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author Adam Chlipala <adam@chlipala.net>
date Wed, 13 Feb 2013 10:24:28 -0500
parents fd6ec9b2dccb
children ed829eaa91b2
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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@488 94 In the third query's output, we see that the occurrence of [Type] that we check is assigned a fresh%\index{universe variable}% _universe variable_ [Top.3]. The output type increments [Top.3] to move up a level in the universe hierarchy. As we write code that uses definitions whose types mention universe variables, unification may refine the values of those variables. Luckily, the user rarely has to worry about the details.
adamc@227 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@479 135 The 1st term has type "Type (* (Top.15)+1 *)"
adam@479 136 which should be coercible to "Set".
adam@343 137 >>
adamc@227 138
adam@343 139 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@227 140
adamc@227 141 Reset id.
adamc@227 142 Definition id (T : Type) (x : T) : T := x.
adamc@227 143 Check id 0.
adamc@227 144 (** %\vspace{-.15in}% [[
adamc@227 145 id 0
adamc@227 146 : nat
adam@302 147 ]]
adam@302 148 *)
adamc@227 149
adamc@227 150 Check id Set.
adamc@227 151 (** %\vspace{-.15in}% [[
adamc@227 152 id Set
adamc@227 153 : Type $ Top.17 ^
adam@302 154 ]]
adam@302 155 *)
adamc@227 156
adamc@227 157 Check id Type.
adamc@227 158 (** %\vspace{-.15in}% [[
adamc@227 159 id Type $ Top.18 ^
adamc@227 160 : Type $ Top.19 ^
adam@302 161 ]]
adam@302 162 *)
adamc@227 163
adamc@227 164 (** 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@227 165 [[
adamc@227 166 Check id id.
adam@343 167 ]]
adamc@227 168
adam@343 169 <<
adamc@227 170 Error: Universe inconsistency (cannot enforce Top.16 < Top.16).
adam@343 171 >>
adamc@227 172
adam@479 173 %\index{universe inconsistency}%This error message reminds us that the universe variable for [T] still exists, even though it is usually hidden. To apply [id] to itself, that variable would need to be less than itself in the type hierarchy. Universe inconsistency error messages announce cases like this one where a term could only type-check by violating an implied constraint over universe variables. Such errors demonstrate that [Type] is _predicative_, where this word has a CIC meaning closely related to its usual mathematical meaning. A predicative system enforces the constraint that, when an object is defined using some sort of quantifier, none of the quantifiers may ever be instantiated with the object itself. %\index{impredicativity}%Impredicativity is associated with popular paradoxes in set theory, involving inconsistent constructions like "the set of all sets that do not contain themselves" (%\index{Russell's paradox}%Russell's paradox). Similar paradoxes would result from uncontrolled impredicativity in Coq. *)
adamc@227 174
adamc@227 175
adamc@227 176 (** ** Inductive Definitions *)
adamc@227 177
adam@505 178 (** Predicativity restrictions also apply to inductive definitions. As an example, let us consider a type of expression trees that allows injection of any native Coq value. The idea is that an [exp T] stands for an encoded expression of type [T].
adamc@227 179 [[
adamc@227 180 Inductive exp : Set -> Set :=
adamc@227 181 | Const : forall T : Set, T -> exp T
adamc@227 182 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 183 | Eq : forall T, exp T -> exp T -> exp bool.
adam@343 184 ]]
adamc@227 185
adam@343 186 <<
adamc@227 187 Error: Large non-propositional inductive types must be in Type.
adam@343 188 >>
adamc@227 189
adam@409 190 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@227 191
adamc@227 192 Inductive exp : Type -> Type :=
adamc@227 193 | Const : forall T, T -> exp T
adamc@227 194 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 195 | Eq : forall T, exp T -> exp T -> exp bool.
adamc@227 196
adam@505 197 (** Note that before we had to include an annotation [: Set] for the variable [T] in [Const]'s type, but we need no annotation now. When the type of a variable is not known, and when that variable is used in a context where only types are allowed, Coq infers that the variable is of type [Type], the right behavior here, though it was wrong for the [Set] version of [exp].
adamc@228 198
adamc@228 199 Our new definition is accepted. We can build some sample expressions. *)
adamc@227 200
adamc@227 201 Check Const 0.
adamc@227 202 (** %\vspace{-.15in}% [[
adamc@227 203 Const 0
adamc@227 204 : exp nat
adam@302 205 ]]
adam@302 206 *)
adamc@227 207
adamc@227 208 Check Pair (Const 0) (Const tt).
adamc@227 209 (** %\vspace{-.15in}% [[
adamc@227 210 Pair (Const 0) (Const tt)
adamc@227 211 : exp (nat * unit)
adam@302 212 ]]
adam@302 213 *)
adamc@227 214
adamc@227 215 Check Eq (Const Set) (Const Type).
adamc@227 216 (** %\vspace{-.15in}% [[
adamc@228 217 Eq (Const Set) (Const Type $ Top.59 ^ )
adamc@227 218 : exp bool
adamc@227 219 ]]
adamc@227 220
adamc@227 221 We can check many expressions, including fancy expressions that include types. However, it is not hard to hit a type-checking wall.
adamc@227 222 [[
adamc@227 223 Check Const (Const O).
adam@343 224 ]]
adamc@227 225
adam@343 226 <<
adamc@227 227 Error: Universe inconsistency (cannot enforce Top.42 < Top.42).
adam@343 228 >>
adamc@227 229
adamc@227 230 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@417 231 (** [[
adamc@227 232 Print exp.
adam@417 233 ]]
adam@444 234 %\vspace{-.15in}%[[
adamc@227 235 Inductive exp
adamc@227 236 : Type $ Top.8 ^ ->
adamc@227 237 Type
adamc@227 238 $ max(0, (Top.11)+1, (Top.14)+1, (Top.15)+1, (Top.19)+1) ^ :=
adamc@227 239 Const : forall T : Type $ Top.11 ^ , T -> exp T
adamc@227 240 | Pair : forall (T1 : Type $ Top.14 ^ ) (T2 : Type $ Top.15 ^ ),
adamc@227 241 exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 242 | Eq : forall T : Type $ Top.19 ^ , exp T -> exp T -> exp bool
adamc@227 243 ]]
adamc@227 244
adam@505 245 We see that the index type of [exp] has been assigned to universe level [Top.8]. In addition, each of the four occurrences of [Type] in the types of the constructors gets its own universe variable. Each of these variables appears explicitly in the type of [exp]. In particular, any type [exp T] lives at a universe level found by incrementing by one the maximum of the four argument variables. Therefore, [exp] _must_ live at a higher universe level than any type which may be passed to one of its constructors. This consequence led to the universe inconsistency.
adamc@227 246
adam@429 247 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@227 248
adamc@227 249 Print Universes.
adamc@227 250 (** %\vspace{-.15in}% [[
adamc@227 251 Top.19 < Top.9 <= Top.8
adamc@227 252 Top.15 < Top.9 <= Top.8 <= Coq.Init.Datatypes.38
adamc@227 253 Top.14 < Top.9 <= Top.8 <= Coq.Init.Datatypes.37
adamc@227 254 Top.11 < Top.9 <= Top.8
adamc@227 255 ]]
adamc@227 256
adam@343 257 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@227 258
adamc@227 259 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@227 260
adam@417 261 (* begin hide *)
adam@437 262 (* begin thide *)
adam@417 263 Inductive prod := pair.
adam@417 264 Reset prod.
adam@437 265 (* end thide *)
adam@417 266 (* end hide *)
adam@417 267
adam@444 268 (** %\vspace{-.3in}%[[
adamc@227 269 Print prod.
adam@417 270 ]]
adam@444 271 %\vspace{-.15in}%[[
adamc@227 272 Inductive prod (A : Type $ Coq.Init.Datatypes.37 ^ )
adamc@227 273 (B : Type $ Coq.Init.Datatypes.38 ^ )
adamc@227 274 : Type $ max(Coq.Init.Datatypes.37, Coq.Init.Datatypes.38) ^ :=
adamc@227 275 pair : A -> B -> A * B
adamc@227 276 ]]
adamc@227 277
adamc@227 278 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@227 279
adamc@227 280 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@227 281
adamc@227 282 %\medskip%
adamc@227 283
adam@505 284 The annotated definition of [prod] reveals something interesting. A type [prod A B] lives at a universe that is the maximum of the universes of [A] and [B]. From our earlier experiments, we might expect that [prod]'s universe would in fact need to be _one higher_ than the maximum. The critical difference is that, in the definition of [prod], [A] and [B] are defined as _parameters_; that is, they appear named to the left of the main colon, rather than appearing (possibly unnamed) to the right.
adamc@227 285
adamc@231 286 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@227 287
adamc@227 288 Check (nat, (Type, Set)).
adamc@227 289 (** %\vspace{-.15in}% [[
adamc@227 290 (nat, (Type $ Top.44 ^ , Set))
adamc@227 291 : Set * (Type $ Top.45 ^ * Type $ Top.46 ^ )
adamc@227 292 ]]
adamc@227 293
adamc@227 294 The same cannot be done with a counterpart to [prod] that does not use parameters. *)
adamc@227 295
adamc@227 296 Inductive prod' : Type -> Type -> Type :=
adamc@227 297 | pair' : forall A B : Type, A -> B -> prod' A B.
adam@444 298 (** %\vspace{-.15in}%[[
adamc@227 299 Check (pair' nat (pair' Type Set)).
adam@343 300 ]]
adamc@227 301
adam@343 302 <<
adamc@227 303 Error: Universe inconsistency (cannot enforce Top.51 < Top.51).
adam@343 304 >>
adamc@227 305
adamc@233 306 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@233 307
adam@343 308 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@233 309
adamc@233 310 Inductive foo (A : Type) : Type :=
adamc@233 311 | Foo : A -> foo A.
adamc@229 312
adamc@229 313 (* begin hide *)
adamc@229 314 Unset Printing Universes.
adamc@229 315 (* end hide *)
adamc@229 316
adamc@233 317 Check foo nat.
adamc@233 318 (** %\vspace{-.15in}% [[
adamc@233 319 foo nat
adamc@233 320 : Set
adam@302 321 ]]
adam@302 322 *)
adamc@233 323
adamc@233 324 Check foo Set.
adamc@233 325 (** %\vspace{-.15in}% [[
adamc@233 326 foo Set
adamc@233 327 : Type
adam@302 328 ]]
adam@302 329 *)
adamc@233 330
adamc@233 331 Check foo True.
adamc@233 332 (** %\vspace{-.15in}% [[
adamc@233 333 foo True
adamc@233 334 : Prop
adamc@233 335 ]]
adamc@233 336
adam@429 337 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@233 338
adamc@233 339 Imitation polymorphism can be confusing in some contexts. For instance, it is what is responsible for this weird behavior. *)
adamc@233 340
adamc@233 341 Inductive bar : Type := Bar : bar.
adamc@233 342
adamc@233 343 Check bar.
adamc@233 344 (** %\vspace{-.15in}% [[
adamc@233 345 bar
adamc@233 346 : Prop
adamc@233 347 ]]
adamc@233 348
adamc@233 349 The type that Coq comes up with may be used in strictly more contexts than the type one might have expected. *)
adamc@233 350
adamc@229 351
adam@388 352 (** ** Deciphering Baffling Messages About Inability to Unify *)
adam@388 353
adam@388 354 (** 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@388 355
adam@388 356 Theorem symmetry : forall A B : Type,
adam@388 357 A = B
adam@388 358 -> B = A.
adam@388 359 intros ? ? H; rewrite H; reflexivity.
adam@388 360 Qed.
adam@388 361
adam@388 362 (** Let us attempt an admittedly silly proof of the following theorem. *)
adam@388 363
adam@388 364 Theorem illustrative_but_silly_detour : unit = unit.
adam@444 365 (** %\vspace{-.25in}%[[
adam@444 366 apply symmetry.
adam@388 367 ]]
adam@388 368 <<
adam@388 369 Error: Impossible to unify "?35 = ?34" with "unit = unit".
adam@388 370 >>
adam@388 371
adam@458 372 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@388 373
adam@510 374 The following command is the secret to getting better error messages in such cases:%\index{Vernacular commands!Set Printing All}% *)
adam@388 375
adam@388 376 Set Printing All.
adam@444 377 (** %\vspace{-.15in}%[[
adam@444 378 apply symmetry.
adam@388 379 ]]
adam@388 380 <<
adam@388 381 Error: Impossible to unify "@eq Type ?46 ?45" with "@eq Set unit unit".
adam@388 382 >>
adam@388 383
adam@398 384 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@388 385
adam@388 386 Abort.
adam@388 387
adam@388 388 (** 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@388 389
adam@388 390 Theorem illustrative_but_silly_detour : (unit : Type) = unit.
adam@388 391 apply symmetry; reflexivity.
adam@388 392 Qed.
adam@388 393
adam@388 394 (** 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@388 395
adam@388 396 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@388 397
adam@388 398 Unset Printing All.
adam@388 399
adam@388 400 Theorem ex_symmetry : (exists x, x = 0) -> (exists x, 0 = x).
adam@435 401 eexists.
adam@388 402 (** %\vspace{-.15in}%[[
adam@388 403 H : exists x : nat, x = 0
adam@388 404 ============================
adam@388 405 0 = ?98
adam@388 406 ]]
adam@388 407 *)
adam@388 408
adam@388 409 destruct H.
adam@388 410 (** %\vspace{-.15in}%[[
adam@388 411 x : nat
adam@388 412 H : x = 0
adam@388 413 ============================
adam@388 414 0 = ?99
adam@388 415 ]]
adam@388 416 *)
adam@388 417
adam@444 418 (** %\vspace{-.2in}%[[
adam@444 419 symmetry; exact H.
adam@388 420 ]]
adam@388 421
adam@388 422 <<
adam@388 423 Error: In environment
adam@388 424 x : nat
adam@388 425 H : x = 0
adam@388 426 The term "H" has type "x = 0" while it is expected to have type
adam@388 427 "?99 = 0".
adam@388 428 >>
adam@388 429
adam@398 430 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@388 431
adam@388 432 Restart.
adam@388 433 destruct 1 as [x]; apply ex_intro with x; symmetry; assumption.
adam@388 434 Qed.
adam@388 435
adam@429 436 (** 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@388 437
adam@388 438
adamc@229 439 (** * The [Prop] Universe *)
adamc@229 440
adam@429 441 (** 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@229 442
adamc@229 443 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@229 444
adamc@229 445 Print sig.
adamc@229 446 (** %\vspace{-.15in}% [[
adamc@229 447 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@229 448 exist : forall x : A, P x -> sig P
adam@302 449 ]]
adam@302 450 *)
adamc@229 451
adamc@229 452 Print ex.
adamc@229 453 (** %\vspace{-.15in}% [[
adamc@229 454 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@229 455 ex_intro : forall x : A, P x -> ex P
adamc@229 456 ]]
adamc@229 457
adamc@229 458 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@229 459
adamc@229 460 Definition projS A (P : A -> Prop) (x : sig P) : A :=
adamc@229 461 match x with
adamc@229 462 | exist v _ => v
adamc@229 463 end.
adamc@229 464
adam@429 465 (* begin hide *)
adam@437 466 (* begin thide *)
adam@429 467 Definition projE := O.
adam@437 468 (* end thide *)
adam@429 469 (* end hide *)
adam@429 470
adamc@229 471 (** We run into trouble with a version that has been changed to work with [ex].
adamc@229 472 [[
adamc@229 473 Definition projE A (P : A -> Prop) (x : ex P) : A :=
adamc@229 474 match x with
adamc@229 475 | ex_intro v _ => v
adamc@229 476 end.
adam@343 477 ]]
adamc@229 478
adam@343 479 <<
adamc@229 480 Error:
adamc@229 481 Incorrect elimination of "x" in the inductive type "ex":
adamc@229 482 the return type has sort "Type" while it should be "Prop".
adamc@229 483 Elimination of an inductive object of sort Prop
adamc@229 484 is not allowed on a predicate in sort Type
adamc@229 485 because proofs can be eliminated only to build proofs.
adam@343 486 >>
adamc@229 487
adam@429 488 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@229 489
adamc@229 490 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@229 491
adam@398 492 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@229 493
adamc@229 494 Definition sym_sig (x : sig (fun n => n = 0)) : sig (fun n => 0 = n) :=
adamc@229 495 match x with
adamc@229 496 | exist n pf => exist _ n (sym_eq pf)
adamc@229 497 end.
adamc@229 498
adamc@229 499 Extraction sym_sig.
adamc@229 500 (** <<
adamc@229 501 (** val sym_sig : nat -> nat **)
adamc@229 502
adamc@229 503 let sym_sig x = x
adamc@229 504 >>
adamc@229 505
adamc@229 506 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@229 507
adamc@229 508 Definition sym_ex (x : ex (fun n => n = 0)) : ex (fun n => 0 = n) :=
adamc@229 509 match x with
adamc@229 510 | ex_intro n pf => ex_intro _ n (sym_eq pf)
adamc@229 511 end.
adamc@229 512
adamc@229 513 Extraction sym_ex.
adamc@229 514 (** <<
adamc@229 515 (** val sym_ex : __ **)
adamc@229 516
adamc@229 517 let sym_ex = __
adamc@229 518 >>
adamc@229 519
adam@435 520 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@229 521
adam@419 522 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@229 523
adam@398 524 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@229 525
adamc@229 526 %\medskip%
adamc@229 527
adam@409 528 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@229 529
adamc@229 530 Check forall P Q : Prop, P \/ Q -> Q \/ P.
adamc@229 531 (** %\vspace{-.15in}% [[
adamc@229 532 forall P Q : Prop, P \/ Q -> Q \/ P
adamc@229 533 : Prop
adamc@229 534 ]]
adamc@229 535
adamc@230 536 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@230 537
adamc@230 538 Impredicativity also allows us to implement a version of our earlier [exp] type that does not suffer from the weakness that we found. *)
adamc@230 539
adamc@230 540 Inductive expP : Type -> Prop :=
adamc@230 541 | ConstP : forall T, T -> expP T
adamc@230 542 | PairP : forall T1 T2, expP T1 -> expP T2 -> expP (T1 * T2)
adamc@230 543 | EqP : forall T, expP T -> expP T -> expP bool.
adamc@230 544
adamc@230 545 Check ConstP 0.
adamc@230 546 (** %\vspace{-.15in}% [[
adamc@230 547 ConstP 0
adamc@230 548 : expP nat
adam@302 549 ]]
adam@302 550 *)
adamc@230 551
adamc@230 552 Check PairP (ConstP 0) (ConstP tt).
adamc@230 553 (** %\vspace{-.15in}% [[
adamc@230 554 PairP (ConstP 0) (ConstP tt)
adamc@230 555 : expP (nat * unit)
adam@302 556 ]]
adam@302 557 *)
adamc@230 558
adamc@230 559 Check EqP (ConstP Set) (ConstP Type).
adamc@230 560 (** %\vspace{-.15in}% [[
adamc@230 561 EqP (ConstP Set) (ConstP Type)
adamc@230 562 : expP bool
adam@302 563 ]]
adam@302 564 *)
adamc@230 565
adamc@230 566 Check ConstP (ConstP O).
adamc@230 567 (** %\vspace{-.15in}% [[
adamc@230 568 ConstP (ConstP 0)
adamc@230 569 : expP (expP nat)
adamc@230 570 ]]
adamc@230 571
adam@287 572 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@230 573
adamc@230 574 Inductive eqPlus : forall T, T -> T -> Prop :=
adamc@230 575 | Base : forall T (x : T), eqPlus x x
adamc@230 576 | Func : forall dom ran (f1 f2 : dom -> ran),
adamc@230 577 (forall x : dom, eqPlus (f1 x) (f2 x))
adamc@230 578 -> eqPlus f1 f2.
adamc@230 579
adamc@230 580 Check (Base 0).
adamc@230 581 (** %\vspace{-.15in}% [[
adamc@230 582 Base 0
adamc@230 583 : eqPlus 0 0
adam@302 584 ]]
adam@302 585 *)
adamc@230 586
adamc@230 587 Check (Func (fun n => n) (fun n => 0 + n) (fun n => Base n)).
adamc@230 588 (** %\vspace{-.15in}% [[
adamc@230 589 Func (fun n : nat => n) (fun n : nat => 0 + n) (fun n : nat => Base n)
adamc@230 590 : eqPlus (fun n : nat => n) (fun n : nat => 0 + n)
adam@302 591 ]]
adam@302 592 *)
adamc@230 593
adamc@230 594 Check (Base (Base 1)).
adamc@230 595 (** %\vspace{-.15in}% [[
adamc@230 596 Base (Base 1)
adamc@230 597 : eqPlus (Base 1) (Base 1)
adam@302 598 ]]
adam@302 599 *)
adamc@230 600
adam@343 601 (** Stating equality facts about proofs may seem baroque, but we have already seen its utility in the chapter on reasoning about equality proofs. *)
adam@343 602
adamc@230 603
adamc@230 604 (** * Axioms *)
adamc@230 605
adam@409 606 (** 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@230 607
adamc@230 608 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@230 609
adamc@230 610 (** ** The Basics *)
adamc@230 611
adam@343 612 (** One simple example of a useful axiom is the %\index{law of the excluded middle}%law of the excluded middle. *)
adamc@230 613
adamc@230 614 Require Import Classical_Prop.
adamc@230 615 Print classic.
adamc@230 616 (** %\vspace{-.15in}% [[
adamc@230 617 *** [ classic : forall P : Prop, P \/ ~ P ]
adamc@230 618 ]]
adamc@230 619
adam@343 620 In the implementation of module [Classical_Prop], this axiom was defined with the command%\index{Vernacular commands!Axiom}% *)
adamc@230 621
adamc@230 622 Axiom classic : forall P : Prop, P \/ ~ P.
adamc@230 623
adam@343 624 (** 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@230 625
adam@458 626 Parameter num : nat.
adam@458 627 Axiom positive : num > 0.
adam@458 628 Reset num.
adamc@230 629
adam@429 630 (** 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@230 631
adam@409 632 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@230 633
adam@287 634 Axiom not_classic : ~ forall P : Prop, P \/ ~ P.
adamc@230 635
adamc@230 636 Theorem uhoh : False.
adam@287 637 generalize classic not_classic; tauto.
adamc@230 638 Qed.
adamc@230 639
adamc@230 640 Theorem uhoh_again : 1 + 1 = 3.
adamc@230 641 destruct uhoh.
adamc@230 642 Qed.
adamc@230 643
adamc@230 644 Reset not_classic.
adamc@230 645
adam@429 646 (** 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@230 647
adam@475 648 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@230 649
adam@398 650 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@230 651
adam@343 652 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@230 653
adamc@230 654 Theorem t1 : forall P : Prop, P -> ~ ~ P.
adamc@230 655 tauto.
adamc@230 656 Qed.
adamc@230 657
adamc@230 658 Print Assumptions t1.
adam@343 659 (** <<
adamc@230 660 Closed under the global context
adam@343 661 >>
adam@302 662 *)
adamc@230 663
adamc@230 664 Theorem t2 : forall P : Prop, ~ ~ P -> P.
adam@444 665 (** %\vspace{-.25in}%[[
adamc@230 666 tauto.
adam@343 667 ]]
adam@343 668 <<
adamc@230 669 Error: tauto failed.
adam@343 670 >>
adam@302 671 *)
adamc@230 672 intro P; destruct (classic P); tauto.
adamc@230 673 Qed.
adamc@230 674
adamc@230 675 Print Assumptions t2.
adamc@230 676 (** %\vspace{-.15in}% [[
adamc@230 677 Axioms:
adamc@230 678 classic : forall P : Prop, P \/ ~ P
adamc@230 679 ]]
adamc@230 680
adam@398 681 It is possible to avoid this dependence in some specific cases, where excluded middle _is_ provable, for decidable families of propositions. *)
adamc@230 682
adam@287 683 Theorem nat_eq_dec : forall n m : nat, n = m \/ n <> m.
adamc@230 684 induction n; destruct m; intuition; generalize (IHn m); intuition.
adamc@230 685 Qed.
adamc@230 686
adamc@230 687 Theorem t2' : forall n m : nat, ~ ~ (n = m) -> n = m.
adam@287 688 intros n m; destruct (nat_eq_dec n m); tauto.
adamc@230 689 Qed.
adamc@230 690
adamc@230 691 Print Assumptions t2'.
adam@343 692 (** <<
adamc@230 693 Closed under the global context
adam@343 694 >>
adamc@230 695
adamc@230 696 %\bigskip%
adamc@230 697
adam@409 698 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@230 699
adamc@230 700 Require Import ProofIrrelevance.
adamc@230 701 Print proof_irrelevance.
adam@458 702
adamc@230 703 (** %\vspace{-.15in}% [[
adamc@230 704 *** [ proof_irrelevance : forall (P : Prop) (p1 p2 : P), p1 = p2 ]
adamc@230 705 ]]
adamc@230 706
adam@458 707 This axiom asserts that any two proofs of the same proposition are equal. Recall this example function from Chapter 6. *)
adamc@230 708
adamc@230 709 (* begin hide *)
adamc@230 710 Lemma zgtz : 0 > 0 -> False.
adamc@230 711 crush.
adamc@230 712 Qed.
adamc@230 713 (* end hide *)
adamc@230 714
adamc@230 715 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@230 716 match n with
adamc@230 717 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@230 718 | S n' => fun _ => n'
adamc@230 719 end.
adamc@230 720
adam@343 721 (** We might want to prove that different proofs of [n > 0] do not lead to different results from our richly typed predecessor function. *)
adamc@230 722
adamc@230 723 Theorem pred_strong1_irrel : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 724 destruct n; crush.
adamc@230 725 Qed.
adamc@230 726
adamc@230 727 (** 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@230 728
adamc@230 729 Theorem pred_strong1_irrel' : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 730 intros; f_equal; apply proof_irrelevance.
adamc@230 731 Qed.
adamc@230 732
adamc@230 733
adamc@230 734 (** %\bigskip%
adamc@230 735
adamc@230 736 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@230 737
adamc@230 738 Require Import Eqdep.
adamc@230 739 Import Eq_rect_eq.
adamc@230 740 Print eq_rect_eq.
adamc@230 741 (** %\vspace{-.15in}% [[
adamc@230 742 *** [ eq_rect_eq :
adamc@230 743 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adamc@230 744 x = eq_rect p Q x p h ]
adamc@230 745 ]]
adamc@230 746
adam@429 747 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@230 748
adam@426 749 Corollary UIP_refl : forall A (x : A) (pf : x = x), pf = eq_refl x.
adam@426 750 intros; replace pf with (eq_rect x (eq x) (eq_refl x) x pf); [
adamc@230 751 symmetry; apply eq_rect_eq
adamc@230 752 | exact (match pf as pf' return match pf' in _ = y return x = y with
adam@426 753 | eq_refl => eq_refl x
adamc@230 754 end = pf' with
adam@426 755 | eq_refl => eq_refl _
adamc@230 756 end) ].
adamc@230 757 Qed.
adamc@230 758
adamc@230 759 Corollary UIP : forall A (x y : A) (pf1 pf2 : x = y), pf1 = pf2.
adamc@230 760 intros; generalize pf1 pf2; subst; intros;
adamc@230 761 match goal with
adamc@230 762 | [ |- ?pf1 = ?pf2 ] => rewrite (UIP_refl pf1); rewrite (UIP_refl pf2); reflexivity
adamc@230 763 end.
adamc@230 764 Qed.
adamc@230 765
adam@436 766 (* begin hide *)
adam@437 767 (* begin thide *)
adam@436 768 Require Eqdep_dec.
adam@437 769 (* end thide *)
adam@436 770 (* end hide *)
adam@436 771
adamc@231 772 (** 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@230 773
adamc@230 774 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@230 775
adamc@230 776 %\bigskip%
adamc@230 777
adamc@230 778 There are two more basic axioms that are often assumed, to avoid complications that do not arise in set theory. *)
adamc@230 779
adamc@230 780 Require Import FunctionalExtensionality.
adamc@230 781 Print functional_extensionality_dep.
adamc@230 782 (** %\vspace{-.15in}% [[
adamc@230 783 *** [ functional_extensionality_dep :
adamc@230 784 forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
adamc@230 785 (forall x : A, f x = g x) -> f = g ]
adamc@230 786
adamc@230 787 ]]
adamc@230 788
adamc@230 789 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@230 790
adam@343 791 A simple corollary shows that the same property applies to predicates. *)
adamc@230 792
adamc@230 793 Corollary predicate_extensionality : forall (A : Type) (B : A -> Prop) (f g : forall x : A, B x),
adamc@230 794 (forall x : A, f x = g x) -> f = g.
adamc@230 795 intros; apply functional_extensionality_dep; assumption.
adamc@230 796 Qed.
adamc@230 797
adam@343 798 (** In some cases, one might prefer to assert this corollary as the axiom, to restrict the consequences to proofs and not programs. *)
adam@343 799
adamc@230 800
adamc@230 801 (** ** Axioms of Choice *)
adamc@230 802
adam@343 803 (** 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@230 804
adam@398 805 First, it is possible to implement a choice operator _without_ axioms in some potentially surprising cases. *)
adamc@230 806
adamc@230 807 Require Import ConstructiveEpsilon.
adamc@230 808 Check constructive_definite_description.
adamc@230 809 (** %\vspace{-.15in}% [[
adamc@230 810 constructive_definite_description
adamc@230 811 : forall (A : Set) (f : A -> nat) (g : nat -> A),
adamc@230 812 (forall x : A, g (f x) = x) ->
adamc@230 813 forall P : A -> Prop,
adam@505 814 (forall x : A, {P x} + { ~ P x}) ->
adamc@230 815 (exists! x : A, P x) -> {x : A | P x}
adam@302 816 ]]
adam@302 817 *)
adamc@230 818
adamc@230 819 Print Assumptions constructive_definite_description.
adam@343 820 (** <<
adamc@230 821 Closed under the global context
adam@343 822 >>
adamc@230 823
adam@398 824 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@230 825
adamc@230 826 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@230 827
adamc@230 828 Require Import ClassicalUniqueChoice.
adamc@230 829 Check dependent_unique_choice.
adamc@230 830 (** %\vspace{-.15in}% [[
adamc@230 831 dependent_unique_choice
adamc@230 832 : forall (A : Type) (B : A -> Type) (R : forall x : A, B x -> Prop),
adamc@230 833 (forall x : A, exists! y : B x, R x y) ->
adam@343 834 exists f : forall x : A, B x,
adam@343 835 forall x : A, R x (f x)
adamc@230 836 ]]
adamc@230 837
adamc@230 838 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@230 839
adam@436 840 (* begin hide *)
adam@437 841 (* begin thide *)
adam@436 842 Require RelationalChoice.
adam@437 843 (* end thide *)
adam@436 844 (* end hide *)
adam@436 845
adamc@230 846 Require Import ClassicalChoice.
adamc@230 847 Check choice.
adamc@230 848 (** %\vspace{-.15in}% [[
adamc@230 849 choice
adamc@230 850 : forall (A B : Type) (R : A -> B -> Prop),
adamc@230 851 (forall x : A, exists y : B, R x y) ->
adamc@230 852 exists f : A -> B, forall x : A, R x (f x)
adam@444 853 ]]
adamc@230 854
adamc@230 855 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@230 856
adamc@230 857 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@230 858
adamc@230 859 Definition choice_Set (A B : Type) (R : A -> B -> Prop) (H : forall x : A, {y : B | R x y})
adamc@230 860 : {f : A -> B | forall x : A, R x (f x)} :=
adamc@230 861 exist (fun f => forall x : A, R x (f x))
adamc@230 862 (fun x => proj1_sig (H x)) (fun x => proj2_sig (H x)).
adamc@230 863
adam@458 864 (** %\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@230 865
adam@505 866 However, when we combine an axiom of choice with the law of the excluded middle, the idea of "choice" becomes more interesting. Excluded middle gives us a highly non-computational way of constructing proofs, but it does not change the computational nature of programs. Thus, the axiom of choice is still giving us a way of translating between two different sorts of "programs," but the input programs (which are proofs) may be written in a rich language that goes beyond normal computability. This combination truly is more than repackaging a function with a different type.
adamc@230 867
adamc@230 868 %\bigskip%
adamc@230 869
adam@505 870 The Coq tools support a command-line flag %\index{impredicative Set}%<<-impredicative-set>>, which modifies Gallina in a more fundamental way by making [Set] impredicative. A term like [forall T : Set, T] has type [Set], and inductive definitions in [Set] may have constructors that quantify over arguments of any types. To maintain consistency, an elimination restriction must be imposed, similarly to the restriction for [Prop]. The restriction only applies to large inductive types, where some constructor quantifies over a type of type [Type]. In such cases, a value in this inductive type may only be pattern-matched over to yield a result type whose type is [Set] or [Prop]. This rule contrasts with the rule for [Prop], where the restriction applies even to non-large inductive types, and where the result type may only have type [Prop].
adamc@230 871
adam@505 872 In old versions of Coq, [Set] was impredicative by default. Later versions make [Set] predicative to avoid inconsistency with some classical axioms. In particular, one should watch out when using impredicative [Set] with axioms of choice. In combination with excluded middle or predicate extensionality, inconsistency can result. Impredicative [Set] can be useful for modeling inherently impredicative mathematical concepts, but almost all Coq developments get by fine without it. *)
adamc@230 873
adamc@230 874 (** ** Axioms and Computation *)
adamc@230 875
adam@398 876 (** 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@230 877
adamc@230 878 Definition cast (x y : Set) (pf : x = y) (v : x) : y :=
adamc@230 879 match pf with
adam@426 880 | eq_refl => v
adamc@230 881 end.
adamc@230 882
adamc@230 883 (** Computation over programs that use [cast] can proceed smoothly. *)
adamc@230 884
adam@426 885 Eval compute in (cast (eq_refl (nat -> nat)) (fun n => S n)) 12.
adam@343 886 (** %\vspace{-.15in}%[[
adamc@230 887 = 13
adamc@230 888 : nat
adam@302 889 ]]
adam@302 890 *)
adamc@230 891
adamc@230 892 (** Things do not go as smoothly when we use [cast] with proofs that rely on axioms. *)
adamc@230 893
adamc@230 894 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 895 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 896 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 897 Qed.
adamc@230 898
adamc@230 899 Eval compute in (cast t3 (fun _ => First)) 12.
adam@444 900 (** %\vspace{-.15in}%[[
adamc@230 901 = match t3 in (_ = P) return P with
adam@426 902 | eq_refl => fun n : nat => First
adamc@230 903 end 12
adamc@230 904 : fin (12 + 1)
adamc@230 905 ]]
adamc@230 906
adam@458 907 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@230 908
adamc@230 909 Reset t3.
adamc@230 910
adamc@230 911 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 912 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 913 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 914 Defined.
adamc@230 915
adamc@230 916 Eval compute in (cast t3 (fun _ => First)) 12.
adam@444 917 (** %\vspace{-.15in}%[[
adamc@230 918 = match
adamc@230 919 match
adamc@230 920 match
adamc@230 921 functional_extensionality
adamc@230 922 ....
adamc@230 923 ]]
adamc@230 924
adam@398 925 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@230 926
adamc@230 927 If we are careful in using tactics to prove an equality, we can still compute with casts over the proof. *)
adamc@230 928
adamc@230 929 Lemma plus1 : forall n, S n = n + 1.
adamc@230 930 induction n; simpl; intuition.
adamc@230 931 Defined.
adamc@230 932
adamc@230 933 Theorem t4 : forall n, fin (S n) = fin (n + 1).
adamc@230 934 intro; f_equal; apply plus1.
adamc@230 935 Defined.
adamc@230 936
adamc@230 937 Eval compute in cast (t4 13) First.
adamc@230 938 (** %\vspace{-.15in}% [[
adamc@230 939 = First
adamc@230 940 : fin (13 + 1)
adam@302 941 ]]
adam@343 942
adam@426 943 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@343 944
adam@344 945
adam@344 946 (** ** Methods for Avoiding Axioms *)
adam@344 947
adam@409 948 (** 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@344 949
adam@429 950 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@344 951
adam@344 952 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@344 953
adam@344 954 Theorem fin_cases : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 955 intros; dep_destruct f; eauto.
adam@344 956 Qed.
adam@344 957
adam@429 958 (* begin hide *)
adam@429 959 Require Import JMeq.
adam@437 960 (* begin thide *)
adam@429 961 Definition jme := (JMeq, JMeq_eq).
adam@437 962 (* end thide *)
adam@429 963 (* end hide *)
adam@429 964
adam@344 965 Print Assumptions fin_cases.
adam@344 966 (** %\vspace{-.15in}%[[
adam@344 967 Axioms:
adam@429 968 JMeq_eq : forall (A : Type) (x y : A), JMeq x y -> x = y
adam@344 969 ]]
adam@344 970
adam@344 971 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@344 972
adam@344 973 (* begin thide *)
adam@344 974 Lemma fin_cases_again' : forall n (f : fin n),
adam@344 975 match n return fin n -> Prop with
adam@344 976 | O => fun _ => False
adam@344 977 | S n' => fun f => f = First \/ exists f', f = Next f'
adam@344 978 end f.
adam@344 979 destruct f; eauto.
adam@344 980 Qed.
adam@344 981
adam@344 982 (** 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@344 983
adam@344 984 Theorem fin_cases_again : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 985 intros; exact (fin_cases_again' f).
adam@344 986 Qed.
adam@344 987 (* end thide *)
adam@344 988
adam@344 989 Print Assumptions fin_cases_again.
adam@344 990 (** %\vspace{-.15in}%
adam@344 991 <<
adam@344 992 Closed under the global context
adam@344 993 >>
adam@344 994
adam@345 995 *)
adam@345 996
adam@345 997 (* begin thide *)
adam@345 998 (** 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@345 999
adam@429 1000 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 1001
adam@345 1002 Definition finOut n (f : fin n) : match n return fin n -> Type with
adam@345 1003 | O => fun _ => Empty_set
adam@345 1004 | _ => fun f => {f' : _ | f = Next f'} + {f = First}
adam@345 1005 end f :=
adam@345 1006 match f with
adam@426 1007 | First _ => inright _ (eq_refl _)
adam@426 1008 | Next _ f' => inleft _ (exist _ f' (eq_refl _))
adam@345 1009 end.
adam@345 1010 (* end thide *)
adam@345 1011
adam@345 1012 (** 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 1013
adam@344 1014 Inductive formula : list Type -> Type :=
adam@344 1015 | Inject : forall Ts, Prop -> formula Ts
adam@344 1016 | VarEq : forall T Ts, T -> formula (T :: Ts)
adam@344 1017 | Lift : forall T Ts, formula Ts -> formula (T :: Ts)
adam@344 1018 | Forall : forall T Ts, formula (T :: Ts) -> formula Ts
adam@344 1019 | And : forall Ts, formula Ts -> formula Ts -> formula Ts.
adam@344 1020
adam@344 1021 (** 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 1022
adam@344 1023 Inductive proof : formula nil -> Prop :=
adam@344 1024 | PInject : forall (P : Prop), P -> proof (Inject nil P)
adam@344 1025 | PAnd : forall p q, proof p -> proof q -> proof (And p q).
adam@344 1026
adam@429 1027 (** Let us prove a lemma showing that a "[P /\ Q -> P]" rule is derivable within the rules of [proof]. *)
adam@344 1028
adam@344 1029 Theorem proj1 : forall p q, proof (And p q) -> proof p.
adam@344 1030 destruct 1.
adam@344 1031 (** %\vspace{-.15in}%[[
adam@344 1032 p : formula nil
adam@344 1033 q : formula nil
adam@344 1034 P : Prop
adam@344 1035 H : P
adam@344 1036 ============================
adam@344 1037 proof p
adam@344 1038 ]]
adam@344 1039 *)
adam@344 1040
adam@344 1041 (** 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 1042
adam@344 1043 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 1044
adam@344 1045 Restart.
adam@344 1046 Require Import Program.
adam@344 1047 intros ? ? H; dependent destruction H; auto.
adam@344 1048 Qed.
adam@344 1049
adam@344 1050 Print Assumptions proj1.
adam@344 1051 (** %\vspace{-.15in}%[[
adam@344 1052 Axioms:
adam@344 1053 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1054 x = eq_rect p Q x p h
adam@344 1055 ]]
adam@344 1056
adam@344 1057 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 1058
adam@344 1059 Lemma proj1_again' : forall r, proof r
adam@344 1060 -> forall p q, r = And p q -> proof p.
adam@344 1061 destruct 1; crush.
adam@344 1062 (** %\vspace{-.15in}%[[
adam@344 1063 H0 : Inject [] P = And p q
adam@344 1064 ============================
adam@344 1065 proof p
adam@344 1066 ]]
adam@344 1067
adam@344 1068 The first goal looks reasonable. Hypothesis [H0] is clearly contradictory, as [discriminate] can show. *)
adam@344 1069
adam@344 1070 discriminate.
adam@344 1071 (** %\vspace{-.15in}%[[
adam@344 1072 H : proof p
adam@344 1073 H1 : And p q = And p0 q0
adam@344 1074 ============================
adam@344 1075 proof p0
adam@344 1076 ]]
adam@344 1077
adam@344 1078 It looks like we are almost done. Hypothesis [H1] gives [p = p0] by injectivity of constructors, and then [H] finishes the case. *)
adam@344 1079
adam@344 1080 injection H1; intros.
adam@344 1081
adam@429 1082 (* begin hide *)
adam@437 1083 (* begin thide *)
adam@429 1084 Definition existT' := existT.
adam@437 1085 (* end thide *)
adam@429 1086 (* end hide *)
adam@429 1087
adam@429 1088 (** Unfortunately, the "equality" that we expected between [p] and [p0] comes in a strange form:
adam@344 1089
adam@344 1090 [[
adam@344 1091 H3 : existT (fun Ts : list Type => formula Ts) []%list p =
adam@344 1092 existT (fun Ts : list Type => formula Ts) []%list p0
adam@344 1093 ============================
adam@344 1094 proof p0
adam@344 1095 ]]
adam@344 1096
adam@345 1097 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 1098
adam@344 1099 How exactly does an axiom come into the picture here? Let us ask [crush] to finish the proof. *)
adam@344 1100
adam@344 1101 crush.
adam@344 1102 Qed.
adam@344 1103
adam@344 1104 Print Assumptions proj1_again'.
adam@344 1105 (** %\vspace{-.15in}%[[
adam@344 1106 Axioms:
adam@344 1107 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1108 x = eq_rect p Q x p h
adam@344 1109 ]]
adam@344 1110
adam@344 1111 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 1112
adam@479 1113 Hope is not lost, however. We can produce an even stranger looking lemma, which gives us the theorem without axioms. As always when we want to do case analysis on a term with a tricky dependent type, the key is to refactor the theorem statement so that every term we [match] on has _variables_ as its type indices; so instead of talking about proofs of [And p q], we talk about proofs of an arbitrary [r], but we only conclude anything interesting when [r] is an [And]. *)
adam@344 1114
adam@344 1115 Lemma proj1_again'' : forall r, proof r
adam@344 1116 -> match r with
adam@344 1117 | And Ps p _ => match Ps return formula Ps -> Prop with
adam@344 1118 | nil => fun p => proof p
adam@344 1119 | _ => fun _ => True
adam@344 1120 end p
adam@344 1121 | _ => True
adam@344 1122 end.
adam@344 1123 destruct 1; auto.
adam@344 1124 Qed.
adam@344 1125
adam@344 1126 Theorem proj1_again : forall p q, proof (And p q) -> proof p.
adam@344 1127 intros ? ? H; exact (proj1_again'' H).
adam@344 1128 Qed.
adam@344 1129
adam@344 1130 Print Assumptions proj1_again.
adam@344 1131 (** <<
adam@344 1132 Closed under the global context
adam@344 1133 >>
adam@344 1134
adam@377 1135 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 1136
adam@377 1137 %\medskip%
adam@377 1138
adam@398 1139 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 1140
adam@377 1141 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 1142
adam@377 1143 Section withTypes.
adam@377 1144 Variable types : list Set.
adam@377 1145
adam@377 1146 (** 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 1147
adam@377 1148 Variable values : hlist (fun x : Set => x) types.
adam@377 1149
adam@377 1150 (** 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 1151
adam@377 1152 Variable natIndex : nat.
adam@377 1153 Variable natIndex_ok : nth_error types natIndex = Some nat.
adam@377 1154
adam@377 1155 (** 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 1156
adam@377 1157 Lemma nth_error_nil : forall A n x,
adam@377 1158 nth_error (@nil A) n = Some x
adam@377 1159 -> False.
adam@377 1160 destruct n; simpl; unfold error; congruence.
adam@377 1161 Defined.
adam@377 1162
adam@377 1163 Implicit Arguments nth_error_nil [A n x].
adam@377 1164
adam@377 1165 Lemma Some_inj : forall A (x y : A),
adam@377 1166 Some x = Some y
adam@377 1167 -> x = y.
adam@377 1168 congruence.
adam@377 1169 Defined.
adam@377 1170
adam@377 1171 Fixpoint getNat (types' : list Set) (values' : hlist (fun x : Set => x) types')
adam@377 1172 (natIndex : nat) : (nth_error types' natIndex = Some nat) -> nat :=
adam@377 1173 match values' with
adam@377 1174 | HNil => fun pf => match nth_error_nil pf with end
adam@377 1175 | HCons t ts x values'' =>
adam@377 1176 match natIndex return nth_error (t :: ts) natIndex = Some nat -> nat with
adam@377 1177 | O => fun pf =>
adam@377 1178 match Some_inj pf in _ = T return T with
adam@426 1179 | eq_refl => x
adam@377 1180 end
adam@377 1181 | S natIndex' => getNat values'' natIndex'
adam@377 1182 end
adam@377 1183 end.
adam@377 1184 End withTypes.
adam@377 1185
adam@377 1186 (** The problem becomes apparent when we experiment with running [getNat] on a concrete [types] list. *)
adam@377 1187
adam@377 1188 Definition myTypes := unit :: nat :: bool :: nil.
adam@377 1189 Definition myValues : hlist (fun x : Set => x) myTypes :=
adam@377 1190 tt ::: 3 ::: false ::: HNil.
adam@377 1191
adam@377 1192 Definition myNatIndex := 1.
adam@377 1193
adam@377 1194 Theorem myNatIndex_ok : nth_error myTypes myNatIndex = Some nat.
adam@377 1195 reflexivity.
adam@377 1196 Defined.
adam@377 1197
adam@377 1198 Eval compute in getNat myValues myNatIndex myNatIndex_ok.
adam@377 1199 (** %\vspace{-.15in}%[[
adam@377 1200 = 3
adam@377 1201 ]]
adam@377 1202
adam@398 1203 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 1204
adam@377 1205 Theorem getNat_is_reasonable : forall pf, getNat myValues myNatIndex pf = 3.
adam@377 1206 intro; compute.
adam@377 1207 (**
adam@377 1208 <<
adam@377 1209 1 subgoal
adam@377 1210 >>
adam@377 1211 %\vspace{-.3in}%[[
adam@377 1212 pf : nth_error myTypes myNatIndex = Some nat
adam@377 1213 ============================
adam@377 1214 match
adam@377 1215 match
adam@377 1216 pf in (_ = y)
adam@377 1217 return (nat = match y with
adam@377 1218 | Some H => H
adam@377 1219 | None => nat
adam@377 1220 end)
adam@377 1221 with
adam@377 1222 | eq_refl => eq_refl
adam@377 1223 end in (_ = T) return T
adam@377 1224 with
adam@377 1225 | eq_refl => 3
adam@377 1226 end = 3
adam@377 1227 ]]
adam@377 1228
adam@377 1229 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 1230
adam@377 1231 Abort.
adam@377 1232
adam@377 1233 (** 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 1234
adam@377 1235 Fixpoint copies A (x : A) (n : nat) : list A :=
adam@377 1236 match n with
adam@377 1237 | O => nil
adam@377 1238 | S n' => x :: copies x n'
adam@377 1239 end.
adam@377 1240
adam@377 1241 Fixpoint update A (ls : list A) (n : nat) (x : A) : list A :=
adam@377 1242 match ls with
adam@377 1243 | nil => copies x n ++ x :: nil
adam@377 1244 | y :: ls' => match n with
adam@377 1245 | O => x :: ls'
adam@377 1246 | S n' => y :: update ls' n' x
adam@377 1247 end
adam@377 1248 end.
adam@377 1249
adam@377 1250 (** Now let us revisit the definition of [getNat]. *)
adam@377 1251
adam@377 1252 Section withTypes'.
adam@377 1253 Variable types : list Set.
adam@377 1254 Variable natIndex : nat.
adam@377 1255
adam@429 1256 (** 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 1257
adam@377 1258 Definition types' := update types natIndex nat.
adam@377 1259
adam@377 1260 Variable values : hlist (fun x : Set => x) types'.
adam@377 1261
adam@377 1262 (** Now a bit of dependent pattern matching helps us rewrite [getNat] in a way that avoids any use of equality proofs. *)
adam@377 1263
adam@378 1264 Fixpoint skipCopies (n : nat)
adam@378 1265 : hlist (fun x : Set => x) (copies nat n ++ nat :: nil) -> nat :=
adam@378 1266 match n with
adam@378 1267 | O => fun vs => hhd vs
adam@378 1268 | S n' => fun vs => skipCopies n' (htl vs)
adam@378 1269 end.
adam@378 1270
adam@377 1271 Fixpoint getNat' (types'' : list Set) (natIndex : nat)
adam@377 1272 : hlist (fun x : Set => x) (update types'' natIndex nat) -> nat :=
adam@377 1273 match types'' with
adam@378 1274 | nil => skipCopies natIndex
adam@377 1275 | t :: types0 =>
adam@377 1276 match natIndex return hlist (fun x : Set => x)
adam@377 1277 (update (t :: types0) natIndex nat) -> nat with
adam@377 1278 | O => fun vs => hhd vs
adam@377 1279 | S natIndex' => fun vs => getNat' types0 natIndex' (htl vs)
adam@377 1280 end
adam@377 1281 end.
adam@377 1282 End withTypes'.
adam@377 1283
adam@398 1284 (** 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 1285
adam@377 1286 Theorem getNat_is_reasonable : getNat' myTypes myNatIndex myValues = 3.
adam@377 1287 reflexivity.
adam@377 1288 Qed.
adam@377 1289
adam@377 1290 (** The same parameters as before work without alteration, and we avoid use of axioms. *)