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author Adam Chlipala <adam@chlipala.net>
date Fri, 08 Jun 2012 11:25:11 -0400
parents 05efde66559d
children 934945edc6b5
rev   line source
adam@377 1 (* Copyright (c) 2009-2012, Adam Chlipala
adamc@227 2 *
adamc@227 3 * This work is licensed under a
adamc@227 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@227 5 * Unported License.
adamc@227 6 * The license text is available at:
adamc@227 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@227 8 *)
adamc@227 9
adamc@227 10 (* begin hide *)
adam@377 11 Require Import List.
adam@377 12
adam@314 13 Require Import DepList CpdtTactics.
adamc@227 14
adamc@227 15 Set Implicit Arguments.
adamc@227 16 (* end hide *)
adamc@227 17
adam@398 18 (** printing $ %({}*% #(<a/>*# *)
adam@398 19 (** printing ^ %*{})% #*<a/>)# *)
adam@398 20
adam@398 21
adamc@227 22
adamc@227 23 (** %\chapter{Universes and Axioms}% *)
adamc@227 24
adam@343 25 (** Many traditional theorems can be proved in Coq without special knowledge of CIC, the logic behind the prover. A development just seems to be using a particular ASCII notation for standard formulas based on %\index{set theory}%set theory. Nonetheless, as we saw in Chapter 4, CIC differs from set theory in starting from fewer orthogonal primitives. It is possible to define the usual logical connectives as derived notions. The foundation of it all is a dependently typed functional programming language, based on dependent function types and inductive type families. By using the facilities of this language directly, we can accomplish some things much more easily than in mainstream math.
adamc@227 26
adam@343 27 %\index{Gallina}%Gallina, which adds features to the more theoretical CIC%~\cite{CIC}%, is the logic implemented in Coq. It has a relatively simple foundation that can be defined rigorously in a page or two of formal proof rules. Still, there are some important subtleties that have practical ramifications. This chapter focuses on those subtleties, avoiding formal metatheory in favor of example code. *)
adamc@227 28
adamc@227 29
adamc@227 30 (** * The [Type] Hierarchy *)
adamc@227 31
adam@343 32 (** %\index{type hierarchy}%Every object in Gallina has a type. *)
adamc@227 33
adamc@227 34 Check 0.
adamc@227 35 (** %\vspace{-.15in}% [[
adamc@227 36 0
adamc@227 37 : nat
adamc@227 38
adamc@227 39 ]]
adamc@227 40
adamc@227 41 It is natural enough that zero be considered as a natural number. *)
adamc@227 42
adamc@227 43 Check nat.
adamc@227 44 (** %\vspace{-.15in}% [[
adamc@227 45 nat
adamc@227 46 : Set
adamc@227 47
adamc@227 48 ]]
adamc@227 49
adam@287 50 From a set theory perspective, it is unsurprising to consider the natural numbers as a %``%#"#set.#"#%''% *)
adamc@227 51
adamc@227 52 Check Set.
adamc@227 53 (** %\vspace{-.15in}% [[
adamc@227 54 Set
adamc@227 55 : Type
adamc@227 56
adamc@227 57 ]]
adamc@227 58
adam@398 59 The type [Set] may be considered as the set of all sets, a concept that set theory handles in terms of %\index{class (in set theory)}%_classes_. In Coq, this more general notion is [Type]. *)
adamc@227 60
adamc@227 61 Check Type.
adamc@227 62 (** %\vspace{-.15in}% [[
adamc@227 63 Type
adamc@227 64 : Type
adamc@227 65
adamc@227 66 ]]
adamc@227 67
adam@343 68 Strangely enough, [Type] appears to be its own type. It is known that polymorphic languages with this property are inconsistent, via %\index{Girard's paradox}%Girard's paradox%~\cite{GirardsParadox}%. That is, using such a language to encode proofs is unwise, because it is possible to %``%#"#prove#"#%''% any proposition. What is really going on here?
adamc@227 69
adam@343 70 Let us repeat some of our queries after toggling a flag related to Coq's printing behavior.%\index{Vernacular commands!Set Printing Universes}% *)
adamc@227 71
adamc@227 72 Set Printing Universes.
adamc@227 73
adamc@227 74 Check nat.
adamc@227 75 (** %\vspace{-.15in}% [[
adamc@227 76 nat
adamc@227 77 : Set
adam@302 78 ]]
adam@398 79 *)
adamc@227 80
adamc@227 81 Check Set.
adamc@227 82 (** %\vspace{-.15in}% [[
adamc@227 83 Set
adamc@227 84 : Type $ (0)+1 ^
adamc@227 85
adam@302 86 ]]
adam@302 87 *)
adamc@227 88
adamc@227 89 Check Type.
adamc@227 90 (** %\vspace{-.15in}% [[
adamc@227 91 Type $ Top.3 ^
adamc@227 92 : Type $ (Top.3)+1 ^
adamc@227 93
adamc@227 94 ]]
adamc@227 95
adam@287 96 Occurrences of [Type] are annotated with some additional information, inside comments. These annotations have to do with the secret behind [Type]: it really stands for an infinite hierarchy of types. The type of [Set] is [Type(0)], the type of [Type(0)] is [Type(1)], the type of [Type(1)] is [Type(2)], and so on. This is how we avoid the %``%#"#[Type : Type]#"#%''% paradox. As a convenience, the universe hierarchy drives Coq's one variety of subtyping. Any term whose type is [Type] at level [i] is automatically also described by [Type] at level [j] when [j > i].
adamc@227 97
adam@398 98 In the outputs of our first [Check] query, we see that the type level of [Set]'s type is [(0)+1]. Here [0] stands for the level of [Set], and we increment it to arrive at the level that _classifies_ [Set].
adamc@227 99
adam@398 100 In the second query's output, we see that the occurrence of [Type] that we check is assigned a fresh %\index{universe variable}%_universe variable_ [Top.3]. The output type increments [Top.3] to move up a level in the universe hierarchy. As we write code that uses definitions whose types mention universe variables, unification may refine the values of those variables. Luckily, the user rarely has to worry about the details.
adamc@227 101
adam@398 102 Another crucial concept in CIC is %\index{predicativity}%_predicativity_. Consider these queries. *)
adamc@227 103
adamc@227 104 Check forall T : nat, fin T.
adamc@227 105 (** %\vspace{-.15in}% [[
adamc@227 106 forall T : nat, fin T
adamc@227 107 : Set
adam@302 108 ]]
adam@302 109 *)
adamc@227 110
adamc@227 111 Check forall T : Set, T.
adamc@227 112 (** %\vspace{-.15in}% [[
adamc@227 113 forall T : Set, T
adamc@227 114 : Type $ max(0, (0)+1) ^
adam@302 115 ]]
adam@302 116 *)
adamc@227 117
adamc@227 118 Check forall T : Type, T.
adamc@227 119 (** %\vspace{-.15in}% [[
adamc@227 120 forall T : Type $ Top.9 ^ , T
adamc@227 121 : Type $ max(Top.9, (Top.9)+1) ^
adamc@227 122
adamc@227 123 ]]
adamc@227 124
adamc@227 125 These outputs demonstrate the rule for determining which universe a [forall] type lives in. In particular, for a type [forall x : T1, T2], we take the maximum of the universes of [T1] and [T2]. In the first example query, both [T1] ([nat]) and [T2] ([fin T]) are in [Set], so the [forall] type is in [Set], too. In the second query, [T1] is [Set], which is at level [(0)+1]; and [T2] is [T], which is at level [0]. Thus, the [forall] exists at the maximum of these two levels. The third example illustrates the same outcome, where we replace [Set] with an occurrence of [Type] that is assigned universe variable [Top.9]. This universe variable appears in the places where [0] appeared in the previous query.
adamc@227 126
adam@287 127 The behind-the-scenes manipulation of universe variables gives us predicativity. Consider this simple definition of a polymorphic identity function, where the first argument [T] will automatically be marked as implicit, since it can be inferred from the type of the second argument [x]. *)
adamc@227 128
adamc@227 129 Definition id (T : Set) (x : T) : T := x.
adamc@227 130
adamc@227 131 Check id 0.
adamc@227 132 (** %\vspace{-.15in}% [[
adamc@227 133 id 0
adamc@227 134 : nat
adamc@227 135
adamc@227 136 Check id Set.
adam@343 137 ]]
adamc@227 138
adam@343 139 <<
adamc@227 140 Error: Illegal application (Type Error):
adamc@227 141 ...
adam@343 142 The 1st term has type "Type (* (Top.15)+1 *)" which should be coercible to "Set".
adam@343 143 >>
adamc@227 144
adam@343 145 The parameter [T] of [id] must be instantiated with a [Set]. The type [nat] is a [Set], but [Set] is not. We can try fixing the problem by generalizing our definition of [id]. *)
adamc@227 146
adamc@227 147 Reset id.
adamc@227 148 Definition id (T : Type) (x : T) : T := x.
adamc@227 149 Check id 0.
adamc@227 150 (** %\vspace{-.15in}% [[
adamc@227 151 id 0
adamc@227 152 : nat
adam@302 153 ]]
adam@302 154 *)
adamc@227 155
adamc@227 156 Check id Set.
adamc@227 157 (** %\vspace{-.15in}% [[
adamc@227 158 id Set
adamc@227 159 : Type $ Top.17 ^
adam@302 160 ]]
adam@302 161 *)
adamc@227 162
adamc@227 163 Check id Type.
adamc@227 164 (** %\vspace{-.15in}% [[
adamc@227 165 id Type $ Top.18 ^
adamc@227 166 : Type $ Top.19 ^
adam@302 167 ]]
adam@302 168 *)
adamc@227 169
adamc@227 170 (** So far so good. As we apply [id] to different [T] values, the inferred index for [T]'s [Type] occurrence automatically moves higher up the type hierarchy.
adamc@227 171 [[
adamc@227 172 Check id id.
adam@343 173 ]]
adamc@227 174
adam@343 175 <<
adamc@227 176 Error: Universe inconsistency (cannot enforce Top.16 < Top.16).
adam@343 177 >>
adamc@227 178
adam@398 179 %\index{universe inconsistency}%This error message reminds us that the universe variable for [T] still exists, even though it is usually hidden. To apply [id] to itself, that variable would need to be less than itself in the type hierarchy. Universe inconsistency error messages announce cases like this one where a term could only type-check by violating an implied constraint over universe variables. Such errors demonstrate that [Type] is _predicative_, where this word has a CIC meaning closely related to its usual mathematical meaning. A predicative system enforces the constraint that, for any object of quantified type, none of those quantifiers may ever be instantiated with the object itself. %\index{impredicativity}%Impredicativity is associated with popular paradoxes in set theory, involving inconsistent constructions like %``%#"#the set of all sets that do not contain themselves#"#%''% (%\index{Russell's paradox}%Russell's paradox). Similar paradoxes would result from uncontrolled impredicativity in Coq. *)
adamc@227 180
adamc@227 181
adamc@227 182 (** ** Inductive Definitions *)
adamc@227 183
adamc@227 184 (** Predicativity restrictions also apply to inductive definitions. As an example, let us consider a type of expression trees that allows injection of any native Coq value. The idea is that an [exp T] stands for a reflected expression of type [T].
adamc@227 185
adamc@227 186 [[
adamc@227 187 Inductive exp : Set -> Set :=
adamc@227 188 | Const : forall T : Set, T -> exp T
adamc@227 189 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 190 | Eq : forall T, exp T -> exp T -> exp bool.
adam@343 191 ]]
adamc@227 192
adam@343 193 <<
adamc@227 194 Error: Large non-propositional inductive types must be in Type.
adam@343 195 >>
adamc@227 196
adam@398 197 This definition is %\index{large inductive types}%_large_ in the sense that at least one of its constructors takes an argument whose type has type [Type]. Coq would be inconsistent if we allowed definitions like this one in their full generality. Instead, we must change [exp] to live in [Type]. We will go even further and move [exp]'s index to [Type] as well. *)
adamc@227 198
adamc@227 199 Inductive exp : Type -> Type :=
adamc@227 200 | Const : forall T, T -> exp T
adamc@227 201 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 202 | Eq : forall T, exp T -> exp T -> exp bool.
adamc@227 203
adamc@228 204 (** Note that before we had to include an annotation [: Set] for the variable [T] in [Const]'s type, but we need no annotation now. When the type of a variable is not known, and when that variable is used in a context where only types are allowed, Coq infers that the variable is of type [Type]. That is the right behavior here, but it was wrong for the [Set] version of [exp].
adamc@228 205
adamc@228 206 Our new definition is accepted. We can build some sample expressions. *)
adamc@227 207
adamc@227 208 Check Const 0.
adamc@227 209 (** %\vspace{-.15in}% [[
adamc@227 210 Const 0
adamc@227 211 : exp nat
adam@302 212 ]]
adam@302 213 *)
adamc@227 214
adamc@227 215 Check Pair (Const 0) (Const tt).
adamc@227 216 (** %\vspace{-.15in}% [[
adamc@227 217 Pair (Const 0) (Const tt)
adamc@227 218 : exp (nat * unit)
adam@302 219 ]]
adam@302 220 *)
adamc@227 221
adamc@227 222 Check Eq (Const Set) (Const Type).
adamc@227 223 (** %\vspace{-.15in}% [[
adamc@228 224 Eq (Const Set) (Const Type $ Top.59 ^ )
adamc@227 225 : exp bool
adamc@227 226
adamc@227 227 ]]
adamc@227 228
adamc@227 229 We can check many expressions, including fancy expressions that include types. However, it is not hard to hit a type-checking wall.
adamc@227 230
adamc@227 231 [[
adamc@227 232 Check Const (Const O).
adam@343 233 ]]
adamc@227 234
adam@343 235 <<
adamc@227 236 Error: Universe inconsistency (cannot enforce Top.42 < Top.42).
adam@343 237 >>
adamc@227 238
adamc@227 239 We are unable to instantiate the parameter [T] of [Const] with an [exp] type. To see why, it is helpful to print the annotated version of [exp]'s inductive definition. *)
adamc@227 240
adamc@227 241 Print exp.
adamc@227 242 (** %\vspace{-.15in}% [[
adamc@227 243 Inductive exp
adamc@227 244 : Type $ Top.8 ^ ->
adamc@227 245 Type
adamc@227 246 $ max(0, (Top.11)+1, (Top.14)+1, (Top.15)+1, (Top.19)+1) ^ :=
adamc@227 247 Const : forall T : Type $ Top.11 ^ , T -> exp T
adamc@227 248 | Pair : forall (T1 : Type $ Top.14 ^ ) (T2 : Type $ Top.15 ^ ),
adamc@227 249 exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 250 | Eq : forall T : Type $ Top.19 ^ , exp T -> exp T -> exp bool
adamc@227 251
adamc@227 252 ]]
adamc@227 253
adam@398 254 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@227 255
adam@343 256 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@227 257
adamc@227 258 Print Universes.
adamc@227 259 (** %\vspace{-.15in}% [[
adamc@227 260 Top.19 < Top.9 <= Top.8
adamc@227 261 Top.15 < Top.9 <= Top.8 <= Coq.Init.Datatypes.38
adamc@227 262 Top.14 < Top.9 <= Top.8 <= Coq.Init.Datatypes.37
adamc@227 263 Top.11 < Top.9 <= Top.8
adamc@227 264
adamc@227 265 ]]
adamc@227 266
adam@343 267 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@227 268
adamc@227 269 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@227 270
adamc@227 271 Print prod.
adamc@227 272 (** %\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
adamc@227 280 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@227 281
adamc@227 282 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@227 283
adamc@227 284 %\medskip%
adamc@227 285
adam@398 286 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@227 287
adamc@231 288 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@227 289
adamc@227 290 Check (nat, (Type, Set)).
adamc@227 291 (** %\vspace{-.15in}% [[
adamc@227 292 (nat, (Type $ Top.44 ^ , Set))
adamc@227 293 : Set * (Type $ Top.45 ^ * Type $ Top.46 ^ )
adamc@227 294 ]]
adamc@227 295
adamc@227 296 The same cannot be done with a counterpart to [prod] that does not use parameters. *)
adamc@227 297
adamc@227 298 Inductive prod' : Type -> Type -> Type :=
adamc@227 299 | pair' : forall A B : Type, A -> B -> prod' A B.
adamc@227 300 (** [[
adamc@227 301 Check (pair' nat (pair' Type Set)).
adam@343 302 ]]
adamc@227 303
adam@343 304 <<
adamc@227 305 Error: Universe inconsistency (cannot enforce Top.51 < Top.51).
adam@343 306 >>
adamc@227 307
adamc@233 308 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@233 309
adam@343 310 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@233 311
adamc@233 312 Inductive foo (A : Type) : Type :=
adamc@233 313 | Foo : A -> foo A.
adamc@229 314
adamc@229 315 (* begin hide *)
adamc@229 316 Unset Printing Universes.
adamc@229 317 (* end hide *)
adamc@229 318
adamc@233 319 Check foo nat.
adamc@233 320 (** %\vspace{-.15in}% [[
adamc@233 321 foo nat
adamc@233 322 : Set
adam@302 323 ]]
adam@302 324 *)
adamc@233 325
adamc@233 326 Check foo Set.
adamc@233 327 (** %\vspace{-.15in}% [[
adamc@233 328 foo Set
adamc@233 329 : Type
adam@302 330 ]]
adam@302 331 *)
adamc@233 332
adamc@233 333 Check foo True.
adamc@233 334 (** %\vspace{-.15in}% [[
adamc@233 335 foo True
adamc@233 336 : Prop
adamc@233 337
adamc@233 338 ]]
adamc@233 339
adam@287 340 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@233 341
adamc@233 342 Imitation polymorphism can be confusing in some contexts. For instance, it is what is responsible for this weird behavior. *)
adamc@233 343
adamc@233 344 Inductive bar : Type := Bar : bar.
adamc@233 345
adamc@233 346 Check bar.
adamc@233 347 (** %\vspace{-.15in}% [[
adamc@233 348 bar
adamc@233 349 : Prop
adamc@233 350 ]]
adamc@233 351
adamc@233 352 The type that Coq comes up with may be used in strictly more contexts than the type one might have expected. *)
adamc@233 353
adamc@229 354
adam@388 355 (** ** Deciphering Baffling Messages About Inability to Unify *)
adam@388 356
adam@388 357 (** 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@388 358
adam@388 359 Theorem symmetry : forall A B : Type,
adam@388 360 A = B
adam@388 361 -> B = A.
adam@388 362 intros ? ? H; rewrite H; reflexivity.
adam@388 363 Qed.
adam@388 364
adam@388 365 (** Let us attempt an admittedly silly proof of the following theorem. *)
adam@388 366
adam@388 367 Theorem illustrative_but_silly_detour : unit = unit.
adam@388 368 (** [[
adam@388 369 apply symmetry.
adam@388 370 ]]
adam@388 371 <<
adam@388 372 Error: Impossible to unify "?35 = ?34" with "unit = unit".
adam@388 373 >>
adam@388 374
adam@398 375 Coq tells us that we cannot, in fact, apply our lemma [symmetry] here, but the error message seems defective. In particular, one might think that [apply] should unify [?35] and [?34] with [unit] to ensure that the unification goes through. In fact, the problem is in a part of the unification problem that is _not_ shown to us in this error message!
adam@388 376
adam@388 377 The following command is the secret to getting better error messages in such cases: *)
adam@388 378
adam@388 379 Set Printing All.
adam@388 380 (** [[
adam@388 381 apply symmetry.
adam@388 382 ]]
adam@388 383 <<
adam@388 384 Error: Impossible to unify "@eq Type ?46 ?45" with "@eq Set unit unit".
adam@388 385 >>
adam@388 386
adam@398 387 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@388 388
adam@388 389 Abort.
adam@388 390
adam@388 391 (** 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@388 392
adam@388 393 Theorem illustrative_but_silly_detour : (unit : Type) = unit.
adam@388 394 apply symmetry; reflexivity.
adam@388 395 Qed.
adam@388 396
adam@388 397 (** 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@388 398
adam@388 399 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@388 400
adam@388 401 Unset Printing All.
adam@388 402
adam@388 403 Theorem ex_symmetry : (exists x, x = 0) -> (exists x, 0 = x).
adam@388 404 econstructor.
adam@388 405 (** %\vspace{-.15in}%[[
adam@388 406 H : exists x : nat, x = 0
adam@388 407 ============================
adam@388 408 0 = ?98
adam@388 409 ]]
adam@388 410 *)
adam@388 411
adam@388 412 destruct H.
adam@388 413 (** %\vspace{-.15in}%[[
adam@388 414 x : nat
adam@388 415 H : x = 0
adam@388 416 ============================
adam@388 417 0 = ?99
adam@388 418 ]]
adam@388 419 *)
adam@388 420
adam@388 421 (** [[
adam@388 422 symmetry; exact H.
adam@388 423 ]]
adam@388 424
adam@388 425 <<
adam@388 426 Error: In environment
adam@388 427 x : nat
adam@388 428 H : x = 0
adam@388 429 The term "H" has type "x = 0" while it is expected to have type
adam@388 430 "?99 = 0".
adam@388 431 >>
adam@388 432
adam@398 433 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@388 434
adam@388 435 Restart.
adam@388 436 destruct 1 as [x]; apply ex_intro with x; symmetry; assumption.
adam@388 437 Qed.
adam@388 438
adam@388 439 (** 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. *)
adam@388 440
adam@388 441
adamc@229 442 (** * The [Prop] Universe *)
adamc@229 443
adam@287 444 (** 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@229 445
adamc@229 446 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@229 447
adamc@229 448 Print sig.
adamc@229 449 (** %\vspace{-.15in}% [[
adamc@229 450 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@229 451 exist : forall x : A, P x -> sig P
adam@302 452 ]]
adam@302 453 *)
adamc@229 454
adamc@229 455 Print ex.
adamc@229 456 (** %\vspace{-.15in}% [[
adamc@229 457 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@229 458 ex_intro : forall x : A, P x -> ex P
adamc@229 459 ]]
adamc@229 460
adamc@229 461 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@229 462
adamc@229 463 Definition projS A (P : A -> Prop) (x : sig P) : A :=
adamc@229 464 match x with
adamc@229 465 | exist v _ => v
adamc@229 466 end.
adamc@229 467
adamc@229 468 (** We run into trouble with a version that has been changed to work with [ex].
adamc@229 469 [[
adamc@229 470 Definition projE A (P : A -> Prop) (x : ex P) : A :=
adamc@229 471 match x with
adamc@229 472 | ex_intro v _ => v
adamc@229 473 end.
adam@343 474 ]]
adamc@229 475
adam@343 476 <<
adamc@229 477 Error:
adamc@229 478 Incorrect elimination of "x" in the inductive type "ex":
adamc@229 479 the return type has sort "Type" while it should be "Prop".
adamc@229 480 Elimination of an inductive object of sort Prop
adamc@229 481 is not allowed on a predicate in sort Type
adamc@229 482 because proofs can be eliminated only to build proofs.
adam@343 483 >>
adamc@229 484
adam@343 485 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@229 486
adamc@229 487 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@229 488
adam@398 489 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@229 490
adamc@229 491 Definition sym_sig (x : sig (fun n => n = 0)) : sig (fun n => 0 = n) :=
adamc@229 492 match x with
adamc@229 493 | exist n pf => exist _ n (sym_eq pf)
adamc@229 494 end.
adamc@229 495
adamc@229 496 Extraction sym_sig.
adamc@229 497 (** <<
adamc@229 498 (** val sym_sig : nat -> nat **)
adamc@229 499
adamc@229 500 let sym_sig x = x
adamc@229 501 >>
adamc@229 502
adamc@229 503 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@229 504
adamc@229 505 Definition sym_ex (x : ex (fun n => n = 0)) : ex (fun n => 0 = n) :=
adamc@229 506 match x with
adamc@229 507 | ex_intro n pf => ex_intro _ n (sym_eq pf)
adamc@229 508 end.
adamc@229 509
adamc@229 510 Extraction sym_ex.
adamc@229 511 (** <<
adamc@229 512 (** val sym_ex : __ **)
adamc@229 513
adamc@229 514 let sym_ex = __
adamc@229 515 >>
adamc@229 516
adam@302 517 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 %\texttt{\_\_}%#<tt>__</tt>#, whose single constructor is %\texttt{\_\_}%#<tt>__</tt>#. Not only are proofs replaced by [__], but proof arguments to functions are also removed completely, as we see here.
adamc@229 518
adam@343 519 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@229 520
adam@398 521 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@229 522
adamc@229 523 %\medskip%
adamc@229 524
adam@398 525 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@229 526
adamc@229 527 Check forall P Q : Prop, P \/ Q -> Q \/ P.
adamc@229 528 (** %\vspace{-.15in}% [[
adamc@229 529 forall P Q : Prop, P \/ Q -> Q \/ P
adamc@229 530 : Prop
adamc@229 531
adamc@229 532 ]]
adamc@229 533
adamc@230 534 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@230 535
adamc@230 536 Impredicativity also allows us to implement a version of our earlier [exp] type that does not suffer from the weakness that we found. *)
adamc@230 537
adamc@230 538 Inductive expP : Type -> Prop :=
adamc@230 539 | ConstP : forall T, T -> expP T
adamc@230 540 | PairP : forall T1 T2, expP T1 -> expP T2 -> expP (T1 * T2)
adamc@230 541 | EqP : forall T, expP T -> expP T -> expP bool.
adamc@230 542
adamc@230 543 Check ConstP 0.
adamc@230 544 (** %\vspace{-.15in}% [[
adamc@230 545 ConstP 0
adamc@230 546 : expP nat
adam@302 547 ]]
adam@302 548 *)
adamc@230 549
adamc@230 550 Check PairP (ConstP 0) (ConstP tt).
adamc@230 551 (** %\vspace{-.15in}% [[
adamc@230 552 PairP (ConstP 0) (ConstP tt)
adamc@230 553 : expP (nat * unit)
adam@302 554 ]]
adam@302 555 *)
adamc@230 556
adamc@230 557 Check EqP (ConstP Set) (ConstP Type).
adamc@230 558 (** %\vspace{-.15in}% [[
adamc@230 559 EqP (ConstP Set) (ConstP Type)
adamc@230 560 : expP bool
adam@302 561 ]]
adam@302 562 *)
adamc@230 563
adamc@230 564 Check ConstP (ConstP O).
adamc@230 565 (** %\vspace{-.15in}% [[
adamc@230 566 ConstP (ConstP 0)
adamc@230 567 : expP (expP nat)
adamc@230 568
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@398 605 (** 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
adamc@230 625 Parameter n : nat.
adamc@230 626 Axiom positive : n > 0.
adamc@230 627 Reset n.
adamc@230 628
adam@287 629 (** 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@398 631 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@398 645 (** 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@398 647 Recall that Coq implements %\index{constructive logic}%_constructive_ logic by default, where excluded middle is not provable. Proofs in constructive logic can be thought of as programs. A [forall] quantifier denotes a dependent function type, and a disjunction denotes a variant type. In such a setting, excluded middle could be interpreted as a decision procedure for arbitrary propositions, which computability theory tells us cannot exist. Thus, constructive logic with excluded middle can no longer be associated with our usual notion of programming.
adamc@230 648
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.
adamc@230 664 (** [[
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@398 697 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.
adamc@230 701 (** %\vspace{-.15in}% [[
adamc@230 702 *** [ proof_irrelevance : forall (P : Prop) (p1 p2 : P), p1 = p2 ]
adamc@230 703 ]]
adamc@230 704
adam@353 705 This axiom asserts that any two proofs of the same proposition are equal. If we replaced [p1 = p2] by [p1 <-> p2], then the statement would be provable. However, equality is a stronger notion than logical equivalence. Recall this example function from Chapter 6. *)
adamc@230 706
adamc@230 707 (* begin hide *)
adamc@230 708 Lemma zgtz : 0 > 0 -> False.
adamc@230 709 crush.
adamc@230 710 Qed.
adamc@230 711 (* end hide *)
adamc@230 712
adamc@230 713 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@230 714 match n with
adamc@230 715 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@230 716 | S n' => fun _ => n'
adamc@230 717 end.
adamc@230 718
adam@343 719 (** We might want to prove that different proofs of [n > 0] do not lead to different results from our richly typed predecessor function. *)
adamc@230 720
adamc@230 721 Theorem pred_strong1_irrel : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 722 destruct n; crush.
adamc@230 723 Qed.
adamc@230 724
adamc@230 725 (** 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@230 726
adamc@230 727 Theorem pred_strong1_irrel' : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 728 intros; f_equal; apply proof_irrelevance.
adamc@230 729 Qed.
adamc@230 730
adamc@230 731
adamc@230 732 (** %\bigskip%
adamc@230 733
adamc@230 734 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@230 735
adamc@230 736 Require Import Eqdep.
adamc@230 737 Import Eq_rect_eq.
adamc@230 738 Print eq_rect_eq.
adamc@230 739 (** %\vspace{-.15in}% [[
adamc@230 740 *** [ eq_rect_eq :
adamc@230 741 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adamc@230 742 x = eq_rect p Q x p h ]
adamc@230 743 ]]
adamc@230 744
adam@343 745 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@230 746
adamc@230 747 Corollary UIP_refl : forall A (x : A) (pf : x = x), pf = refl_equal x.
adamc@230 748 intros; replace pf with (eq_rect x (eq x) (refl_equal x) x pf); [
adamc@230 749 symmetry; apply eq_rect_eq
adamc@230 750 | exact (match pf as pf' return match pf' in _ = y return x = y with
adamc@230 751 | refl_equal => refl_equal x
adamc@230 752 end = pf' with
adamc@230 753 | refl_equal => refl_equal _
adamc@230 754 end) ].
adamc@230 755 Qed.
adamc@230 756
adamc@230 757 Corollary UIP : forall A (x y : A) (pf1 pf2 : x = y), pf1 = pf2.
adamc@230 758 intros; generalize pf1 pf2; subst; intros;
adamc@230 759 match goal with
adamc@230 760 | [ |- ?pf1 = ?pf2 ] => rewrite (UIP_refl pf1); rewrite (UIP_refl pf2); reflexivity
adamc@230 761 end.
adamc@230 762 Qed.
adamc@230 763
adamc@231 764 (** 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@230 765
adamc@230 766 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@230 767
adamc@230 768 %\bigskip%
adamc@230 769
adamc@230 770 There are two more basic axioms that are often assumed, to avoid complications that do not arise in set theory. *)
adamc@230 771
adamc@230 772 Require Import FunctionalExtensionality.
adamc@230 773 Print functional_extensionality_dep.
adamc@230 774 (** %\vspace{-.15in}% [[
adamc@230 775 *** [ functional_extensionality_dep :
adamc@230 776 forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
adamc@230 777 (forall x : A, f x = g x) -> f = g ]
adamc@230 778
adamc@230 779 ]]
adamc@230 780
adamc@230 781 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@230 782
adam@343 783 A simple corollary shows that the same property applies to predicates. *)
adamc@230 784
adamc@230 785 Corollary predicate_extensionality : forall (A : Type) (B : A -> Prop) (f g : forall x : A, B x),
adamc@230 786 (forall x : A, f x = g x) -> f = g.
adamc@230 787 intros; apply functional_extensionality_dep; assumption.
adamc@230 788 Qed.
adamc@230 789
adam@343 790 (** In some cases, one might prefer to assert this corollary as the axiom, to restrict the consequences to proofs and not programs. *)
adam@343 791
adamc@230 792
adamc@230 793 (** ** Axioms of Choice *)
adamc@230 794
adam@343 795 (** 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@230 796
adam@398 797 First, it is possible to implement a choice operator _without_ axioms in some potentially surprising cases. *)
adamc@230 798
adamc@230 799 Require Import ConstructiveEpsilon.
adamc@230 800 Check constructive_definite_description.
adamc@230 801 (** %\vspace{-.15in}% [[
adamc@230 802 constructive_definite_description
adamc@230 803 : forall (A : Set) (f : A -> nat) (g : nat -> A),
adamc@230 804 (forall x : A, g (f x) = x) ->
adamc@230 805 forall P : A -> Prop,
adamc@230 806 (forall x : A, {P x} + {~ P x}) ->
adamc@230 807 (exists! x : A, P x) -> {x : A | P x}
adam@302 808 ]]
adam@302 809 *)
adamc@230 810
adamc@230 811 Print Assumptions constructive_definite_description.
adam@343 812 (** <<
adamc@230 813 Closed under the global context
adam@343 814 >>
adamc@230 815
adam@398 816 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@230 817
adamc@230 818 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@230 819
adamc@230 820 Require Import ClassicalUniqueChoice.
adamc@230 821 Check dependent_unique_choice.
adamc@230 822 (** %\vspace{-.15in}% [[
adamc@230 823 dependent_unique_choice
adamc@230 824 : forall (A : Type) (B : A -> Type) (R : forall x : A, B x -> Prop),
adamc@230 825 (forall x : A, exists! y : B x, R x y) ->
adam@343 826 exists f : forall x : A, B x,
adam@343 827 forall x : A, R x (f x)
adamc@230 828 ]]
adamc@230 829
adamc@230 830 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@230 831
adamc@230 832 Require Import ClassicalChoice.
adamc@230 833 Check choice.
adamc@230 834 (** %\vspace{-.15in}% [[
adamc@230 835 choice
adamc@230 836 : forall (A B : Type) (R : A -> B -> Prop),
adamc@230 837 (forall x : A, exists y : B, R x y) ->
adamc@230 838 exists f : A -> B, forall x : A, R x (f x)
adamc@230 839
adamc@230 840 ]]
adamc@230 841
adamc@230 842 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@230 843
adamc@230 844 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@230 845
adamc@230 846 Definition choice_Set (A B : Type) (R : A -> B -> Prop) (H : forall x : A, {y : B | R x y})
adamc@230 847 : {f : A -> B | forall x : A, R x (f x)} :=
adamc@230 848 exist (fun f => forall x : A, R x (f x))
adamc@230 849 (fun x => proj1_sig (H x)) (fun x => proj2_sig (H x)).
adamc@230 850
adam@287 851 (** Via the Curry-Howard correspondence, this %``%#"#axiom#"#%''% can be taken to have the same meaning as the original. It is implemented trivially as a transformation not much deeper than uncurrying. Thus, we see that the utility of the axioms that we mentioned earlier comes in their usage to build programs from proofs. Normal set theory has no explicit proofs, so the meaning of the usual axiom of choice is subtlely different. In Gallina, the axioms implement a controlled relaxation of the restrictions on information flow from proofs to programs.
adamc@230 852
adam@287 853 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@230 854
adamc@230 855 %\bigskip%
adamc@230 856
adam@343 857 The Coq tools support a command-line flag %\index{impredicative Set}\texttt{%#<tt>#-impredicative-set#</tt>#%}%, 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@230 858
adamc@230 859 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@230 860
adamc@230 861 (** ** Axioms and Computation *)
adamc@230 862
adam@398 863 (** 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@230 864
adamc@230 865 Definition cast (x y : Set) (pf : x = y) (v : x) : y :=
adamc@230 866 match pf with
adamc@230 867 | refl_equal => v
adamc@230 868 end.
adamc@230 869
adamc@230 870 (** Computation over programs that use [cast] can proceed smoothly. *)
adamc@230 871
adamc@230 872 Eval compute in (cast (refl_equal (nat -> nat)) (fun n => S n)) 12.
adam@343 873 (** %\vspace{-.15in}%[[
adamc@230 874 = 13
adamc@230 875 : nat
adam@302 876 ]]
adam@302 877 *)
adamc@230 878
adamc@230 879 (** Things do not go as smoothly when we use [cast] with proofs that rely on axioms. *)
adamc@230 880
adamc@230 881 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 882 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 883 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 884 Qed.
adamc@230 885
adamc@230 886 Eval compute in (cast t3 (fun _ => First)) 12.
adamc@230 887 (** [[
adamc@230 888 = match t3 in (_ = P) return P with
adamc@230 889 | refl_equal => fun n : nat => First
adamc@230 890 end 12
adamc@230 891 : fin (12 + 1)
adamc@230 892 ]]
adamc@230 893
adamc@230 894 Computation gets stuck in a pattern-match on the proof [t3]. The structure of [t3] is not known, so the match cannot proceed. It turns out a more basic problem leads to this particular situation. We ended the proof of [t3] with [Qed], so the definition of [t3] is not available to computation. That is easily fixed. *)
adamc@230 895
adamc@230 896 Reset t3.
adamc@230 897
adamc@230 898 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 899 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 900 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 901 Defined.
adamc@230 902
adamc@230 903 Eval compute in (cast t3 (fun _ => First)) 12.
adamc@230 904 (** [[
adamc@230 905 = match
adamc@230 906 match
adamc@230 907 match
adamc@230 908 functional_extensionality
adamc@230 909 ....
adamc@230 910 ]]
adamc@230 911
adam@398 912 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@230 913
adamc@230 914 If we are careful in using tactics to prove an equality, we can still compute with casts over the proof. *)
adamc@230 915
adamc@230 916 Lemma plus1 : forall n, S n = n + 1.
adamc@230 917 induction n; simpl; intuition.
adamc@230 918 Defined.
adamc@230 919
adamc@230 920 Theorem t4 : forall n, fin (S n) = fin (n + 1).
adamc@230 921 intro; f_equal; apply plus1.
adamc@230 922 Defined.
adamc@230 923
adamc@230 924 Eval compute in cast (t4 13) First.
adamc@230 925 (** %\vspace{-.15in}% [[
adamc@230 926 = First
adamc@230 927 : fin (13 + 1)
adam@302 928 ]]
adam@343 929
adam@343 930 This simple computational reduction hides the use of a recursive function to produce a suitable [refl_equal] proof term. The recursion originates in our use of [induction] in [t4]'s proof. *)
adam@343 931
adam@344 932
adam@344 933 (** ** Methods for Avoiding Axioms *)
adam@344 934
adam@398 935 (** 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@344 936
adam@344 937 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@344 938
adam@344 939 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@344 940
adam@344 941 Theorem fin_cases : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 942 intros; dep_destruct f; eauto.
adam@344 943 Qed.
adam@344 944
adam@344 945 Print Assumptions fin_cases.
adam@344 946 (** %\vspace{-.15in}%[[
adam@344 947 Axioms:
adam@344 948 JMeq.JMeq_eq : forall (A : Type) (x y : A), JMeq.JMeq x y -> x = y
adam@344 949 ]]
adam@344 950
adam@344 951 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@344 952
adam@344 953 (* begin thide *)
adam@344 954 Lemma fin_cases_again' : forall n (f : fin n),
adam@344 955 match n return fin n -> Prop with
adam@344 956 | O => fun _ => False
adam@344 957 | S n' => fun f => f = First \/ exists f', f = Next f'
adam@344 958 end f.
adam@344 959 destruct f; eauto.
adam@344 960 Qed.
adam@344 961
adam@344 962 (** 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@344 963
adam@344 964 Theorem fin_cases_again : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 965 intros; exact (fin_cases_again' f).
adam@344 966 Qed.
adam@344 967 (* end thide *)
adam@344 968
adam@344 969 Print Assumptions fin_cases_again.
adam@344 970 (** %\vspace{-.15in}%
adam@344 971 <<
adam@344 972 Closed under the global context
adam@344 973 >>
adam@344 974
adam@345 975 *)
adam@345 976
adam@345 977 (* begin thide *)
adam@345 978 (** 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@345 979
adam@345 980 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@345 981
adam@345 982 Definition finOut n (f : fin n) : match n return fin n -> Type with
adam@345 983 | O => fun _ => Empty_set
adam@345 984 | _ => fun f => {f' : _ | f = Next f'} + {f = First}
adam@345 985 end f :=
adam@345 986 match f with
adam@345 987 | First _ => inright _ (refl_equal _)
adam@345 988 | Next _ f' => inleft _ (exist _ f' (refl_equal _))
adam@345 989 end.
adam@345 990 (* end thide *)
adam@345 991
adam@345 992 (** 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@344 993
adam@344 994 Inductive formula : list Type -> Type :=
adam@344 995 | Inject : forall Ts, Prop -> formula Ts
adam@344 996 | VarEq : forall T Ts, T -> formula (T :: Ts)
adam@344 997 | Lift : forall T Ts, formula Ts -> formula (T :: Ts)
adam@344 998 | Forall : forall T Ts, formula (T :: Ts) -> formula Ts
adam@344 999 | And : forall Ts, formula Ts -> formula Ts -> formula Ts.
adam@344 1000
adam@344 1001 (** 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 1002
adam@344 1003 Inductive proof : formula nil -> Prop :=
adam@344 1004 | PInject : forall (P : Prop), P -> proof (Inject nil P)
adam@344 1005 | PAnd : forall p q, proof p -> proof q -> proof (And p q).
adam@344 1006
adam@344 1007 (** Let us prove a lemma showing that a %``%#"#[P /\ Q -> P]#"#%''% rule is derivable within the rules of [proof]. *)
adam@344 1008
adam@344 1009 Theorem proj1 : forall p q, proof (And p q) -> proof p.
adam@344 1010 destruct 1.
adam@344 1011 (** %\vspace{-.15in}%[[
adam@344 1012 p : formula nil
adam@344 1013 q : formula nil
adam@344 1014 P : Prop
adam@344 1015 H : P
adam@344 1016 ============================
adam@344 1017 proof p
adam@344 1018 ]]
adam@344 1019 *)
adam@344 1020
adam@344 1021 (** 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 1022
adam@344 1023 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 1024
adam@344 1025 Restart.
adam@344 1026 Require Import Program.
adam@344 1027 intros ? ? H; dependent destruction H; auto.
adam@344 1028 Qed.
adam@344 1029
adam@344 1030 Print Assumptions proj1.
adam@344 1031 (** %\vspace{-.15in}%[[
adam@344 1032 Axioms:
adam@344 1033 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1034 x = eq_rect p Q x p h
adam@344 1035 ]]
adam@344 1036
adam@344 1037 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 1038
adam@344 1039 Lemma proj1_again' : forall r, proof r
adam@344 1040 -> forall p q, r = And p q -> proof p.
adam@344 1041 destruct 1; crush.
adam@344 1042 (** %\vspace{-.15in}%[[
adam@344 1043 H0 : Inject [] P = And p q
adam@344 1044 ============================
adam@344 1045 proof p
adam@344 1046 ]]
adam@344 1047
adam@344 1048 The first goal looks reasonable. Hypothesis [H0] is clearly contradictory, as [discriminate] can show. *)
adam@344 1049
adam@344 1050 discriminate.
adam@344 1051 (** %\vspace{-.15in}%[[
adam@344 1052 H : proof p
adam@344 1053 H1 : And p q = And p0 q0
adam@344 1054 ============================
adam@344 1055 proof p0
adam@344 1056 ]]
adam@344 1057
adam@344 1058 It looks like we are almost done. Hypothesis [H1] gives [p = p0] by injectivity of constructors, and then [H] finishes the case. *)
adam@344 1059
adam@344 1060 injection H1; intros.
adam@344 1061
adam@344 1062 (** Unfortunately, the %``%#"#equality#"#%''% that we expected between [p] and [p0] comes in a strange form:
adam@344 1063
adam@344 1064 [[
adam@344 1065 H3 : existT (fun Ts : list Type => formula Ts) []%list p =
adam@344 1066 existT (fun Ts : list Type => formula Ts) []%list p0
adam@344 1067 ============================
adam@344 1068 proof p0
adam@344 1069 ]]
adam@344 1070
adam@345 1071 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 1072
adam@344 1073 How exactly does an axiom come into the picture here? Let us ask [crush] to finish the proof. *)
adam@344 1074
adam@344 1075 crush.
adam@344 1076 Qed.
adam@344 1077
adam@344 1078 Print Assumptions proj1_again'.
adam@344 1079 (** %\vspace{-.15in}%[[
adam@344 1080 Axioms:
adam@344 1081 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1082 x = eq_rect p Q x p h
adam@344 1083 ]]
adam@344 1084
adam@344 1085 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 1086
adam@344 1087 Hope is not lost, however. We can produce an even stranger looking lemma, which gives us the theorem without axioms. *)
adam@344 1088
adam@344 1089 Lemma proj1_again'' : forall r, proof r
adam@344 1090 -> match r with
adam@344 1091 | And Ps p _ => match Ps return formula Ps -> Prop with
adam@344 1092 | nil => fun p => proof p
adam@344 1093 | _ => fun _ => True
adam@344 1094 end p
adam@344 1095 | _ => True
adam@344 1096 end.
adam@344 1097 destruct 1; auto.
adam@344 1098 Qed.
adam@344 1099
adam@344 1100 Theorem proj1_again : forall p q, proof (And p q) -> proof p.
adam@344 1101 intros ? ? H; exact (proj1_again'' H).
adam@344 1102 Qed.
adam@344 1103
adam@344 1104 Print Assumptions proj1_again.
adam@344 1105 (** <<
adam@344 1106 Closed under the global context
adam@344 1107 >>
adam@344 1108
adam@377 1109 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 1110
adam@377 1111 %\medskip%
adam@377 1112
adam@398 1113 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 1114
adam@377 1115 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 1116
adam@377 1117 Section withTypes.
adam@377 1118 Variable types : list Set.
adam@377 1119
adam@377 1120 (** 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 1121
adam@377 1122 Variable values : hlist (fun x : Set => x) types.
adam@377 1123
adam@377 1124 (** 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 1125
adam@377 1126 Variable natIndex : nat.
adam@377 1127 Variable natIndex_ok : nth_error types natIndex = Some nat.
adam@377 1128
adam@377 1129 (** 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 1130
adam@377 1131 Lemma nth_error_nil : forall A n x,
adam@377 1132 nth_error (@nil A) n = Some x
adam@377 1133 -> False.
adam@377 1134 destruct n; simpl; unfold error; congruence.
adam@377 1135 Defined.
adam@377 1136
adam@377 1137 Implicit Arguments nth_error_nil [A n x].
adam@377 1138
adam@377 1139 Lemma Some_inj : forall A (x y : A),
adam@377 1140 Some x = Some y
adam@377 1141 -> x = y.
adam@377 1142 congruence.
adam@377 1143 Defined.
adam@377 1144
adam@377 1145 Fixpoint getNat (types' : list Set) (values' : hlist (fun x : Set => x) types')
adam@377 1146 (natIndex : nat) : (nth_error types' natIndex = Some nat) -> nat :=
adam@377 1147 match values' with
adam@377 1148 | HNil => fun pf => match nth_error_nil pf with end
adam@377 1149 | HCons t ts x values'' =>
adam@377 1150 match natIndex return nth_error (t :: ts) natIndex = Some nat -> nat with
adam@377 1151 | O => fun pf =>
adam@377 1152 match Some_inj pf in _ = T return T with
adam@377 1153 | refl_equal => x
adam@377 1154 end
adam@377 1155 | S natIndex' => getNat values'' natIndex'
adam@377 1156 end
adam@377 1157 end.
adam@377 1158 End withTypes.
adam@377 1159
adam@377 1160 (** The problem becomes apparent when we experiment with running [getNat] on a concrete [types] list. *)
adam@377 1161
adam@377 1162 Definition myTypes := unit :: nat :: bool :: nil.
adam@377 1163 Definition myValues : hlist (fun x : Set => x) myTypes :=
adam@377 1164 tt ::: 3 ::: false ::: HNil.
adam@377 1165
adam@377 1166 Definition myNatIndex := 1.
adam@377 1167
adam@377 1168 Theorem myNatIndex_ok : nth_error myTypes myNatIndex = Some nat.
adam@377 1169 reflexivity.
adam@377 1170 Defined.
adam@377 1171
adam@377 1172 Eval compute in getNat myValues myNatIndex myNatIndex_ok.
adam@377 1173 (** %\vspace{-.15in}%[[
adam@377 1174 = 3
adam@377 1175 ]]
adam@377 1176
adam@398 1177 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 1178
adam@377 1179 Theorem getNat_is_reasonable : forall pf, getNat myValues myNatIndex pf = 3.
adam@377 1180 intro; compute.
adam@377 1181 (**
adam@377 1182 <<
adam@377 1183 1 subgoal
adam@377 1184 >>
adam@377 1185 %\vspace{-.3in}%[[
adam@377 1186 pf : nth_error myTypes myNatIndex = Some nat
adam@377 1187 ============================
adam@377 1188 match
adam@377 1189 match
adam@377 1190 pf in (_ = y)
adam@377 1191 return (nat = match y with
adam@377 1192 | Some H => H
adam@377 1193 | None => nat
adam@377 1194 end)
adam@377 1195 with
adam@377 1196 | eq_refl => eq_refl
adam@377 1197 end in (_ = T) return T
adam@377 1198 with
adam@377 1199 | eq_refl => 3
adam@377 1200 end = 3
adam@377 1201 ]]
adam@377 1202
adam@377 1203 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 1204
adam@377 1205 Abort.
adam@377 1206
adam@377 1207 (** 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 1208
adam@377 1209 Fixpoint copies A (x : A) (n : nat) : list A :=
adam@377 1210 match n with
adam@377 1211 | O => nil
adam@377 1212 | S n' => x :: copies x n'
adam@377 1213 end.
adam@377 1214
adam@377 1215 Fixpoint update A (ls : list A) (n : nat) (x : A) : list A :=
adam@377 1216 match ls with
adam@377 1217 | nil => copies x n ++ x :: nil
adam@377 1218 | y :: ls' => match n with
adam@377 1219 | O => x :: ls'
adam@377 1220 | S n' => y :: update ls' n' x
adam@377 1221 end
adam@377 1222 end.
adam@377 1223
adam@377 1224 (** Now let us revisit the definition of [getNat]. *)
adam@377 1225
adam@377 1226 Section withTypes'.
adam@377 1227 Variable types : list Set.
adam@377 1228 Variable natIndex : nat.
adam@377 1229
adam@398 1230 (** 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 1231
adam@377 1232 Definition types' := update types natIndex nat.
adam@377 1233
adam@377 1234 Variable values : hlist (fun x : Set => x) types'.
adam@377 1235
adam@377 1236 (** Now a bit of dependent pattern matching helps us rewrite [getNat] in a way that avoids any use of equality proofs. *)
adam@377 1237
adam@378 1238 Fixpoint skipCopies (n : nat)
adam@378 1239 : hlist (fun x : Set => x) (copies nat n ++ nat :: nil) -> nat :=
adam@378 1240 match n with
adam@378 1241 | O => fun vs => hhd vs
adam@378 1242 | S n' => fun vs => skipCopies n' (htl vs)
adam@378 1243 end.
adam@378 1244
adam@377 1245 Fixpoint getNat' (types'' : list Set) (natIndex : nat)
adam@377 1246 : hlist (fun x : Set => x) (update types'' natIndex nat) -> nat :=
adam@377 1247 match types'' with
adam@378 1248 | nil => skipCopies natIndex
adam@377 1249 | t :: types0 =>
adam@377 1250 match natIndex return hlist (fun x : Set => x)
adam@377 1251 (update (t :: types0) natIndex nat) -> nat with
adam@377 1252 | O => fun vs => hhd vs
adam@377 1253 | S natIndex' => fun vs => getNat' types0 natIndex' (htl vs)
adam@377 1254 end
adam@377 1255 end.
adam@377 1256 End withTypes'.
adam@377 1257
adam@398 1258 (** 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 1259
adam@377 1260 Theorem getNat_is_reasonable : getNat' myTypes myNatIndex myValues = 3.
adam@377 1261 reflexivity.
adam@377 1262 Qed.
adam@377 1263
adam@377 1264 (** The same parameters as before work without alteration, and we avoid use of axioms. *)