
adamc@230 923 ....
adamc@230 924 ]]
adamc@230 925
adam@398 926 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@230 927
adamc@230 928 If we are careful in using tactics to prove an equality, we can still compute with casts over the proof. *)
adamc@230 929
adamc@230 930 Lemma plus1 : forall n, S n = n + 1.
adamc@230 931 induction n; simpl; intuition.
adamc@230 932 Defined.
adamc@230 933
adamc@230 934 Theorem t4 : forall n, fin (S n) = fin (n + 1).
adamc@230 935 intro; f_equal; apply plus1.
adamc@230 936 Defined.
adamc@230 937
adamc@230 938 Eval compute in cast (t4 13) First.
adamc@230 939 (** %\vspace{-.15in}% [[
adamc@230 940 = First
adamc@230 941 : fin (13 + 1)
adam@302 942 ]]
adam@343 943
adam@426 944 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@343 945
adam@344 946
adam@344 947 (** ** Methods for Avoiding Axioms *)
adam@344 948
adam@409 949 (** 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@344 950
adam@429 951 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@344 952
adam@344 953 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@344 954
adam@344 955 Theorem fin_cases : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 956 intros; dep_destruct f; eauto.
adam@344 957 Qed.
adam@344 958
adam@429 959 (* begin hide *)
adam@429 960 Require Import JMeq.
adam@437 961 (* begin thide *)
adam@429 962 Definition jme := (JMeq, JMeq_eq).
adam@437 963 (* end thide *)
adam@429 964 (* end hide *)
adam@429 965
adam@344 966 Print Assumptions fin_cases.
adam@344 967 (** %\vspace{-.15in}%[[
adam@344 968 Axioms:
adam@429 969 JMeq_eq : forall (A : Type) (x y : A), JMeq x y -> x = y
adam@344 970 ]]
adam@344 971
adam@344 972 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@344 973
adam@344 974 (* begin thide *)
adam@344 975 Lemma fin_cases_again' : forall n (f : fin n),
adam@344 976 match n return fin n -> Prop with
adam@344 977 | O => fun _ => False
adam@344 978 | S n' => fun f => f = First \/ exists f', f = Next f'
adam@344 979 end f.
adam@344 980 destruct f; eauto.
adam@344 981 Qed.
adam@344 982
adam@344 983 (** 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@344 984
adam@344 985 Theorem fin_cases_again : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 986 intros; exact (fin_cases_again' f).
adam@344 987 Qed.
adam@344 988 (* end thide *)
adam@344 989
adam@344 990 Print Assumptions fin_cases_again.
adam@344 991 (** %\vspace{-.15in}%
adam@344 992 <<
adam@344 993 Closed under the global context
adam@344 994 >>
adam@344 995
adam@345 996 *)
adam@345 997
adam@345 998 (* begin thide *)
adam@345 999 (** 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 1000
adam@429 1001 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 1002
adam@345 1003 Definition finOut n (f : fin n) : match n return fin n -> Type with
adam@345 1004 | O => fun _ => Empty_set
adam@345 1005 | _ => fun f => {f' : _ | f = Next f'} + {f = First}
adam@345 1006 end f :=
adam@345 1007 match f with
adam@426 1008 | First _ => inright _ (eq_refl _)
adam@426 1009 | Next _ f' => inleft _ (exist _ f' (eq_refl _))
adam@345 1010 end.
adam@345 1011 (* end thide *)
adam@345 1012
adam@345 1013 (** 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 1014
adam@344 1015 Inductive formula : list Type -> Type :=
adam@344 1016 | Inject : forall Ts, Prop -> formula Ts
adam@344 1017 | VarEq : forall T Ts, T -> formula (T :: Ts)
adam@344 1018 | Lift : forall T Ts, formula Ts -> formula (T :: Ts)
adam@344 1019 | Forall : forall T Ts, formula (T :: Ts) -> formula Ts
adam@344 1020 | And : forall Ts, formula Ts -> formula Ts -> formula Ts.
adam@344 1021
adam@344 1022 (** 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 1023
adam@344 1024 Inductive proof : formula nil -> Prop :=
adam@344 1025 | PInject : forall (P : Prop), P -> proof (Inject nil P)
adam@344 1026 | PAnd : forall p q, proof p -> proof q -> proof (And p q).
adam@344 1027
adam@429 1028 (** Let us prove a lemma showing that a "[P /\ Q -> P]" rule is derivable within the rules of [proof]. *)
adam@344 1029
adam@344 1030 Theorem proj1 : forall p q, proof (And p q) -> proof p.
adam@344 1031 destruct 1.
adam@344 1032 (** %\vspace{-.15in}%[[
adam@344 1033 p : formula nil
adam@344 1034 q : formula nil
adam@344 1035 P : Prop
adam@344 1036 H : P
adam@344 1037 ============================
adam@344 1038 proof p
adam@344 1039 ]]
adam@344 1040 *)
adam@344 1041
adam@344 1042 (** 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 1043
adam@344 1044 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 1045
adam@344 1046 Restart.
adam@344 1047 Require Import Program.
adam@344 1048 intros ? ? H; dependent destruction H; auto.
adam@344 1049 Qed.
adam@344 1050
adam@344 1051 Print Assumptions proj1.
adam@344 1052 (** %\vspace{-.15in}%[[
adam@344 1053 Axioms:
adam@344 1054 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1055 x = eq_rect p Q x p h
adam@344 1056 ]]
adam@344 1057
adam@344 1058 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 1059
adam@344 1060 Lemma proj1_again' : forall r, proof r
adam@344 1061 -> forall p q, r = And p q -> proof p.
adam@344 1062 destruct 1; crush.
adam@344 1063 (** %\vspace{-.15in}%[[
adam@344 1064 H0 : Inject [] P = And p q
adam@344 1065 ============================
adam@344 1066 proof p
adam@344 1067 ]]
adam@344 1068
adam@344 1069 The first goal looks reasonable. Hypothesis [H0] is clearly contradictory, as [discriminate] can show. *)
adam@344 1070
adam@547 1071 try discriminate. (* Note: Coq 8.6 is now solving this subgoal automatically!
adam@547 1072 * This line left here to keep everything working in
adam@547 1073 * 8.4, 8.5, and 8.6. *)
adam@344 1074 (** %\vspace{-.15in}%[[
adam@344 1075 H : proof p
adam@344 1076 H1 : And p q = And p0 q0
adam@344 1077 ============================
adam@344 1078 proof p0
adam@344 1079 ]]
adam@344 1080
adam@344 1081 It looks like we are almost done. Hypothesis [H1] gives [p = p0] by injectivity of constructors, and then [H] finishes the case. *)
adam@344 1082
adam@344 1083 injection H1; intros.
adam@344 1084
adam@429 1085 (* begin hide *)
adam@437 1086 (* begin thide *)
adam@429 1087 Definition existT' := existT.
adam@437 1088 (* end thide *)
adam@429 1089 (* end hide *)
adam@429 1090
adam@429 1091 (** Unfortunately, the "equality" that we expected between [p] and [p0] comes in a strange form:
adam@344 1092
adam@344 1093 [[
adam@344 1094 H3 : existT (fun Ts : list Type => formula Ts) []%list p =
adam@344 1095 existT (fun Ts : list Type => formula Ts) []%list p0
adam@344 1096 ============================
adam@344 1097 proof p0
adam@344 1098 ]]
adam@344 1099
adam@345 1100 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 1101
adam@344 1102 How exactly does an axiom come into the picture here? Let us ask [crush] to finish the proof. *)
adam@344 1103
adam@344 1104 crush.
adam@344 1105 Qed.
adam@344 1106
adam@344 1107 Print Assumptions proj1_again'.
adam@344 1108 (** %\vspace{-.15in}%[[
adam@344 1109 Axioms:
adam@344 1110 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1111 x = eq_rect p Q x p h
adam@344 1112 ]]
adam@344 1113
adam@344 1114 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 1115
adam@479 1116 Hope is not lost, however. We can produce an even stranger looking lemma, which gives us the theorem without axioms. As always when we want to do case analysis on a term with a tricky dependent type, the key is to refactor the theorem statement so that every term we [match] on has _variables_ as its type indices; so instead of talking about proofs of [And p q], we talk about proofs of an arbitrary [r], but we only conclude anything interesting when [r] is an [And]. *)
adam@344 1117
adam@344 1118 Lemma proj1_again'' : forall r, proof r
adam@344 1119 -> match r with
adam@344 1120 | And Ps p _ => match Ps return formula Ps -> Prop with
adam@344 1121 | nil => fun p => proof p
adam@344 1122 | _ => fun _ => True
adam@344 1123 end p
adam@344 1124 | _ => True
adam@344 1125 end.
adam@344 1126 destruct 1; auto.
adam@344 1127 Qed.
adam@344 1128
adam@344 1129 Theorem proj1_again : forall p q, proof (And p q) -> proof p.
adam@344 1130 intros ? ? H; exact (proj1_again'' H).
adam@344 1131 Qed.
adam@344 1132
adam@344 1133 Print Assumptions proj1_again.
adam@344 1134 (** <<
adam@344 1135 Closed under the global context
adam@344 1136 >>
adam@344 1137
adam@377 1138 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 1139
adam@377 1140 %\medskip%
adam@377 1141
adam@398 1142 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 1143
adam@377 1144 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 1145
adam@377 1146 Section withTypes.
adam@377 1147 Variable types : list Set.
adam@377 1148
adam@377 1149 (** 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 1150
adam@377 1151 Variable values : hlist (fun x : Set => x) types.
adam@377 1152
adam@377 1153 (** 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 1154
adam@377 1155 Variable natIndex : nat.
adam@377 1156 Variable natIndex_ok : nth_error types natIndex = Some nat.
adam@377 1157
adam@377 1158 (** 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 1159
adam@377 1160 Lemma nth_error_nil : forall A n x,
adam@377 1161 nth_error (@nil A) n = Some x
adam@377 1162 -> False.
adam@377 1163 destruct n; simpl; unfold error; congruence.
adam@377 1164 Defined.
adam@377 1165
adam@377 1166 Implicit Arguments nth_error_nil [A n x].
adam@377 1167
adam@377 1168 Lemma Some_inj : forall A (x y : A),
adam@377 1169 Some x = Some y
adam@377 1170 -> x = y.
adam@377 1171 congruence.
adam@377 1172 Defined.
adam@377 1173
adam@377 1174 Fixpoint getNat (types' : list Set) (values' : hlist (fun x : Set => x) types')
adam@377 1175 (natIndex : nat) : (nth_error types' natIndex = Some nat) -> nat :=
adam@377 1176 match values' with
adam@377 1177 | HNil => fun pf => match nth_error_nil pf with end
adam@377 1178 | HCons t ts x values'' =>
adam@377 1179 match natIndex return nth_error (t :: ts) natIndex = Some nat -> nat with
adam@377 1180 | O => fun pf =>
adam@377 1181 match Some_inj pf in _ = T return T with
adam@426 1182 | eq_refl => x
adam@377 1183 end
adam@377 1184 | S natIndex' => getNat values'' natIndex'
adam@377 1185 end
adam@377 1186 end.
adam@377 1187 End withTypes.
adam@377 1188
adam@377 1189 (** The problem becomes apparent when we experiment with running [getNat] on a concrete [types] list. *)
adam@377 1190
adam@377 1191 Definition myTypes := unit :: nat :: bool :: nil.
adam@377 1192 Definition myValues : hlist (fun x : Set => x) myTypes :=
adam@377 1193 tt ::: 3 ::: false ::: HNil.
adam@377 1194
adam@377 1195 Definition myNatIndex := 1.
adam@377 1196
adam@377 1197 Theorem myNatIndex_ok : nth_error myTypes myNatIndex = Some nat.
adam@377 1198 reflexivity.
adam@377 1199 Defined.
adam@377 1200
adam@377 1201 Eval compute in getNat myValues myNatIndex myNatIndex_ok.
adam@377 1202 (** %\vspace{-.15in}%[[
adam@377 1203 = 3
adam@377 1204 ]]
adam@377 1205
adam@398 1206 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 1207
adam@377 1208 Theorem getNat_is_reasonable : forall pf, getNat myValues myNatIndex pf = 3.
adam@377 1209 intro; compute.
adam@377 1210 (**
adam@377 1211 <<
adam@377 1212 1 subgoal
adam@377 1213 >>
adam@377 1214 %\vspace{-.3in}%[[
adam@377 1215 pf : nth_error myTypes myNatIndex = Some nat
adam@377 1216 ============================
adam@377 1217 match
adam@377 1218 match
adam@377 1219 pf in (_ = y)
adam@377 1220 return (nat = match y with
adam@377 1221 | Some H => H
adam@377 1222 | None => nat
adam@377 1223 end)
adam@377 1224 with
adam@377 1225 | eq_refl => eq_refl
adam@377 1226 end in (_ = T) return T
adam@377 1227 with
adam@377 1228 | eq_refl => 3
adam@377 1229 end = 3
adam@377 1230 ]]
adam@377 1231
adam@377 1232 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 1233
adam@377 1234 Abort.
adam@377 1235
adam@377 1236 (** 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 1237
adam@377 1238 Fixpoint copies A (x : A) (n : nat) : list A :=
adam@377 1239 match n with
adam@377 1240 | O => nil
adam@377 1241 | S n' => x :: copies x n'
adam@377 1242 end.
adam@377 1243
adam@377 1244 Fixpoint update A (ls : list A) (n : nat) (x : A) : list A :=
adam@377 1245 match ls with
adam@377 1246 | nil => copies x n ++ x :: nil
adam@377 1247 | y :: ls' => match n with
adam@377 1248 | O => x :: ls'
adam@377 1249 | S n' => y :: update ls' n' x
adam@377 1250 end
adam@377 1251 end.
adam@377 1252
adam@377 1253 (** Now let us revisit the definition of [getNat]. *)
adam@377 1254
adam@377 1255 Section withTypes'.
adam@377 1256 Variable types : list Set.
adam@377 1257 Variable natIndex : nat.
adam@377 1258
adam@429 1259 (** 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 1260
adam@377 1261 Definition types' := update types natIndex nat.
adam@377 1262
adam@377 1263 Variable values : hlist (fun x : Set => x) types'.
adam@377 1264
adam@377 1265 (** Now a bit of dependent pattern matching helps us rewrite [getNat] in a way that avoids any use of equality proofs. *)
adam@377 1266
adam@378 1267 Fixpoint skipCopies (n : nat)
adam@378 1268 : hlist (fun x : Set => x) (copies nat n ++ nat :: nil) -> nat :=
adam@378 1269 match n with
adam@378 1270 | O => fun vs => hhd vs
adam@378 1271 | S n' => fun vs => skipCopies n' (htl vs)
adam@378 1272 end.
adam@378 1273
adam@377 1274 Fixpoint getNat' (types'' : list Set) (natIndex : nat)
adam@377 1275 : hlist (fun x : Set => x) (update types'' natIndex nat) -> nat :=
adam@377 1276 match types'' with
adam@378 1277 | nil => skipCopies natIndex
adam@377 1278 | t :: types0 =>
adam@377 1279 match natIndex return hlist (fun x : Set => x)
adam@377 1280 (update (t :: types0) natIndex nat) -> nat with
adam@377 1281 | O => fun vs => hhd vs
adam@377 1282 | S natIndex' => fun vs => getNat' types0 natIndex' (htl vs)
adam@377 1283 end
adam@377 1284 end.
adam@377 1285 End withTypes'.
adam@377 1286
adam@398 1287 (** 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 1288
adam@377 1289 Theorem getNat_is_reasonable : getNat' myTypes myNatIndex myValues = 3.
adam@377 1290 reflexivity.
adam@377 1291 Qed.
adam@377 1292
adam@377 1293 (** The same parameters as before work without alteration, and we avoid use of axioms. *)