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author Adam Chlipala <adam@chlipala.net>
date Fri, 20 Apr 2012 12:49:47 -0400
parents 057a29f9c773
children 05efde66559d
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
adamc@227 18
adamc@227 19 (** %\chapter{Universes and Axioms}% *)
adamc@227 20
adam@343 21 (** 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 22
adam@343 23 %\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 24
adamc@227 25
adamc@227 26 (** * The [Type] Hierarchy *)
adamc@227 27
adam@343 28 (** %\index{type hierarchy}%Every object in Gallina has a type. *)
adamc@227 29
adamc@227 30 Check 0.
adamc@227 31 (** %\vspace{-.15in}% [[
adamc@227 32 0
adamc@227 33 : nat
adamc@227 34
adamc@227 35 ]]
adamc@227 36
adamc@227 37 It is natural enough that zero be considered as a natural number. *)
adamc@227 38
adamc@227 39 Check nat.
adamc@227 40 (** %\vspace{-.15in}% [[
adamc@227 41 nat
adamc@227 42 : Set
adamc@227 43
adamc@227 44 ]]
adamc@227 45
adam@287 46 From a set theory perspective, it is unsurprising to consider the natural numbers as a %``%#"#set.#"#%''% *)
adamc@227 47
adamc@227 48 Check Set.
adamc@227 49 (** %\vspace{-.15in}% [[
adamc@227 50 Set
adamc@227 51 : Type
adamc@227 52
adamc@227 53 ]]
adamc@227 54
adam@343 55 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)}\textit{%#<i>#classes#</i>#%}%. In Coq, this more general notion is [Type]. *)
adamc@227 56
adamc@227 57 Check Type.
adamc@227 58 (** %\vspace{-.15in}% [[
adamc@227 59 Type
adamc@227 60 : Type
adamc@227 61
adamc@227 62 ]]
adamc@227 63
adam@343 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@302 75 *)
adamc@227 76
adamc@227 77 (** printing $ %({}*% #(<a/>*# *)
adamc@227 78 (** printing ^ %*{})% #*<a/>)# *)
adamc@227 79
adamc@227 80 Check Set.
adamc@227 81 (** %\vspace{-.15in}% [[
adamc@227 82 Set
adamc@227 83 : Type $ (0)+1 ^
adamc@227 84
adam@302 85 ]]
adam@302 86 *)
adamc@227 87
adamc@227 88 Check Type.
adamc@227 89 (** %\vspace{-.15in}% [[
adamc@227 90 Type $ Top.3 ^
adamc@227 91 : Type $ (Top.3)+1 ^
adamc@227 92
adamc@227 93 ]]
adamc@227 94
adam@287 95 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 96
adamc@227 97 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 %\textit{%#<i>#classifies#</i>#%}% [Set].
adamc@227 98
adam@343 99 In the second query's output, we see that the occurrence of [Type] that we check is assigned a fresh %\index{universe variable}\textit{%#<i>#universe variable#</i>#%}% [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@227 100
adam@343 101 Another crucial concept in CIC is %\index{predicativity}\textit{%#<i>#predicativity#</i>#%}%. Consider these queries. *)
adamc@227 102
adamc@227 103 Check forall T : nat, fin T.
adamc@227 104 (** %\vspace{-.15in}% [[
adamc@227 105 forall T : nat, fin T
adamc@227 106 : Set
adam@302 107 ]]
adam@302 108 *)
adamc@227 109
adamc@227 110 Check forall T : Set, T.
adamc@227 111 (** %\vspace{-.15in}% [[
adamc@227 112 forall T : Set, T
adamc@227 113 : Type $ max(0, (0)+1) ^
adam@302 114 ]]
adam@302 115 *)
adamc@227 116
adamc@227 117 Check forall T : Type, T.
adamc@227 118 (** %\vspace{-.15in}% [[
adamc@227 119 forall T : Type $ Top.9 ^ , T
adamc@227 120 : Type $ max(Top.9, (Top.9)+1) ^
adamc@227 121
adamc@227 122 ]]
adamc@227 123
adamc@227 124 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@227 125
adam@287 126 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@227 127
adamc@227 128 Definition id (T : Set) (x : T) : T := x.
adamc@227 129
adamc@227 130 Check id 0.
adamc@227 131 (** %\vspace{-.15in}% [[
adamc@227 132 id 0
adamc@227 133 : nat
adamc@227 134
adamc@227 135 Check id Set.
adam@343 136 ]]
adamc@227 137
adam@343 138 <<
adamc@227 139 Error: Illegal application (Type Error):
adamc@227 140 ...
adam@343 141 The 1st term has type "Type (* (Top.15)+1 *)" which should be coercible to "Set".
adam@343 142 >>
adamc@227 143
adam@343 144 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@227 145
adamc@227 146 Reset id.
adamc@227 147 Definition id (T : Type) (x : T) : T := x.
adamc@227 148 Check id 0.
adamc@227 149 (** %\vspace{-.15in}% [[
adamc@227 150 id 0
adamc@227 151 : nat
adam@302 152 ]]
adam@302 153 *)
adamc@227 154
adamc@227 155 Check id Set.
adamc@227 156 (** %\vspace{-.15in}% [[
adamc@227 157 id Set
adamc@227 158 : Type $ Top.17 ^
adam@302 159 ]]
adam@302 160 *)
adamc@227 161
adamc@227 162 Check id Type.
adamc@227 163 (** %\vspace{-.15in}% [[
adamc@227 164 id Type $ Top.18 ^
adamc@227 165 : Type $ Top.19 ^
adam@302 166 ]]
adam@302 167 *)
adamc@227 168
adamc@227 169 (** 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@227 170 [[
adamc@227 171 Check id id.
adam@343 172 ]]
adamc@227 173
adam@343 174 <<
adamc@227 175 Error: Universe inconsistency (cannot enforce Top.16 < Top.16).
adam@343 176 >>
adamc@227 177
adam@343 178 %\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 %\textit{%#<i>#predicative#</i>#%}%, 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@227 179
adamc@227 180
adamc@227 181 (** ** Inductive Definitions *)
adamc@227 182
adamc@227 183 (** 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@227 184
adamc@227 185 [[
adamc@227 186 Inductive exp : Set -> Set :=
adamc@227 187 | Const : forall T : Set, T -> exp T
adamc@227 188 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 189 | Eq : forall T, exp T -> exp T -> exp bool.
adam@343 190 ]]
adamc@227 191
adam@343 192 <<
adamc@227 193 Error: Large non-propositional inductive types must be in Type.
adam@343 194 >>
adamc@227 195
adam@343 196 This definition is %\index{large inductive types}\textit{%#<i>#large#</i>#%}% 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@227 197
adamc@227 198 Inductive exp : Type -> Type :=
adamc@227 199 | Const : forall T, T -> exp T
adamc@227 200 | Pair : forall T1 T2, exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 201 | Eq : forall T, exp T -> exp T -> exp bool.
adamc@227 202
adamc@228 203 (** 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@228 204
adamc@228 205 Our new definition is accepted. We can build some sample expressions. *)
adamc@227 206
adamc@227 207 Check Const 0.
adamc@227 208 (** %\vspace{-.15in}% [[
adamc@227 209 Const 0
adamc@227 210 : exp nat
adam@302 211 ]]
adam@302 212 *)
adamc@227 213
adamc@227 214 Check Pair (Const 0) (Const tt).
adamc@227 215 (** %\vspace{-.15in}% [[
adamc@227 216 Pair (Const 0) (Const tt)
adamc@227 217 : exp (nat * unit)
adam@302 218 ]]
adam@302 219 *)
adamc@227 220
adamc@227 221 Check Eq (Const Set) (Const Type).
adamc@227 222 (** %\vspace{-.15in}% [[
adamc@228 223 Eq (Const Set) (Const Type $ Top.59 ^ )
adamc@227 224 : exp bool
adamc@227 225
adamc@227 226 ]]
adamc@227 227
adamc@227 228 We can check many expressions, including fancy expressions that include types. However, it is not hard to hit a type-checking wall.
adamc@227 229
adamc@227 230 [[
adamc@227 231 Check Const (Const O).
adam@343 232 ]]
adamc@227 233
adam@343 234 <<
adamc@227 235 Error: Universe inconsistency (cannot enforce Top.42 < Top.42).
adam@343 236 >>
adamc@227 237
adamc@227 238 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@227 239
adamc@227 240 Print exp.
adamc@227 241 (** %\vspace{-.15in}% [[
adamc@227 242 Inductive exp
adamc@227 243 : Type $ Top.8 ^ ->
adamc@227 244 Type
adamc@227 245 $ max(0, (Top.11)+1, (Top.14)+1, (Top.15)+1, (Top.19)+1) ^ :=
adamc@227 246 Const : forall T : Type $ Top.11 ^ , T -> exp T
adamc@227 247 | Pair : forall (T1 : Type $ Top.14 ^ ) (T2 : Type $ Top.15 ^ ),
adamc@227 248 exp T1 -> exp T2 -> exp (T1 * T2)
adamc@227 249 | Eq : forall T : Type $ Top.19 ^ , exp T -> exp T -> exp bool
adamc@227 250
adamc@227 251 ]]
adamc@227 252
adamc@227 253 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] %\textit{%#<i>#must#</i>#%}% 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@227 254
adam@343 255 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@227 256
adamc@227 257 Print Universes.
adamc@227 258 (** %\vspace{-.15in}% [[
adamc@227 259 Top.19 < Top.9 <= Top.8
adamc@227 260 Top.15 < Top.9 <= Top.8 <= Coq.Init.Datatypes.38
adamc@227 261 Top.14 < Top.9 <= Top.8 <= Coq.Init.Datatypes.37
adamc@227 262 Top.11 < Top.9 <= Top.8
adamc@227 263
adamc@227 264 ]]
adamc@227 265
adam@343 266 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@227 267
adamc@227 268 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@227 269
adamc@227 270 Print prod.
adamc@227 271 (** %\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
adamc@227 279 We see that the constraint is enforcing that indices to [exp] must not live in a higher universe level than [B]-indices to [prod]. The next constraint above establishes a symmetric condition for [A].
adamc@227 280
adamc@227 281 Thus it is apparent that Coq maintains a tortuous set of universe variable inequalities behind the scenes. It may look like some functions are polymorphic in the universe levels of their arguments, but what is really happening is imperative updating of a system of constraints, such that all uses of a function are consistent with a global set of universe levels. When the constraint system may not be evolved soundly, we get a universe inconsistency error.
adamc@227 282
adamc@227 283 %\medskip%
adamc@227 284
adamc@227 285 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 %\textit{%#<i>#one higher#</i>#%}% than the maximum. The critical difference is that, in the definition of [prod], [A] and [B] are defined as %\textit{%#<i>#parameters#</i>#%}%; that is, they appear named to the left of the main colon, rather than appearing (possibly unnamed) to the right.
adamc@227 286
adamc@231 287 Parameters are not as flexible as normal inductive type arguments. The range types of all of the constructors of a parameterized type must share the same parameters. Nonetheless, when it is possible to define a polymorphic type in this way, we gain the ability to use the new type family in more ways, without triggering universe inconsistencies. For instance, nested pairs of types are perfectly legal. *)
adamc@227 288
adamc@227 289 Check (nat, (Type, Set)).
adamc@227 290 (** %\vspace{-.15in}% [[
adamc@227 291 (nat, (Type $ Top.44 ^ , Set))
adamc@227 292 : Set * (Type $ Top.45 ^ * Type $ Top.46 ^ )
adamc@227 293 ]]
adamc@227 294
adamc@227 295 The same cannot be done with a counterpart to [prod] that does not use parameters. *)
adamc@227 296
adamc@227 297 Inductive prod' : Type -> Type -> Type :=
adamc@227 298 | pair' : forall A B : Type, A -> B -> prod' A B.
adamc@227 299 (** [[
adamc@227 300 Check (pair' nat (pair' Type Set)).
adam@343 301 ]]
adamc@227 302
adam@343 303 <<
adamc@227 304 Error: Universe inconsistency (cannot enforce Top.51 < Top.51).
adam@343 305 >>
adamc@227 306
adamc@233 307 The key benefit parameters bring us is the ability to avoid quantifying over types in the types of constructors. Such quantification induces less-than constraints, while parameters only introduce less-than-or-equal-to constraints.
adamc@233 308
adam@343 309 Coq includes one more (potentially confusing) feature related to parameters. While Gallina does not support real %\index{universe polymorphism}%universe polymorphism, there is a convenience facility that mimics universe polymorphism in some cases. We can illustrate what this means with a simple example. *)
adamc@233 310
adamc@233 311 Inductive foo (A : Type) : Type :=
adamc@233 312 | Foo : A -> foo A.
adamc@229 313
adamc@229 314 (* begin hide *)
adamc@229 315 Unset Printing Universes.
adamc@229 316 (* end hide *)
adamc@229 317
adamc@233 318 Check foo nat.
adamc@233 319 (** %\vspace{-.15in}% [[
adamc@233 320 foo nat
adamc@233 321 : Set
adam@302 322 ]]
adam@302 323 *)
adamc@233 324
adamc@233 325 Check foo Set.
adamc@233 326 (** %\vspace{-.15in}% [[
adamc@233 327 foo Set
adamc@233 328 : Type
adam@302 329 ]]
adam@302 330 *)
adamc@233 331
adamc@233 332 Check foo True.
adamc@233 333 (** %\vspace{-.15in}% [[
adamc@233 334 foo True
adamc@233 335 : Prop
adamc@233 336
adamc@233 337 ]]
adamc@233 338
adam@287 339 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@233 340
adamc@233 341 Imitation polymorphism can be confusing in some contexts. For instance, it is what is responsible for this weird behavior. *)
adamc@233 342
adamc@233 343 Inductive bar : Type := Bar : bar.
adamc@233 344
adamc@233 345 Check bar.
adamc@233 346 (** %\vspace{-.15in}% [[
adamc@233 347 bar
adamc@233 348 : Prop
adamc@233 349 ]]
adamc@233 350
adamc@233 351 The type that Coq comes up with may be used in strictly more contexts than the type one might have expected. *)
adamc@233 352
adamc@229 353
adam@388 354 (** ** Deciphering Baffling Messages About Inability to Unify *)
adam@388 355
adam@388 356 (** 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@388 357
adam@388 358 Theorem symmetry : forall A B : Type,
adam@388 359 A = B
adam@388 360 -> B = A.
adam@388 361 intros ? ? H; rewrite H; reflexivity.
adam@388 362 Qed.
adam@388 363
adam@388 364 (** Let us attempt an admittedly silly proof of the following theorem. *)
adam@388 365
adam@388 366 Theorem illustrative_but_silly_detour : unit = unit.
adam@388 367 (** [[
adam@388 368 apply symmetry.
adam@388 369 ]]
adam@388 370 <<
adam@388 371 Error: Impossible to unify "?35 = ?34" with "unit = unit".
adam@388 372 >>
adam@388 373
adam@388 374 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 %\emph{%#<i>#not#</i>#%}% shown to us in this error message!
adam@388 375
adam@388 376 The following command is the secret to getting better error messages in such cases: *)
adam@388 377
adam@388 378 Set Printing All.
adam@388 379 (** [[
adam@388 380 apply symmetry.
adam@388 381 ]]
adam@388 382 <<
adam@388 383 Error: Impossible to unify "@eq Type ?46 ?45" with "@eq Set unit unit".
adam@388 384 >>
adam@388 385
adam@388 386 Now we can see the problem: it is the first, %\emph{%#<i>#implicit#</i>#%}% 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@388 387
adam@388 388 Abort.
adam@388 389
adam@388 390 (** 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@388 391
adam@388 392 Theorem illustrative_but_silly_detour : (unit : Type) = unit.
adam@388 393 apply symmetry; reflexivity.
adam@388 394 Qed.
adam@388 395
adam@388 396 (** 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@388 397
adam@388 398 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@388 399
adam@388 400 Unset Printing All.
adam@388 401
adam@388 402 Theorem ex_symmetry : (exists x, x = 0) -> (exists x, 0 = x).
adam@388 403 econstructor.
adam@388 404 (** %\vspace{-.15in}%[[
adam@388 405 H : exists x : nat, x = 0
adam@388 406 ============================
adam@388 407 0 = ?98
adam@388 408 ]]
adam@388 409 *)
adam@388 410
adam@388 411 destruct H.
adam@388 412 (** %\vspace{-.15in}%[[
adam@388 413 x : nat
adam@388 414 H : x = 0
adam@388 415 ============================
adam@388 416 0 = ?99
adam@388 417 ]]
adam@388 418 *)
adam@388 419
adam@388 420 (** [[
adam@388 421 symmetry; exact H.
adam@388 422 ]]
adam@388 423
adam@388 424 <<
adam@388 425 Error: In environment
adam@388 426 x : nat
adam@388 427 H : x = 0
adam@388 428 The term "H" has type "x = 0" while it is expected to have type
adam@388 429 "?99 = 0".
adam@388 430 >>
adam@388 431
adam@388 432 The problem here is that variable [x] was introduced by [destruct] %\emph{%#<i>#after#</i>#%}% we introduced [?99] with [eexists], so the instantiation of [?99] may not mention [x]. A simple reordering of the proof solves the problem. *)
adam@388 433
adam@388 434 Restart.
adam@388 435 destruct 1 as [x]; apply ex_intro with x; symmetry; assumption.
adam@388 436 Qed.
adam@388 437
adam@388 438 (** 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@388 439
adam@388 440
adamc@229 441 (** * The [Prop] Universe *)
adamc@229 442
adam@287 443 (** 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@229 444
adamc@229 445 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@229 446
adamc@229 447 Print sig.
adamc@229 448 (** %\vspace{-.15in}% [[
adamc@229 449 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@229 450 exist : forall x : A, P x -> sig P
adam@302 451 ]]
adam@302 452 *)
adamc@229 453
adamc@229 454 Print ex.
adamc@229 455 (** %\vspace{-.15in}% [[
adamc@229 456 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@229 457 ex_intro : forall x : A, P x -> ex P
adamc@229 458 ]]
adamc@229 459
adamc@229 460 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@229 461
adamc@229 462 Definition projS A (P : A -> Prop) (x : sig P) : A :=
adamc@229 463 match x with
adamc@229 464 | exist v _ => v
adamc@229 465 end.
adamc@229 466
adamc@229 467 (** We run into trouble with a version that has been changed to work with [ex].
adamc@229 468 [[
adamc@229 469 Definition projE A (P : A -> Prop) (x : ex P) : A :=
adamc@229 470 match x with
adamc@229 471 | ex_intro v _ => v
adamc@229 472 end.
adam@343 473 ]]
adamc@229 474
adam@343 475 <<
adamc@229 476 Error:
adamc@229 477 Incorrect elimination of "x" in the inductive type "ex":
adamc@229 478 the return type has sort "Type" while it should be "Prop".
adamc@229 479 Elimination of an inductive object of sort Prop
adamc@229 480 is not allowed on a predicate in sort Type
adamc@229 481 because proofs can be eliminated only to build proofs.
adam@343 482 >>
adamc@229 483
adam@343 484 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@229 485
adamc@229 486 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@229 487
adam@343 488 Recall that %\index{program extraction}%extraction is Coq's facility for translating Coq developments into programs in general-purpose programming languages like OCaml. Extraction %\textit{%#<i>#erases#</i>#%}% proofs and leaves programs intact. A simple example with [sig] and [ex] demonstrates the distinction. *)
adamc@229 489
adamc@229 490 Definition sym_sig (x : sig (fun n => n = 0)) : sig (fun n => 0 = n) :=
adamc@229 491 match x with
adamc@229 492 | exist n pf => exist _ n (sym_eq pf)
adamc@229 493 end.
adamc@229 494
adamc@229 495 Extraction sym_sig.
adamc@229 496 (** <<
adamc@229 497 (** val sym_sig : nat -> nat **)
adamc@229 498
adamc@229 499 let sym_sig x = x
adamc@229 500 >>
adamc@229 501
adamc@229 502 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@229 503
adamc@229 504 Definition sym_ex (x : ex (fun n => n = 0)) : ex (fun n => 0 = n) :=
adamc@229 505 match x with
adamc@229 506 | ex_intro n pf => ex_intro _ n (sym_eq pf)
adamc@229 507 end.
adamc@229 508
adamc@229 509 Extraction sym_ex.
adamc@229 510 (** <<
adamc@229 511 (** val sym_ex : __ **)
adamc@229 512
adamc@229 513 let sym_ex = __
adamc@229 514 >>
adamc@229 515
adam@302 516 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@229 517
adam@343 518 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@229 519
adam@343 520 Many fans of the %\index{Curry-Howard correspondence}%Curry-Howard correspondence support the idea of %\textit{%#<i>#extracting programs from proofs#</i>#%}%. 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@229 521
adamc@229 522 %\medskip%
adamc@229 523
adam@343 524 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}\textit{%#<i>#impredicative#</i>#%}%, as this example shows. *)
adamc@229 525
adamc@229 526 Check forall P Q : Prop, P \/ Q -> Q \/ P.
adamc@229 527 (** %\vspace{-.15in}% [[
adamc@229 528 forall P Q : Prop, P \/ Q -> Q \/ P
adamc@229 529 : Prop
adamc@229 530
adamc@229 531 ]]
adamc@229 532
adamc@230 533 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@230 534
adamc@230 535 Impredicativity also allows us to implement a version of our earlier [exp] type that does not suffer from the weakness that we found. *)
adamc@230 536
adamc@230 537 Inductive expP : Type -> Prop :=
adamc@230 538 | ConstP : forall T, T -> expP T
adamc@230 539 | PairP : forall T1 T2, expP T1 -> expP T2 -> expP (T1 * T2)
adamc@230 540 | EqP : forall T, expP T -> expP T -> expP bool.
adamc@230 541
adamc@230 542 Check ConstP 0.
adamc@230 543 (** %\vspace{-.15in}% [[
adamc@230 544 ConstP 0
adamc@230 545 : expP nat
adam@302 546 ]]
adam@302 547 *)
adamc@230 548
adamc@230 549 Check PairP (ConstP 0) (ConstP tt).
adamc@230 550 (** %\vspace{-.15in}% [[
adamc@230 551 PairP (ConstP 0) (ConstP tt)
adamc@230 552 : expP (nat * unit)
adam@302 553 ]]
adam@302 554 *)
adamc@230 555
adamc@230 556 Check EqP (ConstP Set) (ConstP Type).
adamc@230 557 (** %\vspace{-.15in}% [[
adamc@230 558 EqP (ConstP Set) (ConstP Type)
adamc@230 559 : expP bool
adam@302 560 ]]
adam@302 561 *)
adamc@230 562
adamc@230 563 Check ConstP (ConstP O).
adamc@230 564 (** %\vspace{-.15in}% [[
adamc@230 565 ConstP (ConstP 0)
adamc@230 566 : expP (expP nat)
adamc@230 567
adamc@230 568 ]]
adamc@230 569
adam@287 570 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@230 571
adamc@230 572 Inductive eqPlus : forall T, T -> T -> Prop :=
adamc@230 573 | Base : forall T (x : T), eqPlus x x
adamc@230 574 | Func : forall dom ran (f1 f2 : dom -> ran),
adamc@230 575 (forall x : dom, eqPlus (f1 x) (f2 x))
adamc@230 576 -> eqPlus f1 f2.
adamc@230 577
adamc@230 578 Check (Base 0).
adamc@230 579 (** %\vspace{-.15in}% [[
adamc@230 580 Base 0
adamc@230 581 : eqPlus 0 0
adam@302 582 ]]
adam@302 583 *)
adamc@230 584
adamc@230 585 Check (Func (fun n => n) (fun n => 0 + n) (fun n => Base n)).
adamc@230 586 (** %\vspace{-.15in}% [[
adamc@230 587 Func (fun n : nat => n) (fun n : nat => 0 + n) (fun n : nat => Base n)
adamc@230 588 : eqPlus (fun n : nat => n) (fun n : nat => 0 + n)
adam@302 589 ]]
adam@302 590 *)
adamc@230 591
adamc@230 592 Check (Base (Base 1)).
adamc@230 593 (** %\vspace{-.15in}% [[
adamc@230 594 Base (Base 1)
adamc@230 595 : eqPlus (Base 1) (Base 1)
adam@302 596 ]]
adam@302 597 *)
adamc@230 598
adam@343 599 (** Stating equality facts about proofs may seem baroque, but we have already seen its utility in the chapter on reasoning about equality proofs. *)
adam@343 600
adamc@230 601
adamc@230 602 (** * Axioms *)
adamc@230 603
adam@343 604 (** 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}\textit{%#<i>#axioms#</i>#%}% without proof.
adamc@230 605
adamc@230 606 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@230 607
adamc@230 608 (** ** The Basics *)
adamc@230 609
adam@343 610 (** One simple example of a useful axiom is the %\index{law of the excluded middle}%law of the excluded middle. *)
adamc@230 611
adamc@230 612 Require Import Classical_Prop.
adamc@230 613 Print classic.
adamc@230 614 (** %\vspace{-.15in}% [[
adamc@230 615 *** [ classic : forall P : Prop, P \/ ~ P ]
adamc@230 616 ]]
adamc@230 617
adam@343 618 In the implementation of module [Classical_Prop], this axiom was defined with the command%\index{Vernacular commands!Axiom}% *)
adamc@230 619
adamc@230 620 Axiom classic : forall P : Prop, P \/ ~ P.
adamc@230 621
adam@343 622 (** 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@230 623
adamc@230 624 Parameter n : nat.
adamc@230 625 Axiom positive : n > 0.
adamc@230 626 Reset n.
adamc@230 627
adam@287 628 (** 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@230 629
adam@392 630 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}\textit{%#<i>#inconsistent#</i>#%}%. 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@230 631
adam@287 632 Axiom not_classic : ~ forall P : Prop, P \/ ~ P.
adamc@230 633
adamc@230 634 Theorem uhoh : False.
adam@287 635 generalize classic not_classic; tauto.
adamc@230 636 Qed.
adamc@230 637
adamc@230 638 Theorem uhoh_again : 1 + 1 = 3.
adamc@230 639 destruct uhoh.
adamc@230 640 Qed.
adamc@230 641
adamc@230 642 Reset not_classic.
adamc@230 643
adam@343 644 (** 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 %\textit{%#<i>#model#</i>#%}% 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@230 645
adam@343 646 Recall that Coq implements %\index{constructive logic}\textit{%#<i>#constructive#</i>#%}% 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@230 647
adamc@231 648 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] %\textit{%#<i>#would#</i>#%}% 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@230 649
adam@343 650 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@230 651
adamc@230 652 Theorem t1 : forall P : Prop, P -> ~ ~ P.
adamc@230 653 tauto.
adamc@230 654 Qed.
adamc@230 655
adamc@230 656 Print Assumptions t1.
adam@343 657 (** <<
adamc@230 658 Closed under the global context
adam@343 659 >>
adam@302 660 *)
adamc@230 661
adamc@230 662 Theorem t2 : forall P : Prop, ~ ~ P -> P.
adamc@230 663 (** [[
adamc@230 664 tauto.
adam@343 665 ]]
adam@343 666 <<
adamc@230 667 Error: tauto failed.
adam@343 668 >>
adam@302 669 *)
adamc@230 670 intro P; destruct (classic P); tauto.
adamc@230 671 Qed.
adamc@230 672
adamc@230 673 Print Assumptions t2.
adamc@230 674 (** %\vspace{-.15in}% [[
adamc@230 675 Axioms:
adamc@230 676 classic : forall P : Prop, P \/ ~ P
adamc@230 677 ]]
adamc@230 678
adamc@231 679 It is possible to avoid this dependence in some specific cases, where excluded middle %\textit{%#<i>#is#</i>#%}% provable, for decidable families of propositions. *)
adamc@230 680
adam@287 681 Theorem nat_eq_dec : forall n m : nat, n = m \/ n <> m.
adamc@230 682 induction n; destruct m; intuition; generalize (IHn m); intuition.
adamc@230 683 Qed.
adamc@230 684
adamc@230 685 Theorem t2' : forall n m : nat, ~ ~ (n = m) -> n = m.
adam@287 686 intros n m; destruct (nat_eq_dec n m); tauto.
adamc@230 687 Qed.
adamc@230 688
adamc@230 689 Print Assumptions t2'.
adam@343 690 (** <<
adamc@230 691 Closed under the global context
adam@343 692 >>
adamc@230 693
adamc@230 694 %\bigskip%
adamc@230 695
adam@343 696 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}\textit{%#<i>#proof irrelevance#</i>#%}%, which simplifies proof issues that would not even arise in mainstream math. *)
adamc@230 697
adamc@230 698 Require Import ProofIrrelevance.
adamc@230 699 Print proof_irrelevance.
adamc@230 700 (** %\vspace{-.15in}% [[
adamc@230 701 *** [ proof_irrelevance : forall (P : Prop) (p1 p2 : P), p1 = p2 ]
adamc@230 702 ]]
adamc@230 703
adam@353 704 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@230 705
adamc@230 706 (* begin hide *)
adamc@230 707 Lemma zgtz : 0 > 0 -> False.
adamc@230 708 crush.
adamc@230 709 Qed.
adamc@230 710 (* end hide *)
adamc@230 711
adamc@230 712 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@230 713 match n with
adamc@230 714 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@230 715 | S n' => fun _ => n'
adamc@230 716 end.
adamc@230 717
adam@343 718 (** We might want to prove that different proofs of [n > 0] do not lead to different results from our richly typed predecessor function. *)
adamc@230 719
adamc@230 720 Theorem pred_strong1_irrel : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 721 destruct n; crush.
adamc@230 722 Qed.
adamc@230 723
adamc@230 724 (** 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@230 725
adamc@230 726 Theorem pred_strong1_irrel' : forall n (pf1 pf2 : n > 0), pred_strong1 pf1 = pred_strong1 pf2.
adamc@230 727 intros; f_equal; apply proof_irrelevance.
adamc@230 728 Qed.
adamc@230 729
adamc@230 730
adamc@230 731 (** %\bigskip%
adamc@230 732
adamc@230 733 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@230 734
adamc@230 735 Require Import Eqdep.
adamc@230 736 Import Eq_rect_eq.
adamc@230 737 Print eq_rect_eq.
adamc@230 738 (** %\vspace{-.15in}% [[
adamc@230 739 *** [ eq_rect_eq :
adamc@230 740 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adamc@230 741 x = eq_rect p Q x p h ]
adamc@230 742 ]]
adamc@230 743
adam@343 744 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@230 745
adamc@230 746 Corollary UIP_refl : forall A (x : A) (pf : x = x), pf = refl_equal x.
adamc@230 747 intros; replace pf with (eq_rect x (eq x) (refl_equal x) x pf); [
adamc@230 748 symmetry; apply eq_rect_eq
adamc@230 749 | exact (match pf as pf' return match pf' in _ = y return x = y with
adamc@230 750 | refl_equal => refl_equal x
adamc@230 751 end = pf' with
adamc@230 752 | refl_equal => refl_equal _
adamc@230 753 end) ].
adamc@230 754 Qed.
adamc@230 755
adamc@230 756 Corollary UIP : forall A (x y : A) (pf1 pf2 : x = y), pf1 = pf2.
adamc@230 757 intros; generalize pf1 pf2; subst; intros;
adamc@230 758 match goal with
adamc@230 759 | [ |- ?pf1 = ?pf2 ] => rewrite (UIP_refl pf1); rewrite (UIP_refl pf2); reflexivity
adamc@230 760 end.
adamc@230 761 Qed.
adamc@230 762
adamc@231 763 (** 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@230 764
adamc@230 765 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@230 766
adamc@230 767 %\bigskip%
adamc@230 768
adamc@230 769 There are two more basic axioms that are often assumed, to avoid complications that do not arise in set theory. *)
adamc@230 770
adamc@230 771 Require Import FunctionalExtensionality.
adamc@230 772 Print functional_extensionality_dep.
adamc@230 773 (** %\vspace{-.15in}% [[
adamc@230 774 *** [ functional_extensionality_dep :
adamc@230 775 forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
adamc@230 776 (forall x : A, f x = g x) -> f = g ]
adamc@230 777
adamc@230 778 ]]
adamc@230 779
adamc@230 780 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@230 781
adam@343 782 A simple corollary shows that the same property applies to predicates. *)
adamc@230 783
adamc@230 784 Corollary predicate_extensionality : forall (A : Type) (B : A -> Prop) (f g : forall x : A, B x),
adamc@230 785 (forall x : A, f x = g x) -> f = g.
adamc@230 786 intros; apply functional_extensionality_dep; assumption.
adamc@230 787 Qed.
adamc@230 788
adam@343 789 (** In some cases, one might prefer to assert this corollary as the axiom, to restrict the consequences to proofs and not programs. *)
adam@343 790
adamc@230 791
adamc@230 792 (** ** Axioms of Choice *)
adamc@230 793
adam@343 794 (** 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@230 795
adamc@230 796 First, it is possible to implement a choice operator %\textit{%#<i>#without#</i>#%}% axioms in some potentially surprising cases. *)
adamc@230 797
adamc@230 798 Require Import ConstructiveEpsilon.
adamc@230 799 Check constructive_definite_description.
adamc@230 800 (** %\vspace{-.15in}% [[
adamc@230 801 constructive_definite_description
adamc@230 802 : forall (A : Set) (f : A -> nat) (g : nat -> A),
adamc@230 803 (forall x : A, g (f x) = x) ->
adamc@230 804 forall P : A -> Prop,
adamc@230 805 (forall x : A, {P x} + {~ P x}) ->
adamc@230 806 (exists! x : A, P x) -> {x : A | P x}
adam@302 807 ]]
adam@302 808 *)
adamc@230 809
adamc@230 810 Print Assumptions constructive_definite_description.
adam@343 811 (** <<
adamc@230 812 Closed under the global context
adam@343 813 >>
adamc@230 814
adamc@231 815 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 %\textit{%#<i>#unique#</i>#%}% 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@230 816
adamc@230 817 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@230 818
adamc@230 819 Require Import ClassicalUniqueChoice.
adamc@230 820 Check dependent_unique_choice.
adamc@230 821 (** %\vspace{-.15in}% [[
adamc@230 822 dependent_unique_choice
adamc@230 823 : forall (A : Type) (B : A -> Type) (R : forall x : A, B x -> Prop),
adamc@230 824 (forall x : A, exists! y : B x, R x y) ->
adam@343 825 exists f : forall x : A, B x,
adam@343 826 forall x : A, R x (f x)
adamc@230 827 ]]
adamc@230 828
adamc@230 829 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@230 830
adamc@230 831 Require Import ClassicalChoice.
adamc@230 832 Check choice.
adamc@230 833 (** %\vspace{-.15in}% [[
adamc@230 834 choice
adamc@230 835 : forall (A B : Type) (R : A -> B -> Prop),
adamc@230 836 (forall x : A, exists y : B, R x y) ->
adamc@230 837 exists f : A -> B, forall x : A, R x (f x)
adamc@230 838
adamc@230 839 ]]
adamc@230 840
adamc@230 841 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@230 842
adamc@230 843 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@230 844
adamc@230 845 Definition choice_Set (A B : Type) (R : A -> B -> Prop) (H : forall x : A, {y : B | R x y})
adamc@230 846 : {f : A -> B | forall x : A, R x (f x)} :=
adamc@230 847 exist (fun f => forall x : A, R x (f x))
adamc@230 848 (fun x => proj1_sig (H x)) (fun x => proj2_sig (H x)).
adamc@230 849
adam@287 850 (** 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@230 851
adam@287 852 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@230 853
adamc@230 854 %\bigskip%
adamc@230 855
adam@343 856 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@230 857
adamc@230 858 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@230 859
adamc@230 860 (** ** Axioms and Computation *)
adamc@230 861
adamc@230 862 (** One additional axiom-related wrinkle arises from an aspect of Gallina that is very different from set theory: a notion of %\textit{%#<i>#computational equivalence#</i>#%}% 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@230 863
adamc@230 864 Definition cast (x y : Set) (pf : x = y) (v : x) : y :=
adamc@230 865 match pf with
adamc@230 866 | refl_equal => v
adamc@230 867 end.
adamc@230 868
adamc@230 869 (** Computation over programs that use [cast] can proceed smoothly. *)
adamc@230 870
adamc@230 871 Eval compute in (cast (refl_equal (nat -> nat)) (fun n => S n)) 12.
adam@343 872 (** %\vspace{-.15in}%[[
adamc@230 873 = 13
adamc@230 874 : nat
adam@302 875 ]]
adam@302 876 *)
adamc@230 877
adamc@230 878 (** Things do not go as smoothly when we use [cast] with proofs that rely on axioms. *)
adamc@230 879
adamc@230 880 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 881 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 882 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 883 Qed.
adamc@230 884
adamc@230 885 Eval compute in (cast t3 (fun _ => First)) 12.
adamc@230 886 (** [[
adamc@230 887 = match t3 in (_ = P) return P with
adamc@230 888 | refl_equal => fun n : nat => First
adamc@230 889 end 12
adamc@230 890 : fin (12 + 1)
adamc@230 891 ]]
adamc@230 892
adamc@230 893 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@230 894
adamc@230 895 Reset t3.
adamc@230 896
adamc@230 897 Theorem t3 : (forall n : nat, fin (S n)) = (forall n : nat, fin (n + 1)).
adamc@230 898 change ((forall n : nat, (fun n => fin (S n)) n) = (forall n : nat, (fun n => fin (n + 1)) n));
adamc@230 899 rewrite (functional_extensionality (fun n => fin (n + 1)) (fun n => fin (S n))); crush.
adamc@230 900 Defined.
adamc@230 901
adamc@230 902 Eval compute in (cast t3 (fun _ => First)) 12.
adamc@230 903 (** [[
adamc@230 904 = match
adamc@230 905 match
adamc@230 906 match
adamc@230 907 functional_extensionality
adamc@230 908 ....
adamc@230 909 ]]
adamc@230 910
adamc@230 911 We elide most of the details. A very unwieldy tree of nested matches on equality proofs appears. This time evaluation really %\textit{%#<i>#is#</i>#%}% stuck on a use of an axiom.
adamc@230 912
adamc@230 913 If we are careful in using tactics to prove an equality, we can still compute with casts over the proof. *)
adamc@230 914
adamc@230 915 Lemma plus1 : forall n, S n = n + 1.
adamc@230 916 induction n; simpl; intuition.
adamc@230 917 Defined.
adamc@230 918
adamc@230 919 Theorem t4 : forall n, fin (S n) = fin (n + 1).
adamc@230 920 intro; f_equal; apply plus1.
adamc@230 921 Defined.
adamc@230 922
adamc@230 923 Eval compute in cast (t4 13) First.
adamc@230 924 (** %\vspace{-.15in}% [[
adamc@230 925 = First
adamc@230 926 : fin (13 + 1)
adam@302 927 ]]
adam@343 928
adam@343 929 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@343 930
adam@344 931
adam@344 932 (** ** Methods for Avoiding Axioms *)
adam@344 933
adam@344 934 (** 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}\emph{%#<i>#trusted code base#</i>#%}%. 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@344 935
adam@344 936 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@344 937
adam@344 938 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@344 939
adam@344 940 Theorem fin_cases : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 941 intros; dep_destruct f; eauto.
adam@344 942 Qed.
adam@344 943
adam@344 944 Print Assumptions fin_cases.
adam@344 945 (** %\vspace{-.15in}%[[
adam@344 946 Axioms:
adam@344 947 JMeq.JMeq_eq : forall (A : Type) (x y : A), JMeq.JMeq x y -> x = y
adam@344 948 ]]
adam@344 949
adam@344 950 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@344 951
adam@344 952 (* begin thide *)
adam@344 953 Lemma fin_cases_again' : forall n (f : fin n),
adam@344 954 match n return fin n -> Prop with
adam@344 955 | O => fun _ => False
adam@344 956 | S n' => fun f => f = First \/ exists f', f = Next f'
adam@344 957 end f.
adam@344 958 destruct f; eauto.
adam@344 959 Qed.
adam@344 960
adam@344 961 (** 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@344 962
adam@344 963 Theorem fin_cases_again : forall n (f : fin (S n)), f = First \/ exists f', f = Next f'.
adam@344 964 intros; exact (fin_cases_again' f).
adam@344 965 Qed.
adam@344 966 (* end thide *)
adam@344 967
adam@344 968 Print Assumptions fin_cases_again.
adam@344 969 (** %\vspace{-.15in}%
adam@344 970 <<
adam@344 971 Closed under the global context
adam@344 972 >>
adam@344 973
adam@345 974 *)
adam@345 975
adam@345 976 (* begin thide *)
adam@345 977 (** 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@345 978
adam@345 979 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@345 980
adam@345 981 Definition finOut n (f : fin n) : match n return fin n -> Type with
adam@345 982 | O => fun _ => Empty_set
adam@345 983 | _ => fun f => {f' : _ | f = Next f'} + {f = First}
adam@345 984 end f :=
adam@345 985 match f with
adam@345 986 | First _ => inright _ (refl_equal _)
adam@345 987 | Next _ f' => inleft _ (exist _ f' (refl_equal _))
adam@345 988 end.
adam@345 989 (* end thide *)
adam@345 990
adam@345 991 (** 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@344 992
adam@344 993 Inductive formula : list Type -> Type :=
adam@344 994 | Inject : forall Ts, Prop -> formula Ts
adam@344 995 | VarEq : forall T Ts, T -> formula (T :: Ts)
adam@344 996 | Lift : forall T Ts, formula Ts -> formula (T :: Ts)
adam@344 997 | Forall : forall T Ts, formula (T :: Ts) -> formula Ts
adam@344 998 | And : forall Ts, formula Ts -> formula Ts -> formula Ts.
adam@344 999
adam@344 1000 (** 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 1001
adam@344 1002 Inductive proof : formula nil -> Prop :=
adam@344 1003 | PInject : forall (P : Prop), P -> proof (Inject nil P)
adam@344 1004 | PAnd : forall p q, proof p -> proof q -> proof (And p q).
adam@344 1005
adam@344 1006 (** Let us prove a lemma showing that a %``%#"#[P /\ Q -> P]#"#%''% rule is derivable within the rules of [proof]. *)
adam@344 1007
adam@344 1008 Theorem proj1 : forall p q, proof (And p q) -> proof p.
adam@344 1009 destruct 1.
adam@344 1010 (** %\vspace{-.15in}%[[
adam@344 1011 p : formula nil
adam@344 1012 q : formula nil
adam@344 1013 P : Prop
adam@344 1014 H : P
adam@344 1015 ============================
adam@344 1016 proof p
adam@344 1017 ]]
adam@344 1018 *)
adam@344 1019
adam@344 1020 (** 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 1021
adam@344 1022 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 1023
adam@344 1024 Restart.
adam@344 1025 Require Import Program.
adam@344 1026 intros ? ? H; dependent destruction H; auto.
adam@344 1027 Qed.
adam@344 1028
adam@344 1029 Print Assumptions proj1.
adam@344 1030 (** %\vspace{-.15in}%[[
adam@344 1031 Axioms:
adam@344 1032 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1033 x = eq_rect p Q x p h
adam@344 1034 ]]
adam@344 1035
adam@344 1036 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 1037
adam@344 1038 Lemma proj1_again' : forall r, proof r
adam@344 1039 -> forall p q, r = And p q -> proof p.
adam@344 1040 destruct 1; crush.
adam@344 1041 (** %\vspace{-.15in}%[[
adam@344 1042 H0 : Inject [] P = And p q
adam@344 1043 ============================
adam@344 1044 proof p
adam@344 1045 ]]
adam@344 1046
adam@344 1047 The first goal looks reasonable. Hypothesis [H0] is clearly contradictory, as [discriminate] can show. *)
adam@344 1048
adam@344 1049 discriminate.
adam@344 1050 (** %\vspace{-.15in}%[[
adam@344 1051 H : proof p
adam@344 1052 H1 : And p q = And p0 q0
adam@344 1053 ============================
adam@344 1054 proof p0
adam@344 1055 ]]
adam@344 1056
adam@344 1057 It looks like we are almost done. Hypothesis [H1] gives [p = p0] by injectivity of constructors, and then [H] finishes the case. *)
adam@344 1058
adam@344 1059 injection H1; intros.
adam@344 1060
adam@344 1061 (** Unfortunately, the %``%#"#equality#"#%''% that we expected between [p] and [p0] comes in a strange form:
adam@344 1062
adam@344 1063 [[
adam@344 1064 H3 : existT (fun Ts : list Type => formula Ts) []%list p =
adam@344 1065 existT (fun Ts : list Type => formula Ts) []%list p0
adam@344 1066 ============================
adam@344 1067 proof p0
adam@344 1068 ]]
adam@344 1069
adam@345 1070 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 1071
adam@344 1072 How exactly does an axiom come into the picture here? Let us ask [crush] to finish the proof. *)
adam@344 1073
adam@344 1074 crush.
adam@344 1075 Qed.
adam@344 1076
adam@344 1077 Print Assumptions proj1_again'.
adam@344 1078 (** %\vspace{-.15in}%[[
adam@344 1079 Axioms:
adam@344 1080 eq_rect_eq : forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p),
adam@344 1081 x = eq_rect p Q x p h
adam@344 1082 ]]
adam@344 1083
adam@344 1084 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 1085
adam@344 1086 Hope is not lost, however. We can produce an even stranger looking lemma, which gives us the theorem without axioms. *)
adam@344 1087
adam@344 1088 Lemma proj1_again'' : forall r, proof r
adam@344 1089 -> match r with
adam@344 1090 | And Ps p _ => match Ps return formula Ps -> Prop with
adam@344 1091 | nil => fun p => proof p
adam@344 1092 | _ => fun _ => True
adam@344 1093 end p
adam@344 1094 | _ => True
adam@344 1095 end.
adam@344 1096 destruct 1; auto.
adam@344 1097 Qed.
adam@344 1098
adam@344 1099 Theorem proj1_again : forall p q, proof (And p q) -> proof p.
adam@344 1100 intros ? ? H; exact (proj1_again'' H).
adam@344 1101 Qed.
adam@344 1102
adam@344 1103 Print Assumptions proj1_again.
adam@344 1104 (** <<
adam@344 1105 Closed under the global context
adam@344 1106 >>
adam@344 1107
adam@377 1108 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 1109
adam@377 1110 %\medskip%
adam@377 1111
adam@377 1112 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 %\emph{%#<i>#compute#</i>#%}%. 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 1113
adam@377 1114 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 1115
adam@377 1116 Section withTypes.
adam@377 1117 Variable types : list Set.
adam@377 1118
adam@377 1119 (** 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 1120
adam@377 1121 Variable values : hlist (fun x : Set => x) types.
adam@377 1122
adam@377 1123 (** 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 1124
adam@377 1125 Variable natIndex : nat.
adam@377 1126 Variable natIndex_ok : nth_error types natIndex = Some nat.
adam@377 1127
adam@377 1128 (** 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 1129
adam@377 1130 Lemma nth_error_nil : forall A n x,
adam@377 1131 nth_error (@nil A) n = Some x
adam@377 1132 -> False.
adam@377 1133 destruct n; simpl; unfold error; congruence.
adam@377 1134 Defined.
adam@377 1135
adam@377 1136 Implicit Arguments nth_error_nil [A n x].
adam@377 1137
adam@377 1138 Lemma Some_inj : forall A (x y : A),
adam@377 1139 Some x = Some y
adam@377 1140 -> x = y.
adam@377 1141 congruence.
adam@377 1142 Defined.
adam@377 1143
adam@377 1144 Fixpoint getNat (types' : list Set) (values' : hlist (fun x : Set => x) types')
adam@377 1145 (natIndex : nat) : (nth_error types' natIndex = Some nat) -> nat :=
adam@377 1146 match values' with
adam@377 1147 | HNil => fun pf => match nth_error_nil pf with end
adam@377 1148 | HCons t ts x values'' =>
adam@377 1149 match natIndex return nth_error (t :: ts) natIndex = Some nat -> nat with
adam@377 1150 | O => fun pf =>
adam@377 1151 match Some_inj pf in _ = T return T with
adam@377 1152 | refl_equal => x
adam@377 1153 end
adam@377 1154 | S natIndex' => getNat values'' natIndex'
adam@377 1155 end
adam@377 1156 end.
adam@377 1157 End withTypes.
adam@377 1158
adam@377 1159 (** The problem becomes apparent when we experiment with running [getNat] on a concrete [types] list. *)
adam@377 1160
adam@377 1161 Definition myTypes := unit :: nat :: bool :: nil.
adam@377 1162 Definition myValues : hlist (fun x : Set => x) myTypes :=
adam@377 1163 tt ::: 3 ::: false ::: HNil.
adam@377 1164
adam@377 1165 Definition myNatIndex := 1.
adam@377 1166
adam@377 1167 Theorem myNatIndex_ok : nth_error myTypes myNatIndex = Some nat.
adam@377 1168 reflexivity.
adam@377 1169 Defined.
adam@377 1170
adam@377 1171 Eval compute in getNat myValues myNatIndex myNatIndex_ok.
adam@377 1172 (** %\vspace{-.15in}%[[
adam@377 1173 = 3
adam@377 1174 ]]
adam@377 1175
adam@377 1176 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] %\emph{%#<i>#independently#</i>#%}% of a specific proof. *)
adam@377 1177
adam@377 1178 Theorem getNat_is_reasonable : forall pf, getNat myValues myNatIndex pf = 3.
adam@377 1179 intro; compute.
adam@377 1180 (**
adam@377 1181 <<
adam@377 1182 1 subgoal
adam@377 1183 >>
adam@377 1184 %\vspace{-.3in}%[[
adam@377 1185 pf : nth_error myTypes myNatIndex = Some nat
adam@377 1186 ============================
adam@377 1187 match
adam@377 1188 match
adam@377 1189 pf in (_ = y)
adam@377 1190 return (nat = match y with
adam@377 1191 | Some H => H
adam@377 1192 | None => nat
adam@377 1193 end)
adam@377 1194 with
adam@377 1195 | eq_refl => eq_refl
adam@377 1196 end in (_ = T) return T
adam@377 1197 with
adam@377 1198 | eq_refl => 3
adam@377 1199 end = 3
adam@377 1200 ]]
adam@377 1201
adam@377 1202 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 1203
adam@377 1204 Abort.
adam@377 1205
adam@377 1206 (** 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 1207
adam@377 1208 Fixpoint copies A (x : A) (n : nat) : list A :=
adam@377 1209 match n with
adam@377 1210 | O => nil
adam@377 1211 | S n' => x :: copies x n'
adam@377 1212 end.
adam@377 1213
adam@377 1214 Fixpoint update A (ls : list A) (n : nat) (x : A) : list A :=
adam@377 1215 match ls with
adam@377 1216 | nil => copies x n ++ x :: nil
adam@377 1217 | y :: ls' => match n with
adam@377 1218 | O => x :: ls'
adam@377 1219 | S n' => y :: update ls' n' x
adam@377 1220 end
adam@377 1221 end.
adam@377 1222
adam@377 1223 (** Now let us revisit the definition of [getNat]. *)
adam@377 1224
adam@377 1225 Section withTypes'.
adam@377 1226 Variable types : list Set.
adam@377 1227 Variable natIndex : nat.
adam@377 1228
adam@377 1229 (** Here is the trick: instead of asserting properties about the list [types], we build a %``%#"#new#"#%''% list that is %\emph{%#<i>#guaranteed by construction#</i>#%}% to have those properties. *)
adam@377 1230
adam@377 1231 Definition types' := update types natIndex nat.
adam@377 1232
adam@377 1233 Variable values : hlist (fun x : Set => x) types'.
adam@377 1234
adam@377 1235 (** Now a bit of dependent pattern matching helps us rewrite [getNat] in a way that avoids any use of equality proofs. *)
adam@377 1236
adam@378 1237 Fixpoint skipCopies (n : nat)
adam@378 1238 : hlist (fun x : Set => x) (copies nat n ++ nat :: nil) -> nat :=
adam@378 1239 match n with
adam@378 1240 | O => fun vs => hhd vs
adam@378 1241 | S n' => fun vs => skipCopies n' (htl vs)
adam@378 1242 end.
adam@378 1243
adam@377 1244 Fixpoint getNat' (types'' : list Set) (natIndex : nat)
adam@377 1245 : hlist (fun x : Set => x) (update types'' natIndex nat) -> nat :=
adam@377 1246 match types'' with
adam@378 1247 | nil => skipCopies natIndex
adam@377 1248 | t :: types0 =>
adam@377 1249 match natIndex return hlist (fun x : Set => x)
adam@377 1250 (update (t :: types0) natIndex nat) -> nat with
adam@377 1251 | O => fun vs => hhd vs
adam@377 1252 | S natIndex' => fun vs => getNat' types0 natIndex' (htl vs)
adam@377 1253 end
adam@377 1254 end.
adam@377 1255 End withTypes'.
adam@377 1256
adam@377 1257 (** Now the surprise comes in how easy it is to %\emph{%#<i>#use#</i>#%}% [getNat']. While typing works by modification of a types list, we can choose parameters so that the modification has no effect. *)
adam@377 1258
adam@377 1259 Theorem getNat_is_reasonable : getNat' myTypes myNatIndex myValues = 3.
adam@377 1260 reflexivity.
adam@377 1261 Qed.
adam@377 1262
adam@377 1263 (** The same parameters as before work without alteration, and we avoid use of axioms. *)