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author Adam Chlipala <adam@chlipala.net>
date Mon, 22 Oct 2012 14:23:52 -0400
parents b750ec0a8edb
children 40a9a36844d6
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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 It is natural enough that zero be considered as a natural number. *)
adamc@227 41
adamc@227 42 Check nat.
adamc@227 43 (** %\vspace{-.15in}% [[
adamc@227 44 nat
adamc@227 45 : Set
adamc@227 46 ]]
adamc@227 47
adam@429 48 From a set theory perspective, it is unsurprising to consider the natural numbers as a "set." *)
adamc@227 49
adamc@227 50 Check Set.
adamc@227 51 (** %\vspace{-.15in}% [[
adamc@227 52 Set
adamc@227 53 : Type
adamc@227 54 ]]
adamc@227 55
adam@409 56 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 57
adamc@227 58 Check Type.
adamc@227 59 (** %\vspace{-.15in}% [[
adamc@227 60 Type
adamc@227 61 : Type
adamc@227 62 ]]
adamc@227 63
adam@429 64 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 65
adam@343 66 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 67
adamc@227 68 Set Printing Universes.
adamc@227 69
adamc@227 70 Check nat.
adamc@227 71 (** %\vspace{-.15in}% [[
adamc@227 72 nat
adamc@227 73 : Set
adam@302 74 ]]
adam@398 75 *)
adamc@227 76
adamc@227 77 Check Set.
adamc@227 78 (** %\vspace{-.15in}% [[
adamc@227 79 Set
adamc@227 80 : Type $ (0)+1 ^
adam@302 81 ]]
adam@302 82 *)
adamc@227 83
adamc@227 84 Check Type.
adamc@227 85 (** %\vspace{-.15in}% [[
adamc@227 86 Type $ Top.3 ^
adamc@227 87 : Type $ (Top.3)+1 ^
adamc@227 88 ]]
adamc@227 89
adam@429 90 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 91
adam@398 92 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 93
adam@409 94 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 95
adam@409 96 Another crucial concept in CIC is%\index{predicativity}% _predicativity_. Consider these queries. *)
adamc@227 97
adamc@227 98 Check forall T : nat, fin T.
adamc@227 99 (** %\vspace{-.15in}% [[
adamc@227 100 forall T : nat, fin T
adamc@227 101 : Set
adam@302 102 ]]
adam@302 103 *)
adamc@227 104
adamc@227 105 Check forall T : Set, T.
adamc@227 106 (** %\vspace{-.15in}% [[
adamc@227 107 forall T : Set, T
adamc@227 108 : Type $ max(0, (0)+1) ^
adam@302 109 ]]
adam@302 110 *)
adamc@227 111
adamc@227 112 Check forall T : Type, T.
adamc@227 113 (** %\vspace{-.15in}% [[
adamc@227 114 forall T : Type $ Top.9 ^ , T
adamc@227 115 : Type $ max(Top.9, (Top.9)+1) ^
adamc@227 116 ]]
adamc@227 117
adamc@227 118 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@227 119
adam@287 120 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@227 121
adamc@227 122 Definition id (T : Set) (x : T) : T := x.
adamc@227 123
adamc@227 124 Check id 0.
adamc@227 125 (** %\vspace{-.15in}% [[
adamc@227 126 id 0
adamc@227 127 : nat
adamc@227 128
adamc@227 129 Check id Set.
adam@343 130 ]]
adamc@227 131
adam@343 132 <<
adamc@227 133 Error: Illegal application (Type Error):
adamc@227 134 ...
adam@343 135 The 1st term has type "Type (* (Top.15)+1 *)" which should be coercible to "Set".
adam@343 136 >>
adamc@227 137
adam@343 138 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@227 139
adamc@227 140 Reset id.
adamc@227 141 Definition id (T : Type) (x : T) : T := x.
adamc@227 142 Check id 0.
adamc@227 143 (** %\vspace{-.15in}% [[
adamc@227 144 id 0
adamc@227 145 : nat
adam@302 146 ]]
adam@302 147 *)
adamc@227 148
adamc@227 149 Check id Set.
adamc@227 150 (** %\vspace{-.15in}% [[
adamc@227 151 id Set
adamc@227 152 : Type $ Top.17 ^
adam@302 153 ]]
adam@302 154 *)
adamc@227 155
adamc@227 156 Check id Type.
adamc@227 157 (** %\vspace{-.15in}% [[
adamc@227 158 id Type $ Top.18 ^
adamc@227 159 : Type $ Top.19 ^
adam@302 160 ]]
adam@302 161 *)
adamc@227 162
adamc@227 163 (** 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@227 164 [[
adamc@227 165 Check id id.
adam@343 166 ]]
adamc@227 167
adam@343 168 <<
adamc@227 169 Error: Universe inconsistency (cannot enforce Top.16 < Top.16).
adam@343 170 >>
adamc@227 171
adam@429 172 %\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@227 173
adamc@227 174
adamc@227 175 (** ** Inductive Definitions *)
adamc@227 176
adamc@227 177 (** 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@227 178 [[
adamc@227 179 Inductive exp : Set -> Set :=
adamc@227 180 | Const : forall T : Set, T -> exp T
adamc@227 181 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 182 | Eq : forall T, exp T -> exp T -> exp bool.
adam@343 183 ]]
adamc@227 184
adam@343 185 <<
adamc@227 186 Error: Large non-propositional inductive types must be in Type.
adam@343 187 >>
adamc@227 188
adam@409 189 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@227 190
adamc@227 191 Inductive exp : Type -> Type :=
adamc@227 192 | Const : forall T, T -> exp T
adamc@227 193 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 194 | Eq : forall T, exp T -> exp T -> exp bool.
adamc@227 195
adamc@228 196 (** 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@228 197
adamc@228 198 Our new definition is accepted. We can build some sample expressions. *)
adamc@227 199
adamc@227 200 Check Const 0.
adamc@227 201 (** %\vspace{-.15in}% [[
adamc@227 202 Const 0
adamc@227 203 : exp nat
adam@302 204 ]]
adam@302 205 *)
adamc@227 206
adamc@227 207 Check Pair (Const 0) (Const tt).
adamc@227 208 (** %\vspace{-.15in}% [[
adamc@227 209 Pair (Const 0) (Const tt)
adamc@227 210 : exp (nat * unit)
adam@302 211 ]]
adam@302 212 *)
adamc@227 213
adamc@227 214 Check Eq (Const Set) (Const Type).
adamc@227 215 (** %\vspace{-.15in}% [[
adamc@228 216 Eq (Const Set) (Const Type $ Top.59 ^ )
adamc@227 217 : exp bool
adamc@227 218 ]]
adamc@227 219
adamc@227 220 We can check many expressions, including fancy expressions that include types. However, it is not hard to hit a type-checking wall.
adamc@227 221 [[
adamc@227 222 Check Const (Const O).
adam@343 223 ]]
adamc@227 224
adam@343 225 <<
adamc@227 226 Error: Universe inconsistency (cannot enforce Top.42 < Top.42).
adam@343 227 >>
adamc@227 228
adamc@227 229 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@417 230 (** [[
adamc@227 231 Print exp.
adam@417 232 ]]
adam@444 233 %\vspace{-.15in}%[[
adamc@227 234 Inductive exp
adamc@227 235 : Type $ Top.8 ^ ->
adamc@227 236 Type
adamc@227 237 $ max(0, (Top.11)+1, (Top.14)+1, (Top.15)+1, (Top.19)+1) ^ :=
adamc@227 238 Const : forall T : Type $ Top.11 ^ , T -> exp T
adamc@227 239 | Pair : forall (T1 : Type $ Top.14 ^ ) (T2 : Type $ Top.15 ^ ),
adamc@227 240 exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 241 | Eq : forall T : Type $ Top.19 ^ , exp T -> exp T -> exp bool
adamc@227 242 ]]
adamc@227 243
adam@398 244 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@227 245
adam@429 246 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@227 247
adamc@227 248 Print Universes.
adamc@227 249 (** %\vspace{-.15in}% [[
adamc@227 250 Top.19 < Top.9 <= Top.8
adamc@227 251 Top.15 < Top.9 <= Top.8 <= Coq.Init.Datatypes.38
adamc@227 252 Top.14 < Top.9 <= Top.8 <= Coq.Init.Datatypes.37
adamc@227 253 Top.11 < Top.9 <= Top.8
adamc@227 254 ]]
adamc@227 255
adam@343 256 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@227 257
adamc@227 258 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@227 259
adam@417 260 (* begin hide *)
adam@437 261 (* begin thide *)
adam@417 262 Inductive prod := pair.
adam@417 263 Reset prod.
adam@437 264 (* end thide *)
adam@417 265 (* end hide *)
adam@417 266
adam@444 267 (** %\vspace{-.3in}%[[
adamc@227 268 Print prod.
adam@417 269 ]]
adam@444 270 %\vspace{-.15in}%[[
adamc@227 271 Inductive prod (A : Type $ Coq.Init.Datatypes.37 ^ )
adamc@227 272 (B : Type $ Coq.Init.Datatypes.38 ^ )
adamc@227 273 : Type $ max(Coq.Init.Datatypes.37, Coq.Init.Datatypes.38) ^ :=
adamc@227 274 pair : A -> B -> A * B
adamc@227 275 ]]
adamc@227 276
adamc@227 277 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@227 278
adamc@227 279 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@227 280
adamc@227 281 %\medskip%
adamc@227 282
adam@398 283 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@227 284
adamc@231 285 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@227 286
adamc@227 287 Check (nat, (Type, Set)).
adamc@227 288 (** %\vspace{-.15in}% [[
adamc@227 289 (nat, (Type $ Top.44 ^ , Set))
adamc@227 290 : Set * (Type $ Top.45 ^ * Type $ Top.46 ^ )
adamc@227 291 ]]
adamc@227 292
adamc@227 293 The same cannot be done with a counterpart to [prod] that does not use parameters. *)
adamc@227 294
adamc@227 295 Inductive prod' : Type -> Type -> Type :=
adamc@227 296 | pair' : forall A B : Type, A -> B -> prod' A B.
adam@444 297 (** %\vspace{-.15in}%[[
adamc@227 298 Check (pair' nat (pair' Type Set)).
adam@343 299 ]]
adamc@227 300
adam@343 301 <<
adamc@227 302 Error: Universe inconsistency (cannot enforce Top.51 < Top.51).
adam@343 303 >>
adamc@227 304
adamc@233 305 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@233 306
adam@343 307 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@233 308
adamc@233 309 Inductive foo (A : Type) : Type :=
adamc@233 310 | Foo : A -> foo A.
adamc@229 311
adamc@229 312 (* begin hide *)
adamc@229 313 Unset Printing Universes.
adamc@229 314 (* end hide *)
adamc@229 315
adamc@233 316 Check foo nat.
adamc@233 317 (** %\vspace{-.15in}% [[
adamc@233 318 foo nat
adamc@233 319 : Set
adam@302 320 ]]
adam@302 321 *)
adamc@233 322
adamc@233 323 Check foo Set.
adamc@233 324 (** %\vspace{-.15in}% [[
adamc@233 325 foo Set
adamc@233 326 : Type
adam@302 327 ]]
adam@302 328 *)
adamc@233 329
adamc@233 330 Check foo True.
adamc@233 331 (** %\vspace{-.15in}% [[
adamc@233 332 foo True
adamc@233 333 : Prop
adamc@233 334 ]]
adamc@233 335
adam@429 336 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@233 337
adamc@233 338 Imitation polymorphism can be confusing in some contexts. For instance, it is what is responsible for this weird behavior. *)
adamc@233 339
adamc@233 340 Inductive bar : Type := Bar : bar.
adamc@233 341
adamc@233 342 Check bar.
adamc@233 343 (** %\vspace{-.15in}% [[
adamc@233 344 bar
adamc@233 345 : Prop
adamc@233 346 ]]
adamc@233 347
adamc@233 348 The type that Coq comes up with may be used in strictly more contexts than the type one might have expected. *)
adamc@233 349
adamc@229 350
adam@388 351 (** ** Deciphering Baffling Messages About Inability to Unify *)
adam@388 352
adam@388 353 (** 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@388 354
adam@388 355 Theorem symmetry : forall A B : Type,
adam@388 356 A = B
adam@388 357 -> B = A.
adam@388 358 intros ? ? H; rewrite H; reflexivity.
adam@388 359 Qed.
adam@388 360
adam@388 361 (** Let us attempt an admittedly silly proof of the following theorem. *)
adam@388 362
adam@388 363 Theorem illustrative_but_silly_detour : unit = unit.
adam@444 364 (** %\vspace{-.25in}%[[
adam@444 365 apply symmetry.
adam@388 366 ]]
adam@388 367 <<
adam@388 368 Error: Impossible to unify "?35 = ?34" with "unit = unit".
adam@388 369 >>
adam@388 370
adam@458 371 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@388 372
adam@388 373 The following command is the secret to getting better error messages in such cases: *)
adam@388 374
adam@388 375 Set Printing All.
adam@444 376 (** %\vspace{-.15in}%[[
adam@444 377 apply symmetry.
adam@388 378 ]]
adam@388 379 <<
adam@388 380 Error: Impossible to unify "@eq Type ?46 ?45" with "@eq Set unit unit".
adam@388 381 >>
adam@388 382
adam@398 383 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@388 384
adam@388 385 Abort.
adam@388 386
adam@388 387 (** 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@388 388
adam@388 389 Theorem illustrative_but_silly_detour : (unit : Type) = unit.
adam@388 390 apply symmetry; reflexivity.
adam@388 391 Qed.
adam@388 392
adam@388 393 (** 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@388 394
adam@388 395 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@388 396
adam@388 397 Unset Printing All.
adam@388 398
adam@388 399 Theorem ex_symmetry : (exists x, x = 0) -> (exists x, 0 = x).
adam@435 400 eexists.
adam@388 401 (** %\vspace{-.15in}%[[
adam@388 402 H : exists x : nat, x = 0
adam@388 403 ============================
adam@388 404 0 = ?98
adam@388 405 ]]
adam@388 406 *)
adam@388 407
adam@388 408 destruct H.
adam@388 409 (** %\vspace{-.15in}%[[
adam@388 410 x : nat
adam@388 411 H : x = 0
adam@388 412 ============================
adam@388 413 0 = ?99
adam@388 414 ]]
adam@388 415 *)
adam@388 416
adam@444 417 (** %\vspace{-.2in}%[[
adam@444 418 symmetry; exact H.
adam@388 419 ]]
adam@388 420
adam@388 421 <<
adam@388 422 Error: In environment
adam@388 423 x : nat
adam@388 424 H : x = 0
adam@388 425 The term "H" has type "x = 0" while it is expected to have type
adam@388 426 "?99 = 0".
adam@388 427 >>
adam@388 428
adam@398 429 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@388 430
adam@388 431 Restart.
adam@388 432 destruct 1 as [x]; apply ex_intro with x; symmetry; assumption.
adam@388 433 Qed.
adam@388 434
adam@429 435 (** 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@388 436
adam@388 437
adamc@229 438 (** * The [Prop] Universe *)
adamc@229 439
adam@429 440 (** 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@229 441
adamc@229 442 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@229 443
adamc@229 444 Print sig.
adamc@229 445 (** %\vspace{-.15in}% [[
adamc@229 446 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@229 447 exist : forall x : A, P x -> sig P
adam@302 448 ]]
adam@302 449 *)
adamc@229 450
adamc@229 451 Print ex.
adamc@229 452 (** %\vspace{-.15in}% [[
adamc@229 453 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@229 454 ex_intro : forall x : A, P x -> ex P
adamc@229 455 ]]
adamc@229 456
adamc@229 457 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@229 458
adamc@229 459 Definition projS A (P : A -> Prop) (x : sig P) : A :=
adamc@229 460 match x with
adamc@229 461 | exist v _ => v
adamc@229 462 end.
adamc@229 463
adam@429 464 (* begin hide *)
adam@437 465 (* begin thide *)
adam@429 466 Definition projE := O.
adam@437 467 (* end thide *)
adam@429 468 (* end hide *)
adam@429 469
adamc@229 470 (** We run into trouble with a version that has been changed to work with [ex].
adamc@229 471 [[
adamc@229 472 Definition projE A (P : A -> Prop) (x : ex P) : A :=
adamc@229 473 match x with
adamc@229 474 | ex_intro v _ => v
adamc@229 475 end.
adam@343 476 ]]
adamc@229 477
adam@343 478 <<
adamc@229 479 Error:
adamc@229 480 Incorrect elimination of "x" in the inductive type "ex":
adamc@229 481 the return type has sort "Type" while it should be "Prop".
adamc@229 482 Elimination of an inductive object of sort Prop
adamc@229 483 is not allowed on a predicate in sort Type
adamc@229 484 because proofs can be eliminated only to build proofs.
adam@343 485 >>
adamc@229 486
adam@429 487 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@229 488
adamc@229 489 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@229 490
adam@398 491 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@229 492
adamc@229 493 Definition sym_sig (x : sig (fun n => n = 0)) : sig (fun n => 0 = n) :=
adamc@229 494 match x with
adamc@229 495 | exist n pf => exist _ n (sym_eq pf)
adamc@229 496 end.
adamc@229 497
adamc@229 498 Extraction sym_sig.
adamc@229 499 (** <<
adamc@229 500 (** val sym_sig : nat -> nat **)
adamc@229 501
adamc@229 502 let sym_sig x = x
adamc@229 503 >>
adamc@229 504
adamc@229 505 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@229 506
adamc@229 507 Definition sym_ex (x : ex (fun n => n = 0)) : ex (fun n => 0 = n) :=
adamc@229 508 match x with
adamc@229 509 | ex_intro n pf => ex_intro _ n (sym_eq pf)
adamc@229 510 end.
adamc@229 511
adamc@229 512 Extraction sym_ex.
adamc@229 513 (** <<
adamc@229 514 (** val sym_ex : __ **)
adamc@229 515
adamc@229 516 let sym_ex = __
adamc@229 517 >>
adamc@229 518
adam@435 519 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@229 520
adam@419 521 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@229 522
adam@398 523 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@229 524
adamc@229 525 %\medskip%
adamc@229 526
adam@409 527 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@229 528
adamc@229 529 Check forall P Q : Prop, P \/ Q -> Q \/ P.
adamc@229 530 (** %\vspace{-.15in}% [[
adamc@229 531 forall P Q : Prop, P \/ Q -> Q \/ P
adamc@229 532 : Prop
adamc@229 533 ]]
adamc@229 534
adamc@230 535 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@230 536
adamc@230 537 Impredicativity also allows us to implement a version of our earlier [exp] type that does not suffer from the weakness that we found. *)
adamc@230 538
adamc@230 539 Inductive expP : Type -> Prop :=
adamc@230 540 | ConstP : forall T, T -> expP T
adamc@230 541 | PairP : forall T1 T2, expP T1 -> expP T2 -> expP (T1 * T2)
adamc@230 542 | EqP : forall T, expP T -> expP T -> expP bool.
adamc@230 543
adamc@230 544 Check ConstP 0.
adamc@230 545 (** %\vspace{-.15in}% [[
adamc@230 546 ConstP 0
adamc@230 547 : expP nat
adam@302 548 ]]
adam@302 549 *)
adamc@230 550
adamc@230 551 Check PairP (ConstP 0) (ConstP tt).
adamc@230 552 (** %\vspace{-.15in}% [[
adamc@230 553 PairP (ConstP 0) (ConstP tt)
adamc@230 554 : expP (nat * unit)
adam@302 555 ]]
adam@302 556 *)
adamc@230 557
adamc@230 558 Check EqP (ConstP Set) (ConstP Type).
adamc@230 559 (** %\vspace{-.15in}% [[
adamc@230 560 EqP (ConstP Set) (ConstP Type)
adamc@230 561 : expP bool
adam@302 562 ]]
adam@302 563 *)
adamc@230 564
adamc@230 565 Check ConstP (ConstP O).
adamc@230 566 (** %\vspace{-.15in}% [[
adamc@230 567 ConstP (ConstP 0)
adamc@230 568 : expP (expP nat)
adamc@230 569 ]]
adamc@230 570
adam@287 571 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@230 572
adamc@230 573 Inductive eqPlus : forall T, T -> T -> Prop :=
adamc@230 574 | Base : forall T (x : T), eqPlus x x
adamc@230 575 | Func : forall dom ran (f1 f2 : dom -> ran),
adamc@230 576 (forall x : dom, eqPlus (f1 x) (f2 x))
adamc@230 577 -> eqPlus f1 f2.
adamc@230 578
adamc@230 579 Check (Base 0).
adamc@230 580 (** %\vspace{-.15in}% [[
adamc@230 581 Base 0
adamc@230 582 : eqPlus 0 0
adam@302 583 ]]
adam@302 584 *)
adamc@230 585
adamc@230 586 Check (Func (fun n => n) (fun n => 0 + n) (fun n => Base n)).
adamc@230 587 (** %\vspace{-.15in}% [[
adamc@230 588 Func (fun n : nat => n) (fun n : nat => 0 + n) (fun n : nat => Base n)
adamc@230 589 : eqPlus (fun n : nat => n) (fun n : nat => 0 + n)
adam@302 590 ]]
adam@302 591 *)
adamc@230 592
adamc@230 593 Check (Base (Base 1)).
adamc@230 594 (** %\vspace{-.15in}% [[
adamc@230 595 Base (Base 1)
adamc@230 596 : eqPlus (Base 1) (Base 1)
adam@302 597 ]]
adam@302 598 *)
adamc@230 599
adam@343 600 (** Stating equality facts about proofs may seem baroque, but we have already seen its utility in the chapter on reasoning about equality proofs. *)
adam@343 601
adamc@230 602
adamc@230 603 (** * Axioms *)
adamc@230 604
adam@409 605 (** 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@230 606
adamc@230 607 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@230 608
adamc@230 609 (** ** The Basics *)
adamc@230 610
adam@343 611 (** One simple example of a useful axiom is the %\index{law of the excluded middle}%law of the excluded middle. *)
adamc@230 612
adamc@230 613 Require Import Classical_Prop.
adamc@230 614 Print classic.
adamc@230 615 (** %\vspace{-.15in}% [[
adamc@230 616 *** [ classic : forall P : Prop, P \/ ~ P ]
adamc@230 617 ]]
adamc@230 618
adam@343 619 In the implementation of module [Classical_Prop], this axiom was defined with the command%\index{Vernacular commands!Axiom}% *)
adamc@230 620
adamc@230 621 Axiom classic : forall P : Prop, P \/ ~ P.
adamc@230 622
adam@343 623 (** 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@230 624
adam@458 625 Parameter num : nat.
adam@458 626 Axiom positive : num > 0.
adam@458 627 Reset num.
adamc@230 628
adam@429 629 (** 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@230 630
adam@409 631 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@230 632
adam@287 633 Axiom not_classic : ~ forall P : Prop, P \/ ~ P.
adamc@230 634
adamc@230 635 Theorem uhoh : False.
adam@287 636 generalize classic not_classic; tauto.
adamc@230 637 Qed.
adamc@230 638
adamc@230 639 Theorem uhoh_again : 1 + 1 = 3.
adamc@230 640 destruct uhoh.
adamc@230 641 Qed.
adamc@230 642
adamc@230 643 Reset not_classic.
adamc@230 644
adam@429 645 (** 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@230 646
adam@475 647 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@230 648
adam@398 649 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@230 650
adam@343 651 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@230 652
adamc@230 653 Theorem t1 : forall P : Prop, P -> ~ ~ P.
adamc@230 654 tauto.
adamc@230 655 Qed.
adamc@230 656
adamc@230 657 Print Assumptions t1.
adam@343 658 (** <<
adamc@230 659 Closed under the global context
adam@343 660 >>
adam@302 661 *)
adamc@230 662
adamc@230 663 Theorem t2 : forall P : Prop, ~ ~ P -> P.
adam@444 664 (** %\vspace{-.25in}%[[
adamc@230 665 tauto.
adam@343 666 ]]
adam@343 667 <<
adamc@230 668 Error: tauto failed.
adam@343 669 >>
adam@302 670 *)
adamc@230 671 intro P; destruct (classic P); tauto.
adamc@230 672 Qed.
adamc@230 673
adamc@230 674 Print Assumptions t2.
adamc@230 675 (** %\vspace{-.15in}% [[
adamc@230 676 Axioms:
adamc@230 677 classic : forall P : Prop, P \/ ~ P
adamc@230 678 ]]
adamc@230 679
adam@398 680 It is possible to avoid this dependence in some specific cases, where excluded middle _is_ provable, for decidable families of propositions. *)
adamc@230 681
adam@287 682 Theorem nat_eq_dec : forall n m : nat, n = m \/ n <> m.
adamc@230 683 induction n; destruct m; intuition; generalize (IHn m); intuition.
adamc@230 684 Qed.
adamc@230 685
adamc@230 686 Theorem t2' : forall n m : nat, ~ ~ (n = m) -> n = m.
adam@287 687 intros n m; destruct (nat_eq_dec n m); tauto.
adamc@230 688 Qed.
adamc@230 689
adamc@230 690 Print Assumptions t2'.
adam@343 691 (** <<
adamc@230 692 Closed under the global context
adam@343 693 >>
adamc@230 694
adamc@230 695 %\bigskip%
adamc@230 696
adam@409 697 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@230 698
adamc@230 699 Require Import ProofIrrelevance.
adamc@230 700 Print proof_irrelevance.
adam@458 701
adamc@230 702 (** %\vspace{-.15in}% [[
adamc@230 703 *** [ proof_irrelevance : forall (P : Prop) (p1 p2 : P), p1 = p2 ]
adamc@230 704 ]]
adamc@230 705
adam@458 706 This axiom asserts that any two proofs of the same proposition are equal. Recall this example function from Chapter 6. *)
adamc@230 707
adamc@230 708 (* begin hide *)
adamc@230 709 Lemma zgtz : 0 > 0 -> False.
adamc@230 710 crush.
adamc@230 711 Qed.
adamc@230 712 (* end hide *)
adamc@230 713
adamc@230 714 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@230 715 match n with
adamc@230 716 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@230 717 | S n' => fun _ => n'
adamc@230 718 end.
adamc@230 719
adam@343 720 (** We might want to prove that different proofs of [n > 0] do not lead to different results from our richly typed predecessor function. *)
adamc@230 721
adamc@230 722 Theorem pred_strong1_irrel : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 723 destruct n; crush.
adamc@230 724 Qed.
adamc@230 725
adamc@230 726 (** 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@230 727
adamc@230 728 Theorem pred_strong1_irrel' : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 729 intros; f_equal; apply proof_irrelevance.
adamc@230 730 Qed.
adamc@230 731
adamc@230 732
adamc@230 733 (** %\bigskip%
adamc@230 734
adamc@230 735 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@230 736
adamc@230 737 Require Import Eqdep.
adamc@230 738 Import Eq_rect_eq.
adamc@230 739 Print eq_rect_eq.
adamc@230 740 (** %\vspace{-.15in}% [[
adamc@230 741 *** [ eq_rect_eq :
adamc@230 742 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adamc@230 743 x = eq_rect p Q x p h ]
adamc@230 744 ]]
adamc@230 745
adam@429 746 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@230 747
adam@426 748 Corollary UIP_refl : forall A (x : A) (pf : x = x), pf = eq_refl x.
adam@426 749 intros; replace pf with (eq_rect x (eq x) (eq_refl x) x pf); [
adamc@230 750 symmetry; apply eq_rect_eq
adamc@230 751 | exact (match pf as pf' return match pf' in _ = y return x = y with
adam@426 752 | eq_refl => eq_refl x
adamc@230 753 end = pf' with
adam@426 754 | eq_refl => eq_refl _
adamc@230 755 end) ].
adamc@230 756 Qed.
adamc@230 757
adamc@230 758 Corollary UIP : forall A (x y : A) (pf1 pf2 : x = y), pf1 = pf2.
adamc@230 759 intros; generalize pf1 pf2; subst; intros;
adamc@230 760 match goal with
adamc@230 761 | [ |- ?pf1 = ?pf2 ] => rewrite (UIP_refl pf1); rewrite (UIP_refl pf2); reflexivity
adamc@230 762 end.
adamc@230 763 Qed.
adamc@230 764
adam@436 765 (* begin hide *)
adam@437 766 (* begin thide *)
adam@436 767 Require Eqdep_dec.
adam@437 768 (* end thide *)
adam@436 769 (* end hide *)
adam@436 770
adamc@231 771 (** 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@230 772
adamc@230 773 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@230 774
adamc@230 775 %\bigskip%
adamc@230 776
adamc@230 777 There are two more basic axioms that are often assumed, to avoid complications that do not arise in set theory. *)
adamc@230 778
adamc@230 779 Require Import FunctionalExtensionality.
adamc@230 780 Print functional_extensionality_dep.
adamc@230 781 (** %\vspace{-.15in}% [[
adamc@230 782 *** [ functional_extensionality_dep :
adamc@230 783 forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
adamc@230 784 (forall x : A, f x = g x) -> f = g ]
adamc@230 785
adamc@230 786 ]]
adamc@230 787
adamc@230 788 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@230 789
adam@343 790 A simple corollary shows that the same property applies to predicates. *)
adamc@230 791
adamc@230 792 Corollary predicate_extensionality : forall (A : Type) (B : A -> Prop) (f g : forall x : A, B x),
adamc@230 793 (forall x : A, f x = g x) -> f = g.
adamc@230 794 intros; apply functional_extensionality_dep; assumption.
adamc@230 795 Qed.
adamc@230 796
adam@343 797 (** In some cases, one might prefer to assert this corollary as the axiom, to restrict the consequences to proofs and not programs. *)
adam@343 798
adamc@230 799
adamc@230 800 (** ** Axioms of Choice *)
adamc@230 801
adam@343 802 (** 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@230 803
adam@398 804 First, it is possible to implement a choice operator _without_ axioms in some potentially surprising cases. *)
adamc@230 805
adamc@230 806 Require Import ConstructiveEpsilon.
adamc@230 807 Check constructive_definite_description.
adamc@230 808 (** %\vspace{-.15in}% [[
adamc@230 809 constructive_definite_description
adamc@230 810 : forall (A : Set) (f : A -> nat) (g : nat -> A),
adamc@230 811 (forall x : A, g (f x) = x) ->
adamc@230 812 forall P : A -> Prop,
adamc@230 813 (forall x : A, {P x} + {~ P x}) ->
adamc@230 814 (exists! x : A, P x) -> {x : A | P x}
adam@302 815 ]]
adam@302 816 *)
adamc@230 817
adamc@230 818 Print Assumptions constructive_definite_description.
adam@343 819 (** <<
adamc@230 820 Closed under the global context
adam@343 821 >>
adamc@230 822
adam@398 823 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@230 824
adamc@230 825 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@230 826
adamc@230 827 Require Import ClassicalUniqueChoice.
adamc@230 828 Check dependent_unique_choice.
adamc@230 829 (** %\vspace{-.15in}% [[
adamc@230 830 dependent_unique_choice
adamc@230 831 : forall (A : Type) (B : A -> Type) (R : forall x : A, B x -> Prop),
adamc@230 832 (forall x : A, exists! y : B x, R x y) ->
adam@343 833 exists f : forall x : A, B x,
adam@343 834 forall x : A, R x (f x)
adamc@230 835 ]]
adamc@230 836
adamc@230 837 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@230 838
adam@436 839 (* begin hide *)
adam@437 840 (* begin thide *)
adam@436 841 Require RelationalChoice.
adam@437 842 (* end thide *)
adam@436 843 (* end hide *)
adam@436 844
adamc@230 845 Require Import ClassicalChoice.
adamc@230 846 Check choice.
adamc@230 847 (** %\vspace{-.15in}% [[
adamc@230 848 choice
adamc@230 849 : forall (A B : Type) (R : A -> B -> Prop),
adamc@230 850 (forall x : A, exists y : B, R x y) ->
adamc@230 851 exists f : A -> B, forall x : A, R x (f x)
adam@444 852 ]]
adamc@230 853
adamc@230 854 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@230 855
adamc@230 856 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@230 857
adamc@230 858 Definition choice_Set (A B : Type) (R : A -> B -> Prop) (H : forall x : A, {y : B | R x y})
adamc@230 859 : {f : A -> B | forall x : A, R x (f x)} :=
adamc@230 860 exist (fun f => forall x : A, R x (f x))
adamc@230 861 (fun x => proj1_sig (H x)) (fun x => proj2_sig (H x)).
adamc@230 862
adam@458 863 (** %\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@230 864
adam@429 865 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@230 866
adamc@230 867 %\bigskip%
adamc@230 868
adam@429 869 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@230 870
adamc@230 871 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@230 872
adamc@230 873 (** ** Axioms and Computation *)
adamc@230 874
adam@398 875 (** 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@230 876
adamc@230 877 Definition cast (x y : Set) (pf : x = y) (v : x) : y :=
adamc@230 878 match pf with
adam@426 879 | eq_refl => v
adamc@230 880 end.
adamc@230 881
adamc@230 882 (** Computation over programs that use [cast] can proceed smoothly. *)
adamc@230 883
adam@426 884 Eval compute in (cast (eq_refl (nat -> nat)) (fun n => S n)) 12.
adam@343 885 (** %\vspace{-.15in}%[[
adamc@230 886 = 13
adamc@230 887 : nat
adam@302 888 ]]
adam@302 889 *)
adamc@230 890
adamc@230 891 (** Things do not go as smoothly when we use [cast] with proofs that rely on axioms. *)
adamc@230 892
adamc@230 893 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 894 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 895 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 896 Qed.
adamc@230 897
adamc@230 898 Eval compute in (cast t3 (fun _ => First)) 12.
adam@444 899 (** %\vspace{-.15in}%[[
adamc@230 900 = match t3 in (_ = P) return P with
adam@426 901 | eq_refl => fun n : nat => First
adamc@230 902 end 12
adamc@230 903 : fin (12 + 1)
adamc@230 904 ]]
adamc@230 905
adam@458 906 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@230 907
adamc@230 908 Reset t3.
adamc@230 909
adamc@230 910 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 911 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 912 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 913 Defined.
adamc@230 914
adamc@230 915 Eval compute in (cast t3 (fun _ => First)) 12.
adam@444 916 (** %\vspace{-.15in}%[[
adamc@230 917 = match
adamc@230 918 match
adamc@230 919 match
adamc@230 920 functional_extensionality
adamc@230 921 ....
adamc@230 922 ]]
adamc@230 923
adam@398 924 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@230 925
adamc@230 926 If we are careful in using tactics to prove an equality, we can still compute with casts over the proof. *)
adamc@230 927
adamc@230 928 Lemma plus1 : forall n, S n = n + 1.
adamc@230 929 induction n; simpl; intuition.
adamc@230 930 Defined.
adamc@230 931
adamc@230 932 Theorem t4 : forall n, fin (S n) = fin (n + 1).
adamc@230 933 intro; f_equal; apply plus1.
adamc@230 934 Defined.
adamc@230 935
adamc@230 936 Eval compute in cast (t4 13) First.
adamc@230 937 (** %\vspace{-.15in}% [[
adamc@230 938 = First
adamc@230 939 : fin (13 + 1)
adam@302 940 ]]
adam@343 941
adam@426 942 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@343 943
adam@344 944
adam@344 945 (** ** Methods for Avoiding Axioms *)
adam@344 946
adam@409 947 (** 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@344 948
adam@429 949 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@344 950
adam@344 951 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@344 952
adam@344 953 Theorem fin_cases : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 954 intros; dep_destruct f; eauto.
adam@344 955 Qed.
adam@344 956
adam@429 957 (* begin hide *)
adam@429 958 Require Import JMeq.
adam@437 959 (* begin thide *)
adam@429 960 Definition jme := (JMeq, JMeq_eq).
adam@437 961 (* end thide *)
adam@429 962 (* end hide *)
adam@429 963
adam@344 964 Print Assumptions fin_cases.
adam@344 965 (** %\vspace{-.15in}%[[
adam@344 966 Axioms:
adam@429 967 JMeq_eq : forall (A : Type) (x y : A), JMeq x y -> x = y
adam@344 968 ]]
adam@344 969
adam@344 970 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@344 971
adam@344 972 (* begin thide *)
adam@344 973 Lemma fin_cases_again' : forall n (f : fin n),
adam@344 974 match n return fin n -> Prop with
adam@344 975 | O => fun _ => False
adam@344 976 | S n' => fun f => f = First \/ exists f', f = Next f'
adam@344 977 end f.
adam@344 978 destruct f; eauto.
adam@344 979 Qed.
adam@344 980
adam@344 981 (** 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@344 982
adam@344 983 Theorem fin_cases_again : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 984 intros; exact (fin_cases_again' f).
adam@344 985 Qed.
adam@344 986 (* end thide *)
adam@344 987
adam@344 988 Print Assumptions fin_cases_again.
adam@344 989 (** %\vspace{-.15in}%
adam@344 990 <<
adam@344 991 Closed under the global context
adam@344 992 >>
adam@344 993
adam@345 994 *)
adam@345 995
adam@345 996 (* begin thide *)
adam@345 997 (** 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@345 998
adam@429 999 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 1000
adam@345 1001 Definition finOut n (f : fin n) : match n return fin n -> Type with
adam@345 1002 | O => fun _ => Empty_set
adam@345 1003 | _ => fun f => {f' : _ | f = Next f'} + {f = First}
adam@345 1004 end f :=
adam@345 1005 match f with
adam@426 1006 | First _ => inright _ (eq_refl _)
adam@426 1007 | Next _ f' => inleft _ (exist _ f' (eq_refl _))
adam@345 1008 end.
adam@345 1009 (* end thide *)
adam@345 1010
adam@345 1011 (** 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 1012
adam@344 1013 Inductive formula : list Type -> Type :=
adam@344 1014 | Inject : forall Ts, Prop -> formula Ts
adam@344 1015 | VarEq : forall T Ts, T -> formula (T :: Ts)
adam@344 1016 | Lift : forall T Ts, formula Ts -> formula (T :: Ts)
adam@344 1017 | Forall : forall T Ts, formula (T :: Ts) -> formula Ts
adam@344 1018 | And : forall Ts, formula Ts -> formula Ts -> formula Ts.
adam@344 1019
adam@344 1020 (** 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 1021
adam@344 1022 Inductive proof : formula nil -> Prop :=
adam@344 1023 | PInject : forall (P : Prop), P -> proof (Inject nil P)
adam@344 1024 | PAnd : forall p q, proof p -> proof q -> proof (And p q).
adam@344 1025
adam@429 1026 (** Let us prove a lemma showing that a "[P /\ Q -> P]" rule is derivable within the rules of [proof]. *)
adam@344 1027
adam@344 1028 Theorem proj1 : forall p q, proof (And p q) -> proof p.
adam@344 1029 destruct 1.
adam@344 1030 (** %\vspace{-.15in}%[[
adam@344 1031 p : formula nil
adam@344 1032 q : formula nil
adam@344 1033 P : Prop
adam@344 1034 H : P
adam@344 1035 ============================
adam@344 1036 proof p
adam@344 1037 ]]
adam@344 1038 *)
adam@344 1039
adam@344 1040 (** 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 1041
adam@344 1042 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 1043
adam@344 1044 Restart.
adam@344 1045 Require Import Program.
adam@344 1046 intros ? ? H; dependent destruction H; auto.
adam@344 1047 Qed.
adam@344 1048
adam@344 1049 Print Assumptions proj1.
adam@344 1050 (** %\vspace{-.15in}%[[
adam@344 1051 Axioms:
adam@344 1052 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1053 x = eq_rect p Q x p h
adam@344 1054 ]]
adam@344 1055
adam@344 1056 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 1057
adam@344 1058 Lemma proj1_again' : forall r, proof r
adam@344 1059 -> forall p q, r = And p q -> proof p.
adam@344 1060 destruct 1; crush.
adam@344 1061 (** %\vspace{-.15in}%[[
adam@344 1062 H0 : Inject [] P = And p q
adam@344 1063 ============================
adam@344 1064 proof p
adam@344 1065 ]]
adam@344 1066
adam@344 1067 The first goal looks reasonable. Hypothesis [H0] is clearly contradictory, as [discriminate] can show. *)
adam@344 1068
adam@344 1069 discriminate.
adam@344 1070 (** %\vspace{-.15in}%[[
adam@344 1071 H : proof p
adam@344 1072 H1 : And p q = And p0 q0
adam@344 1073 ============================
adam@344 1074 proof p0
adam@344 1075 ]]
adam@344 1076
adam@344 1077 It looks like we are almost done. Hypothesis [H1] gives [p = p0] by injectivity of constructors, and then [H] finishes the case. *)
adam@344 1078
adam@344 1079 injection H1; intros.
adam@344 1080
adam@429 1081 (* begin hide *)
adam@437 1082 (* begin thide *)
adam@429 1083 Definition existT' := existT.
adam@437 1084 (* end thide *)
adam@429 1085 (* end hide *)
adam@429 1086
adam@429 1087 (** Unfortunately, the "equality" that we expected between [p] and [p0] comes in a strange form:
adam@344 1088
adam@344 1089 [[
adam@344 1090 H3 : existT (fun Ts : list Type => formula Ts) []%list p =
adam@344 1091 existT (fun Ts : list Type => formula Ts) []%list p0
adam@344 1092 ============================
adam@344 1093 proof p0
adam@344 1094 ]]
adam@344 1095
adam@345 1096 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 1097
adam@344 1098 How exactly does an axiom come into the picture here? Let us ask [crush] to finish the proof. *)
adam@344 1099
adam@344 1100 crush.
adam@344 1101 Qed.
adam@344 1102
adam@344 1103 Print Assumptions proj1_again'.
adam@344 1104 (** %\vspace{-.15in}%[[
adam@344 1105 Axioms:
adam@344 1106 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1107 x = eq_rect p Q x p h
adam@344 1108 ]]
adam@344 1109
adam@344 1110 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 1111
adam@344 1112 Hope is not lost, however. We can produce an even stranger looking lemma, which gives us the theorem without axioms. *)
adam@344 1113
adam@344 1114 Lemma proj1_again'' : forall r, proof r
adam@344 1115 -> match r with
adam@344 1116 | And Ps p _ => match Ps return formula Ps -> Prop with
adam@344 1117 | nil => fun p => proof p
adam@344 1118 | _ => fun _ => True
adam@344 1119 end p
adam@344 1120 | _ => True
adam@344 1121 end.
adam@344 1122 destruct 1; auto.
adam@344 1123 Qed.
adam@344 1124
adam@344 1125 Theorem proj1_again : forall p q, proof (And p q) -> proof p.
adam@344 1126 intros ? ? H; exact (proj1_again'' H).
adam@344 1127 Qed.
adam@344 1128
adam@344 1129 Print Assumptions proj1_again.
adam@344 1130 (** <<
adam@344 1131 Closed under the global context
adam@344 1132 >>
adam@344 1133
adam@377 1134 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 1135
adam@377 1136 %\medskip%
adam@377 1137
adam@398 1138 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 1139
adam@377 1140 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 1141
adam@377 1142 Section withTypes.
adam@377 1143 Variable types : list Set.
adam@377 1144
adam@377 1145 (** 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 1146
adam@377 1147 Variable values : hlist (fun x : Set => x) types.
adam@377 1148
adam@377 1149 (** 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 1150
adam@377 1151 Variable natIndex : nat.
adam@377 1152 Variable natIndex_ok : nth_error types natIndex = Some nat.
adam@377 1153
adam@377 1154 (** 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 1155
adam@377 1156 Lemma nth_error_nil : forall A n x,
adam@377 1157 nth_error (@nil A) n = Some x
adam@377 1158 -> False.
adam@377 1159 destruct n; simpl; unfold error; congruence.
adam@377 1160 Defined.
adam@377 1161
adam@377 1162 Implicit Arguments nth_error_nil [A n x].
adam@377 1163
adam@377 1164 Lemma Some_inj : forall A (x y : A),
adam@377 1165 Some x = Some y
adam@377 1166 -> x = y.
adam@377 1167 congruence.
adam@377 1168 Defined.
adam@377 1169
adam@377 1170 Fixpoint getNat (types' : list Set) (values' : hlist (fun x : Set => x) types')
adam@377 1171 (natIndex : nat) : (nth_error types' natIndex = Some nat) -> nat :=
adam@377 1172 match values' with
adam@377 1173 | HNil => fun pf => match nth_error_nil pf with end
adam@377 1174 | HCons t ts x values'' =>
adam@377 1175 match natIndex return nth_error (t :: ts) natIndex = Some nat -> nat with
adam@377 1176 | O => fun pf =>
adam@377 1177 match Some_inj pf in _ = T return T with
adam@426 1178 | eq_refl => x
adam@377 1179 end
adam@377 1180 | S natIndex' => getNat values'' natIndex'
adam@377 1181 end
adam@377 1182 end.
adam@377 1183 End withTypes.
adam@377 1184
adam@377 1185 (** The problem becomes apparent when we experiment with running [getNat] on a concrete [types] list. *)
adam@377 1186
adam@377 1187 Definition myTypes := unit :: nat :: bool :: nil.
adam@377 1188 Definition myValues : hlist (fun x : Set => x) myTypes :=
adam@377 1189 tt ::: 3 ::: false ::: HNil.
adam@377 1190
adam@377 1191 Definition myNatIndex := 1.
adam@377 1192
adam@377 1193 Theorem myNatIndex_ok : nth_error myTypes myNatIndex = Some nat.
adam@377 1194 reflexivity.
adam@377 1195 Defined.
adam@377 1196
adam@377 1197 Eval compute in getNat myValues myNatIndex myNatIndex_ok.
adam@377 1198 (** %\vspace{-.15in}%[[
adam@377 1199 = 3
adam@377 1200 ]]
adam@377 1201
adam@398 1202 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 1203
adam@377 1204 Theorem getNat_is_reasonable : forall pf, getNat myValues myNatIndex pf = 3.
adam@377 1205 intro; compute.
adam@377 1206 (**
adam@377 1207 <<
adam@377 1208 1 subgoal
adam@377 1209 >>
adam@377 1210 %\vspace{-.3in}%[[
adam@377 1211 pf : nth_error myTypes myNatIndex = Some nat
adam@377 1212 ============================
adam@377 1213 match
adam@377 1214 match
adam@377 1215 pf in (_ = y)
adam@377 1216 return (nat = match y with
adam@377 1217 | Some H => H
adam@377 1218 | None => nat
adam@377 1219 end)
adam@377 1220 with
adam@377 1221 | eq_refl => eq_refl
adam@377 1222 end in (_ = T) return T
adam@377 1223 with
adam@377 1224 | eq_refl => 3
adam@377 1225 end = 3
adam@377 1226 ]]
adam@377 1227
adam@377 1228 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 1229
adam@377 1230 Abort.
adam@377 1231
adam@377 1232 (** 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 1233
adam@377 1234 Fixpoint copies A (x : A) (n : nat) : list A :=
adam@377 1235 match n with
adam@377 1236 | O => nil
adam@377 1237 | S n' => x :: copies x n'
adam@377 1238 end.
adam@377 1239
adam@377 1240 Fixpoint update A (ls : list A) (n : nat) (x : A) : list A :=
adam@377 1241 match ls with
adam@377 1242 | nil => copies x n ++ x :: nil
adam@377 1243 | y :: ls' => match n with
adam@377 1244 | O => x :: ls'
adam@377 1245 | S n' => y :: update ls' n' x
adam@377 1246 end
adam@377 1247 end.
adam@377 1248
adam@377 1249 (** Now let us revisit the definition of [getNat]. *)
adam@377 1250
adam@377 1251 Section withTypes'.
adam@377 1252 Variable types : list Set.
adam@377 1253 Variable natIndex : nat.
adam@377 1254
adam@429 1255 (** 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 1256
adam@377 1257 Definition types' := update types natIndex nat.
adam@377 1258
adam@377 1259 Variable values : hlist (fun x : Set => x) types'.
adam@377 1260
adam@377 1261 (** Now a bit of dependent pattern matching helps us rewrite [getNat] in a way that avoids any use of equality proofs. *)
adam@377 1262
adam@378 1263 Fixpoint skipCopies (n : nat)
adam@378 1264 : hlist (fun x : Set => x) (copies nat n ++ nat :: nil) -> nat :=
adam@378 1265 match n with
adam@378 1266 | O => fun vs => hhd vs
adam@378 1267 | S n' => fun vs => skipCopies n' (htl vs)
adam@378 1268 end.
adam@378 1269
adam@377 1270 Fixpoint getNat' (types'' : list Set) (natIndex : nat)
adam@377 1271 : hlist (fun x : Set => x) (update types'' natIndex nat) -> nat :=
adam@377 1272 match types'' with
adam@378 1273 | nil => skipCopies natIndex
adam@377 1274 | t :: types0 =>
adam@377 1275 match natIndex return hlist (fun x : Set => x)
adam@377 1276 (update (t :: types0) natIndex nat) -> nat with
adam@377 1277 | O => fun vs => hhd vs
adam@377 1278 | S natIndex' => fun vs => getNat' types0 natIndex' (htl vs)
adam@377 1279 end
adam@377 1280 end.
adam@377 1281 End withTypes'.
adam@377 1282
adam@398 1283 (** 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 1284
adam@377 1285 Theorem getNat_is_reasonable : getNat' myTypes myNatIndex myValues = 3.
adam@377 1286 reflexivity.
adam@377 1287 Qed.
adam@377 1288
adam@377 1289 (** The same parameters as before work without alteration, and we avoid use of axioms. *)