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author | Adam Chlipala <adam@chlipala.net> |
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date | Fri, 10 Nov 2017 19:02:44 -0500 |

parents | d65e9c1c9041 |

children |

rev | line source |
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adam@534 | 1 (* Copyright (c) 2008-2012, 2015, Adam Chlipala |

adamc@118 | 2 * |

adamc@118 | 3 * This work is licensed under a |

adamc@118 | 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 |

adamc@118 | 5 * Unported License. |

adamc@118 | 6 * The license text is available at: |

adamc@118 | 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/ |

adamc@118 | 8 *) |

adamc@118 | 9 |

adamc@118 | 10 (* begin hide *) |

adamc@120 | 11 Require Import Eqdep JMeq List. |

adamc@118 | 12 |

adam@534 | 13 Require Import Cpdt.CpdtTactics. |

adamc@118 | 14 |

adamc@118 | 15 Set Implicit Arguments. |

adam@534 | 16 Set Asymmetric Patterns. |

adamc@118 | 17 (* end hide *) |

adamc@118 | 18 |

adamc@118 | 19 |

adamc@118 | 20 (** %\chapter{Reasoning About Equality Proofs}% *) |

adamc@118 | 21 |

adam@456 | 22 (** In traditional mathematics, the concept of equality is usually taken as a given. On the other hand, in type theory, equality is a very contentious subject. There are at least three different notions of equality that are important in Coq, and researchers are actively investigating new definitions of what it means for two terms to be equal. Even once we fix a notion of equality, there are inevitably tricky issues that arise in proving properties of programs that manipulate equality proofs explicitly. In this chapter, I will focus on design patterns for circumventing these tricky issues, and I will introduce the different notions of equality as they are germane. *) |

adamc@118 | 23 |

adamc@118 | 24 |

adamc@122 | 25 (** * The Definitional Equality *) |

adamc@122 | 26 |

adam@435 | 27 (** We have seen many examples so far where proof goals follow "by computation." That is, we apply computational reduction rules to reduce the goal to a normal form, at which point it follows trivially. Exactly when this works and when it does not depends on the details of Coq's%\index{definitional equality}% _definitional equality_. This is an untyped binary relation appearing in the formal metatheory of CIC. CIC contains a typing rule allowing the conclusion [E : T] from the premise [E : T'] and a proof that [T] and [T'] are definitionally equal. |

adamc@122 | 28 |

adam@364 | 29 The %\index{tactics!cbv}%[cbv] tactic will help us illustrate the rules of Coq's definitional equality. We redefine the natural number predecessor function in a somewhat convoluted way and construct a manual proof that it returns [0] when applied to [1]. *) |

adamc@122 | 30 |

adamc@122 | 31 Definition pred' (x : nat) := |

adamc@122 | 32 match x with |

adamc@122 | 33 | O => O |

adamc@122 | 34 | S n' => let y := n' in y |

adamc@122 | 35 end. |

adamc@122 | 36 |

adamc@122 | 37 Theorem reduce_me : pred' 1 = 0. |

adamc@218 | 38 |

adamc@124 | 39 (* begin thide *) |

adam@364 | 40 (** CIC follows the traditions of lambda calculus in associating reduction rules with Greek letters. Coq can certainly be said to support the familiar alpha reduction rule, which allows capture-avoiding renaming of bound variables, but we never need to apply alpha explicitly, since Coq uses a de Bruijn representation%~\cite{DeBruijn}% that encodes terms canonically. |

adamc@122 | 41 |

adam@364 | 42 The %\index{delta reduction}%delta rule is for unfolding global definitions. We can use it here to unfold the definition of [pred']. We do this with the [cbv] tactic, which takes a list of reduction rules and makes as many call-by-value reduction steps as possible, using only those rules. There is an analogous tactic %\index{tactics!lazy}%[lazy] for call-by-need reduction. *) |

adamc@122 | 43 |

adamc@122 | 44 cbv delta. |

adam@364 | 45 (** %\vspace{-.15in}%[[ |

adamc@122 | 46 ============================ |

adamc@122 | 47 (fun x : nat => match x with |

adamc@122 | 48 | 0 => 0 |

adamc@122 | 49 | S n' => let y := n' in y |

adamc@122 | 50 end) 1 = 0 |

adamc@122 | 51 ]] |

adamc@122 | 52 |

adam@364 | 53 At this point, we want to apply the famous %\index{beta reduction}%beta reduction of lambda calculus, to simplify the application of a known function abstraction. *) |

adamc@122 | 54 |

adamc@122 | 55 cbv beta. |

adam@364 | 56 (** %\vspace{-.15in}%[[ |

adamc@122 | 57 ============================ |

adamc@122 | 58 match 1 with |

adamc@122 | 59 | 0 => 0 |

adamc@122 | 60 | S n' => let y := n' in y |

adamc@122 | 61 end = 0 |

adamc@122 | 62 ]] |

adamc@122 | 63 |

adam@364 | 64 Next on the list is the %\index{iota reduction}%iota reduction, which simplifies a single [match] term by determining which pattern matches. *) |

adamc@122 | 65 |

adamc@122 | 66 cbv iota. |

adam@364 | 67 (** %\vspace{-.15in}%[[ |

adamc@122 | 68 ============================ |

adamc@122 | 69 (fun n' : nat => let y := n' in y) 0 = 0 |

adamc@122 | 70 ]] |

adamc@122 | 71 |

adamc@122 | 72 Now we need another beta reduction. *) |

adamc@122 | 73 |

adamc@122 | 74 cbv beta. |

adam@364 | 75 (** %\vspace{-.15in}%[[ |

adamc@122 | 76 ============================ |

adamc@122 | 77 (let y := 0 in y) = 0 |

adamc@122 | 78 ]] |

adamc@122 | 79 |

adam@364 | 80 The final reduction rule is %\index{zeta reduction}%zeta, which replaces a [let] expression by its body with the appropriate term substituted. *) |

adamc@122 | 81 |

adamc@122 | 82 cbv zeta. |

adam@364 | 83 (** %\vspace{-.15in}%[[ |

adamc@122 | 84 ============================ |

adamc@122 | 85 0 = 0 |

adam@302 | 86 ]] |

adam@302 | 87 *) |

adamc@122 | 88 |

adamc@122 | 89 reflexivity. |

adamc@122 | 90 Qed. |

adamc@124 | 91 (* end thide *) |

adamc@122 | 92 |

adam@365 | 93 (** The [beta] reduction rule applies to recursive functions as well, and its behavior may be surprising in some instances. For instance, we can run some simple tests using the reduction strategy [compute], which applies all applicable rules of the definitional equality. *) |

adam@365 | 94 |

adam@365 | 95 Definition id (n : nat) := n. |

adam@365 | 96 |

adam@365 | 97 Eval compute in fun x => id x. |

adam@365 | 98 (** %\vspace{-.15in}%[[ |

adam@365 | 99 = fun x : nat => x |

adam@365 | 100 ]] |

adam@365 | 101 *) |

adam@365 | 102 |

adam@365 | 103 Fixpoint id' (n : nat) := n. |

adam@365 | 104 |

adam@365 | 105 Eval compute in fun x => id' x. |

adam@365 | 106 (** %\vspace{-.15in}%[[ |

adam@365 | 107 = fun x : nat => (fix id' (n : nat) : nat := n) x |

adam@365 | 108 ]] |

adam@365 | 109 |

adam@479 | 110 By running [compute], we ask Coq to run reduction steps until no more apply, so why do we see an application of a known function, where clearly no beta reduction has been performed? The answer has to do with ensuring termination of all Gallina programs. One candidate rule would say that we apply recursive definitions wherever possible. However, this would clearly lead to nonterminating reduction sequences, since the function may appear fully applied within its own definition, and we would %\%naive%{}%ly "simplify" such applications immediately. Instead, Coq only applies the beta rule for a recursive function when _the top-level structure of the recursive argument is known_. For [id'] above, we have only one argument [n], so clearly it is the recursive argument, and the top-level structure of [n] is known when the function is applied to [O] or to some [S e] term. The variable [x] is neither, so reduction is blocked. |

adam@365 | 111 |

adam@365 | 112 What are recursive arguments in general? Every recursive function is compiled by Coq to a %\index{Gallina terms!fix}%[fix] expression, for anonymous definition of recursive functions. Further, every [fix] with multiple arguments has one designated as the recursive argument via a [struct] annotation. The recursive argument is the one that must decrease across recursive calls, to appease Coq's termination checker. Coq will generally infer which argument is recursive, though we may also specify it manually, if we want to tweak reduction behavior. For instance, consider this definition of a function to add two lists of [nat]s elementwise: *) |

adam@365 | 113 |

adam@365 | 114 Fixpoint addLists (ls1 ls2 : list nat) : list nat := |

adam@365 | 115 match ls1, ls2 with |

adam@365 | 116 | n1 :: ls1' , n2 :: ls2' => n1 + n2 :: addLists ls1' ls2' |

adam@365 | 117 | _, _ => nil |

adam@365 | 118 end. |

adam@365 | 119 |

adam@365 | 120 (** By default, Coq chooses [ls1] as the recursive argument. We can see that [ls2] would have been another valid choice. The choice has a critical effect on reduction behavior, as these two examples illustrate: *) |

adam@365 | 121 |

adam@365 | 122 Eval compute in fun ls => addLists nil ls. |

adam@365 | 123 (** %\vspace{-.15in}%[[ |

adam@365 | 124 = fun _ : list nat => nil |

adam@365 | 125 ]] |

adam@365 | 126 *) |

adam@365 | 127 |

adam@365 | 128 Eval compute in fun ls => addLists ls nil. |

adam@365 | 129 (** %\vspace{-.15in}%[[ |

adam@365 | 130 = fun ls : list nat => |

adam@365 | 131 (fix addLists (ls1 ls2 : list nat) : list nat := |

adam@365 | 132 match ls1 with |

adam@365 | 133 | nil => nil |

adam@365 | 134 | n1 :: ls1' => |

adam@365 | 135 match ls2 with |

adam@365 | 136 | nil => nil |

adam@365 | 137 | n2 :: ls2' => |

adam@365 | 138 (fix plus (n m : nat) : nat := |

adam@365 | 139 match n with |

adam@365 | 140 | 0 => m |

adam@365 | 141 | S p => S (plus p m) |

adam@365 | 142 end) n1 n2 :: addLists ls1' ls2' |

adam@365 | 143 end |

adam@365 | 144 end) ls nil |

adam@365 | 145 ]] |

adam@365 | 146 |

adam@427 | 147 The outer application of the [fix] expression for [addLists] was only simplified in the first case, because in the second case the recursive argument is [ls], whose top-level structure is not known. |

adam@365 | 148 |

adam@427 | 149 The opposite behavior pertains to a version of [addLists] with [ls2] marked as recursive. *) |

adam@365 | 150 |

adam@365 | 151 Fixpoint addLists' (ls1 ls2 : list nat) {struct ls2} : list nat := |

adam@365 | 152 match ls1, ls2 with |

adam@365 | 153 | n1 :: ls1' , n2 :: ls2' => n1 + n2 :: addLists' ls1' ls2' |

adam@365 | 154 | _, _ => nil |

adam@365 | 155 end. |

adam@365 | 156 |

adam@427 | 157 (* begin hide *) |

adam@437 | 158 (* begin thide *) |

adam@437 | 159 Definition foo := (@eq, plus). |

adam@437 | 160 (* end thide *) |

adam@427 | 161 (* end hide *) |

adam@427 | 162 |

adam@365 | 163 Eval compute in fun ls => addLists' ls nil. |

adam@365 | 164 (** %\vspace{-.15in}%[[ |

adam@365 | 165 = fun ls : list nat => match ls with |

adam@365 | 166 | nil => nil |

adam@365 | 167 | _ :: _ => nil |

adam@365 | 168 end |

adam@365 | 169 ]] |

adam@365 | 170 |

adam@365 | 171 We see that all use of recursive functions has been eliminated, though the term has not quite simplified to [nil]. We could get it to do so by switching the order of the [match] discriminees in the definition of [addLists']. |

adam@365 | 172 |

adam@365 | 173 Recall that co-recursive definitions have a dual rule: a co-recursive call only simplifies when it is the discriminee of a [match]. This condition is built into the beta rule for %\index{Gallina terms!cofix}%[cofix], the anonymous form of [CoFixpoint]. |

adam@365 | 174 |

adam@365 | 175 %\medskip% |

adam@365 | 176 |

adam@407 | 177 The standard [eq] relation is critically dependent on the definitional equality. The relation [eq] is often called a%\index{propositional equality}% _propositional equality_, because it reifies definitional equality as a proposition that may or may not hold. Standard axiomatizations of an equality predicate in first-order logic define equality in terms of properties it has, like reflexivity, symmetry, and transitivity. In contrast, for [eq] in Coq, those properties are implicit in the properties of the definitional equality, which are built into CIC's metatheory and the implementation of Gallina. We could add new rules to the definitional equality, and [eq] would keep its definition and methods of use. |

adamc@122 | 178 |

adam@427 | 179 This all may make it sound like the choice of [eq]'s definition is unimportant. To the contrary, in this chapter, we will see examples where alternate definitions may simplify proofs. Before that point, I will introduce proof methods for goals that use proofs of the standard propositional equality "as data." *) |

adamc@122 | 180 |

adamc@122 | 181 |

adamc@118 | 182 (** * Heterogeneous Lists Revisited *) |

adamc@118 | 183 |

adam@480 | 184 (** One of our example dependent data structures from the last chapter (code repeated below) was the heterogeneous list and its associated "cursor" type. The recursive version poses some special challenges related to equality proofs, since it uses such proofs in its definition of [fmember] types. *) |

adamc@118 | 185 |

adamc@118 | 186 Section fhlist. |

adamc@118 | 187 Variable A : Type. |

adamc@118 | 188 Variable B : A -> Type. |

adamc@118 | 189 |

adamc@118 | 190 Fixpoint fhlist (ls : list A) : Type := |

adamc@118 | 191 match ls with |

adamc@118 | 192 | nil => unit |

adamc@118 | 193 | x :: ls' => B x * fhlist ls' |

adamc@118 | 194 end%type. |

adamc@118 | 195 |

adamc@118 | 196 Variable elm : A. |

adamc@118 | 197 |

adamc@118 | 198 Fixpoint fmember (ls : list A) : Type := |

adamc@118 | 199 match ls with |

adamc@118 | 200 | nil => Empty_set |

adamc@118 | 201 | x :: ls' => (x = elm) + fmember ls' |

adamc@118 | 202 end%type. |

adamc@118 | 203 |

adamc@118 | 204 Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm := |

adamc@118 | 205 match ls return fhlist ls -> fmember ls -> B elm with |

adamc@118 | 206 | nil => fun _ idx => match idx with end |

adamc@118 | 207 | _ :: ls' => fun mls idx => |

adamc@118 | 208 match idx with |

adamc@118 | 209 | inl pf => match pf with |

adam@426 | 210 | eq_refl => fst mls |

adamc@118 | 211 end |

adamc@118 | 212 | inr idx' => fhget ls' (snd mls) idx' |

adamc@118 | 213 end |

adamc@118 | 214 end. |

adamc@118 | 215 End fhlist. |

adamc@118 | 216 |

adamc@118 | 217 Implicit Arguments fhget [A B elm ls]. |

adamc@118 | 218 |

adam@427 | 219 (* begin hide *) |

adam@437 | 220 (* begin thide *) |

adam@427 | 221 Definition map := O. |

adam@437 | 222 (* end thide *) |

adam@427 | 223 (* end hide *) |

adam@427 | 224 |

adamc@118 | 225 (** We can define a [map]-like function for [fhlist]s. *) |

adamc@118 | 226 |

adamc@118 | 227 Section fhlist_map. |

adamc@118 | 228 Variables A : Type. |

adamc@118 | 229 Variables B C : A -> Type. |

adamc@118 | 230 Variable f : forall x, B x -> C x. |

adamc@118 | 231 |

adamc@118 | 232 Fixpoint fhmap (ls : list A) : fhlist B ls -> fhlist C ls := |

adamc@118 | 233 match ls return fhlist B ls -> fhlist C ls with |

adamc@118 | 234 | nil => fun _ => tt |

adamc@118 | 235 | _ :: _ => fun hls => (f (fst hls), fhmap _ (snd hls)) |

adamc@118 | 236 end. |

adamc@118 | 237 |

adamc@118 | 238 Implicit Arguments fhmap [ls]. |

adamc@118 | 239 |

adam@427 | 240 (* begin hide *) |

adam@437 | 241 (* begin thide *) |

adam@427 | 242 Definition ilist := O. |

adam@427 | 243 Definition get := O. |

adam@427 | 244 Definition imap := O. |

adam@437 | 245 (* end thide *) |

adam@427 | 246 (* end hide *) |

adam@427 | 247 |

adam@502 | 248 (** For the inductive versions of the [ilist] definitions, we proved a lemma about the interaction of [get] and [imap]. It was a strategic choice not to attempt such a proof for the definitions that we just gave, which sets us on a collision course with the problems that are the subject of this chapter. *) |

adamc@118 | 249 |

adamc@118 | 250 Variable elm : A. |

adamc@118 | 251 |

adam@479 | 252 Theorem fhget_fhmap : forall ls (mem : fmember elm ls) (hls : fhlist B ls), |

adamc@118 | 253 fhget (fhmap hls) mem = f (fhget hls mem). |

adam@298 | 254 (* begin hide *) |

adam@298 | 255 induction ls; crush; case a0; reflexivity. |

adam@298 | 256 (* end hide *) |

adam@364 | 257 (** %\vspace{-.2in}%[[ |

adamc@118 | 258 induction ls; crush. |

adam@298 | 259 ]] |

adamc@118 | 260 |

adam@364 | 261 %\vspace{-.15in}%In Coq 8.2, one subgoal remains at this point. Coq 8.3 has added some tactic improvements that enable [crush] to complete all of both inductive cases. To introduce the basics of reasoning about equality, it will be useful to review what was necessary in Coq 8.2. |

adam@297 | 262 |

adam@297 | 263 Part of our single remaining subgoal is: |

adamc@118 | 264 [[ |

adamc@118 | 265 a0 : a = elm |

adamc@118 | 266 ============================ |

adamc@118 | 267 match a0 in (_ = a2) return (C a2) with |

adam@426 | 268 | eq_refl => f a1 |

adamc@118 | 269 end = f match a0 in (_ = a2) return (B a2) with |

adam@426 | 270 | eq_refl => a1 |

adamc@118 | 271 end |

adamc@118 | 272 ]] |

adamc@118 | 273 |

adam@502 | 274 This seems like a trivial enough obligation. The equality proof [a0] must be [eq_refl], the only constructor of [eq]. Therefore, both the [match]es reduce to the point where the conclusion follows by reflexivity. |

adamc@118 | 275 [[ |

adamc@118 | 276 destruct a0. |

adamc@118 | 277 ]] |

adamc@118 | 278 |

adam@364 | 279 << |

adam@364 | 280 User error: Cannot solve a second-order unification problem |

adam@364 | 281 >> |

adam@364 | 282 |

adam@502 | 283 This is one of Coq's standard error messages for informing us of a failure in its heuristics for attempting an instance of an undecidable problem about dependent typing. We might try to nudge things in the right direction by stating the lemma that we believe makes the conclusion trivial. |

adamc@118 | 284 [[ |

adam@426 | 285 assert (a0 = eq_refl _). |

adam@364 | 286 ]] |

adamc@118 | 287 |

adam@364 | 288 << |

adam@426 | 289 The term "eq_refl ?98" has type "?98 = ?98" |

adamc@118 | 290 while it is expected to have type "a = elm" |

adam@364 | 291 >> |

adamc@118 | 292 |

adamc@118 | 293 In retrospect, the problem is not so hard to see. Reflexivity proofs only show [x = x] for particular values of [x], whereas here we are thinking in terms of a proof of [a = elm], where the two sides of the equality are not equal syntactically. Thus, the essential lemma we need does not even type-check! |

adamc@118 | 294 |

adam@427 | 295 Is it time to throw in the towel? Luckily, the answer is "no." In this chapter, we will see several useful patterns for proving obligations like this. |

adamc@118 | 296 |

adam@407 | 297 For this particular example, the solution is surprisingly straightforward. The [destruct] tactic has a simpler sibling [case] which should behave identically for any inductive type with one constructor of no arguments. |

adam@297 | 298 [[ |

adamc@118 | 299 case a0. |

adam@297 | 300 |

adamc@118 | 301 ============================ |

adamc@118 | 302 f a1 = f a1 |

adamc@118 | 303 ]] |

adamc@118 | 304 |

adam@297 | 305 It seems that [destruct] was trying to be too smart for its own good. |

adam@297 | 306 [[ |

adamc@118 | 307 reflexivity. |

adam@302 | 308 ]] |

adam@364 | 309 %\vspace{-.2in}% *) |

adamc@118 | 310 Qed. |

adamc@124 | 311 (* end thide *) |

adamc@118 | 312 |

adamc@118 | 313 (** It will be helpful to examine the proof terms generated by this sort of strategy. A simpler example illustrates what is going on. *) |

adamc@118 | 314 |

adam@426 | 315 Lemma lemma1 : forall x (pf : x = elm), O = match pf with eq_refl => O end. |

adamc@124 | 316 (* begin thide *) |

adamc@118 | 317 simple destruct pf; reflexivity. |

adamc@118 | 318 Qed. |

adamc@124 | 319 (* end thide *) |

adamc@118 | 320 |

adam@364 | 321 (** The tactic %\index{tactics!simple destruct}%[simple destruct pf] is a convenient form for applying [case]. It runs [intro] to bring into scope all quantified variables up to its argument. *) |

adamc@118 | 322 |

adamc@118 | 323 Print lemma1. |

adamc@218 | 324 (** %\vspace{-.15in}% [[ |

adamc@118 | 325 lemma1 = |

adamc@118 | 326 fun (x : A) (pf : x = elm) => |

adamc@118 | 327 match pf as e in (_ = y) return (0 = match e with |

adam@426 | 328 | eq_refl => 0 |

adamc@118 | 329 end) with |

adam@426 | 330 | eq_refl => eq_refl 0 |

adamc@118 | 331 end |

adamc@118 | 332 : forall (x : A) (pf : x = elm), 0 = match pf with |

adam@426 | 333 | eq_refl => 0 |

adamc@118 | 334 end |

adamc@218 | 335 |

adamc@118 | 336 ]] |

adamc@118 | 337 |

adamc@118 | 338 Using what we know about shorthands for [match] annotations, we can write this proof in shorter form manually. *) |

adamc@118 | 339 |

adamc@124 | 340 (* begin thide *) |

adam@456 | 341 Definition lemma1' (x : A) (pf : x = elm) := |

adam@456 | 342 match pf return (0 = match pf with |

adam@456 | 343 | eq_refl => 0 |

adam@456 | 344 end) with |

adam@456 | 345 | eq_refl => eq_refl 0 |

adam@456 | 346 end. |

adamc@124 | 347 (* end thide *) |

adamc@118 | 348 |

adam@398 | 349 (** Surprisingly, what seems at first like a _simpler_ lemma is harder to prove. *) |

adamc@118 | 350 |

adam@426 | 351 Lemma lemma2 : forall (x : A) (pf : x = x), O = match pf with eq_refl => O end. |

adamc@124 | 352 (* begin thide *) |

adam@364 | 353 (** %\vspace{-.25in}%[[ |

adamc@118 | 354 simple destruct pf. |

adam@364 | 355 ]] |

adamc@205 | 356 |

adam@364 | 357 << |

adamc@118 | 358 User error: Cannot solve a second-order unification problem |

adam@364 | 359 >> |

adam@302 | 360 *) |

adamc@118 | 361 Abort. |

adamc@118 | 362 |

adamc@118 | 363 (** Nonetheless, we can adapt the last manual proof to handle this theorem. *) |

adamc@118 | 364 |

adamc@124 | 365 (* begin thide *) |

adamc@124 | 366 Definition lemma2 := |

adamc@118 | 367 fun (x : A) (pf : x = x) => |

adamc@118 | 368 match pf return (0 = match pf with |

adam@426 | 369 | eq_refl => 0 |

adamc@118 | 370 end) with |

adam@426 | 371 | eq_refl => eq_refl 0 |

adamc@118 | 372 end. |

adamc@124 | 373 (* end thide *) |

adamc@118 | 374 |

adamc@118 | 375 (** We can try to prove a lemma that would simplify proofs of many facts like [lemma2]: *) |

adamc@118 | 376 |

adam@427 | 377 (* begin hide *) |

adam@437 | 378 (* begin thide *) |

adam@427 | 379 Definition lemma3' := O. |

adam@437 | 380 (* end thide *) |

adam@427 | 381 (* end hide *) |

adam@427 | 382 |

adam@426 | 383 Lemma lemma3 : forall (x : A) (pf : x = x), pf = eq_refl x. |

adamc@124 | 384 (* begin thide *) |

adam@364 | 385 (** %\vspace{-.25in}%[[ |

adamc@118 | 386 simple destruct pf. |

adam@364 | 387 ]] |

adamc@205 | 388 |

adam@364 | 389 << |

adamc@118 | 390 User error: Cannot solve a second-order unification problem |

adam@364 | 391 >> |

adam@364 | 392 %\vspace{-.15in}%*) |

adamc@218 | 393 |

adamc@118 | 394 Abort. |

adamc@118 | 395 |

adamc@118 | 396 (** This time, even our manual attempt fails. |

adamc@118 | 397 [[ |

adamc@118 | 398 Definition lemma3' := |

adamc@118 | 399 fun (x : A) (pf : x = x) => |

adam@426 | 400 match pf as pf' in (_ = x') return (pf' = eq_refl x') with |

adam@426 | 401 | eq_refl => eq_refl _ |

adamc@118 | 402 end. |

adam@364 | 403 ]] |

adamc@118 | 404 |

adam@364 | 405 << |

adam@426 | 406 The term "eq_refl x'" has type "x' = x'" while it is expected to have type |

adamc@118 | 407 "x = x'" |

adam@364 | 408 >> |

adamc@118 | 409 |

adam@427 | 410 The type error comes from our [return] annotation. In that annotation, the [as]-bound variable [pf'] has type [x = x'], referring to the [in]-bound variable [x']. To do a dependent [match], we _must_ choose a fresh name for the second argument of [eq]. We are just as constrained to use the "real" value [x] for the first argument. Thus, within the [return] clause, the proof we are matching on _must_ equate two non-matching terms, which makes it impossible to equate that proof with reflexivity. |

adamc@118 | 411 |

adam@398 | 412 Nonetheless, it turns out that, with one catch, we _can_ prove this lemma. *) |

adamc@118 | 413 |

adam@426 | 414 Lemma lemma3 : forall (x : A) (pf : x = x), pf = eq_refl x. |

adamc@118 | 415 intros; apply UIP_refl. |

adamc@118 | 416 Qed. |

adamc@118 | 417 |

adamc@118 | 418 Check UIP_refl. |

adamc@218 | 419 (** %\vspace{-.15in}% [[ |

adamc@118 | 420 UIP_refl |

adam@426 | 421 : forall (U : Type) (x : U) (p : x = x), p = eq_refl x |

adamc@118 | 422 ]] |

adamc@118 | 423 |

adam@480 | 424 The theorem %\index{Gallina terms!UIP\_refl}%[UIP_refl] comes from the [Eqdep] module of the standard library. (Its name uses the acronym "UIP" for "unicity of identity proofs.") Do the Coq authors know of some clever trick for building such proofs that we have not seen yet? If they do, they did not use it for this proof. Rather, the proof is based on an _axiom_, the term [eq_rect_eq] below. *) |

adam@427 | 425 |

adam@436 | 426 (* begin hide *) |

adam@436 | 427 Import Eq_rect_eq. |

adam@436 | 428 (* end hide *) |

adamc@118 | 429 Print eq_rect_eq. |

adamc@218 | 430 (** %\vspace{-.15in}% [[ |

adam@436 | 431 *** [ eq_rect_eq : |

adam@436 | 432 forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p), |

adam@436 | 433 x = eq_rect p Q x p h ] |

adamc@118 | 434 ]] |

adamc@118 | 435 |

adam@502 | 436 The axiom %\index{Gallina terms!eq\_rect\_eq}%[eq_rect_eq] states a "fact" that seems like common sense, once the notation is deciphered. The term [eq_rect] is the automatically generated recursion principle for [eq]. Calling [eq_rect] is another way of [match]ing on an equality proof. The proof we match on is the argument [h], and [x] is the body of the [match]. The statement of [eq_rect_eq] just says that [match]es on proofs of [p = p], for any [p], are superfluous and may be removed. We can see this intuition better in code by asking Coq to simplify the theorem statement with the [compute] reduction strategy. *) |

adam@427 | 437 |

adam@427 | 438 (* begin hide *) |

adam@437 | 439 (* begin thide *) |

adam@427 | 440 Definition False' := False. |

adam@437 | 441 (* end thide *) |

adam@427 | 442 (* end hide *) |

adamc@118 | 443 |

adam@364 | 444 Eval compute in (forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p), |

adam@364 | 445 x = eq_rect p Q x p h). |

adam@364 | 446 (** %\vspace{-.15in}%[[ |

adam@364 | 447 = forall (U : Type) (p : U) (Q : U -> Type) (x : Q p) (h : p = p), |

adam@364 | 448 x = match h in (_ = y) return (Q y) with |

adam@364 | 449 | eq_refl => x |

adam@364 | 450 end |

adam@364 | 451 ]] |

adam@364 | 452 |

adam@427 | 453 Perhaps surprisingly, we cannot prove [eq_rect_eq] from within Coq. This proposition is introduced as an %\index{axioms}%axiom; that is, a proposition asserted as true without proof. We cannot assert just any statement without proof. Adding [False] as an axiom would allow us to prove any proposition, for instance, defeating the point of using a proof assistant. In general, we need to be sure that we never assert _inconsistent_ sets of axioms. A set of axioms is inconsistent if its conjunction implies [False]. For the case of [eq_rect_eq], consistency has been verified outside of Coq via "informal" metatheory%~\cite{AxiomK}%, in a study that also established unprovability of the axiom in CIC. |

adamc@118 | 454 |

adamc@118 | 455 This axiom is equivalent to another that is more commonly known and mentioned in type theory circles. *) |

adamc@118 | 456 |

adam@427 | 457 (* begin hide *) |

adam@437 | 458 (* begin thide *) |

adam@437 | 459 Definition Streicher_K' := UIP_refl__Streicher_K. |

adam@437 | 460 (* end thide *) |

adam@427 | 461 (* end hide *) |

adam@427 | 462 |

adam@480 | 463 Check Streicher_K. |

adamc@124 | 464 (* end thide *) |

adamc@218 | 465 (** %\vspace{-.15in}% [[ |

adam@480 | 466 Streicher_K |

adamc@118 | 467 : forall (U : Type) (x : U) (P : x = x -> Prop), |

adam@480 | 468 P eq_refl -> forall p : x = x, P p |

adamc@118 | 469 ]] |

adamc@118 | 470 |

adam@480 | 471 This is the opaquely named %\index{axiom K}%"Streicher's axiom K," which says that a predicate on properly typed equality proofs holds of all such proofs if it holds of reflexivity. *) |

adamc@118 | 472 |

adamc@118 | 473 End fhlist_map. |

adamc@118 | 474 |

adam@436 | 475 (* begin hide *) |

adam@437 | 476 (* begin thide *) |

adam@436 | 477 Require Eqdep_dec. |

adam@437 | 478 (* end thide *) |

adam@436 | 479 (* end hide *) |

adam@436 | 480 |

adam@364 | 481 (** It is worth remarking that it is possible to avoid axioms altogether for equalities on types with decidable equality. The [Eqdep_dec] module of the standard library contains a parametric proof of [UIP_refl] for such cases. To simplify presentation, we will stick with the axiom version in the rest of this chapter. *) |

adam@364 | 482 |

adamc@119 | 483 |

adamc@119 | 484 (** * Type-Casts in Theorem Statements *) |

adamc@119 | 485 |

adamc@119 | 486 (** Sometimes we need to use tricks with equality just to state the theorems that we care about. To illustrate, we start by defining a concatenation function for [fhlist]s. *) |

adamc@119 | 487 |

adamc@119 | 488 Section fhapp. |

adamc@119 | 489 Variable A : Type. |

adamc@119 | 490 Variable B : A -> Type. |

adamc@119 | 491 |

adamc@218 | 492 Fixpoint fhapp (ls1 ls2 : list A) |

adamc@119 | 493 : fhlist B ls1 -> fhlist B ls2 -> fhlist B (ls1 ++ ls2) := |

adamc@218 | 494 match ls1 with |

adamc@119 | 495 | nil => fun _ hls2 => hls2 |

adamc@119 | 496 | _ :: _ => fun hls1 hls2 => (fst hls1, fhapp _ _ (snd hls1) hls2) |

adamc@119 | 497 end. |

adamc@119 | 498 |

adamc@119 | 499 Implicit Arguments fhapp [ls1 ls2]. |

adamc@119 | 500 |

adamc@124 | 501 (* EX: Prove that fhapp is associative. *) |

adamc@124 | 502 (* begin thide *) |

adamc@124 | 503 |

adamc@119 | 504 (** We might like to prove that [fhapp] is associative. |

adamc@119 | 505 [[ |

adam@479 | 506 Theorem fhapp_assoc : forall ls1 ls2 ls3 |

adamc@119 | 507 (hls1 : fhlist B ls1) (hls2 : fhlist B ls2) (hls3 : fhlist B ls3), |

adamc@119 | 508 fhapp hls1 (fhapp hls2 hls3) = fhapp (fhapp hls1 hls2) hls3. |

adam@364 | 509 ]] |

adamc@119 | 510 |

adam@364 | 511 << |

adamc@119 | 512 The term |

adamc@119 | 513 "fhapp (ls1:=ls1 ++ ls2) (ls2:=ls3) (fhapp (ls1:=ls1) (ls2:=ls2) hls1 hls2) |

adamc@119 | 514 hls3" has type "fhlist B ((ls1 ++ ls2) ++ ls3)" |

adamc@119 | 515 while it is expected to have type "fhlist B (ls1 ++ ls2 ++ ls3)" |

adam@364 | 516 >> |

adamc@119 | 517 |

adam@407 | 518 This first cut at the theorem statement does not even type-check. We know that the two [fhlist] types appearing in the error message are always equal, by associativity of normal list append, but this fact is not apparent to the type checker. This stems from the fact that Coq's equality is%\index{intensional type theory}% _intensional_, in the sense that type equality theorems can never be applied after the fact to get a term to type-check. Instead, we need to make use of equality explicitly in the theorem statement. *) |

adamc@119 | 519 |

adam@479 | 520 Theorem fhapp_assoc : forall ls1 ls2 ls3 |

adamc@119 | 521 (pf : (ls1 ++ ls2) ++ ls3 = ls1 ++ (ls2 ++ ls3)) |

adamc@119 | 522 (hls1 : fhlist B ls1) (hls2 : fhlist B ls2) (hls3 : fhlist B ls3), |

adamc@119 | 523 fhapp hls1 (fhapp hls2 hls3) |

adamc@119 | 524 = match pf in (_ = ls) return fhlist _ ls with |

adam@426 | 525 | eq_refl => fhapp (fhapp hls1 hls2) hls3 |

adamc@119 | 526 end. |

adamc@119 | 527 induction ls1; crush. |

adamc@119 | 528 |

adamc@119 | 529 (** The first remaining subgoal looks trivial enough: |

adamc@119 | 530 [[ |

adamc@119 | 531 ============================ |

adamc@119 | 532 fhapp (ls1:=ls2) (ls2:=ls3) hls2 hls3 = |

adamc@119 | 533 match pf in (_ = ls) return (fhlist B ls) with |

adam@426 | 534 | eq_refl => fhapp (ls1:=ls2) (ls2:=ls3) hls2 hls3 |

adamc@119 | 535 end |

adamc@119 | 536 ]] |

adamc@119 | 537 |

adamc@119 | 538 We can try what worked in previous examples. |

adamc@119 | 539 [[ |

adamc@119 | 540 case pf. |

adam@364 | 541 ]] |

adamc@119 | 542 |

adam@364 | 543 << |

adamc@119 | 544 User error: Cannot solve a second-order unification problem |

adam@364 | 545 >> |

adamc@119 | 546 |

adamc@119 | 547 It seems we have reached another case where it is unclear how to use a dependent [match] to implement case analysis on our proof. The [UIP_refl] theorem can come to our rescue again. *) |

adamc@119 | 548 |

adamc@119 | 549 rewrite (UIP_refl _ _ pf). |

adamc@119 | 550 (** [[ |

adamc@119 | 551 ============================ |

adamc@119 | 552 fhapp (ls1:=ls2) (ls2:=ls3) hls2 hls3 = |

adamc@119 | 553 fhapp (ls1:=ls2) (ls2:=ls3) hls2 hls3 |

adam@302 | 554 ]] |

adam@302 | 555 *) |

adamc@119 | 556 |

adamc@119 | 557 reflexivity. |

adamc@119 | 558 |

adamc@119 | 559 (** Our second subgoal is trickier. |

adamc@119 | 560 [[ |

adamc@119 | 561 pf : a :: (ls1 ++ ls2) ++ ls3 = a :: ls1 ++ ls2 ++ ls3 |

adamc@119 | 562 ============================ |

adamc@119 | 563 (a0, |

adamc@119 | 564 fhapp (ls1:=ls1) (ls2:=ls2 ++ ls3) b |

adamc@119 | 565 (fhapp (ls1:=ls2) (ls2:=ls3) hls2 hls3)) = |

adamc@119 | 566 match pf in (_ = ls) return (fhlist B ls) with |

adam@426 | 567 | eq_refl => |

adamc@119 | 568 (a0, |

adamc@119 | 569 fhapp (ls1:=ls1 ++ ls2) (ls2:=ls3) |

adamc@119 | 570 (fhapp (ls1:=ls1) (ls2:=ls2) b hls2) hls3) |

adamc@119 | 571 end |

adamc@119 | 572 |

adamc@119 | 573 rewrite (UIP_refl _ _ pf). |

adam@364 | 574 ]] |

adamc@119 | 575 |

adam@364 | 576 << |

adamc@119 | 577 The term "pf" has type "a :: (ls1 ++ ls2) ++ ls3 = a :: ls1 ++ ls2 ++ ls3" |

adamc@119 | 578 while it is expected to have type "?556 = ?556" |

adam@364 | 579 >> |

adamc@119 | 580 |

adamc@119 | 581 We can only apply [UIP_refl] on proofs of equality with syntactically equal operands, which is not the case of [pf] here. We will need to manipulate the form of this subgoal to get us to a point where we may use [UIP_refl]. A first step is obtaining a proof suitable to use in applying the induction hypothesis. Inversion on the structure of [pf] is sufficient for that. *) |

adamc@119 | 582 |

adamc@119 | 583 injection pf; intro pf'. |

adamc@119 | 584 (** [[ |

adamc@119 | 585 pf : a :: (ls1 ++ ls2) ++ ls3 = a :: ls1 ++ ls2 ++ ls3 |

adamc@119 | 586 pf' : (ls1 ++ ls2) ++ ls3 = ls1 ++ ls2 ++ ls3 |

adamc@119 | 587 ============================ |

adamc@119 | 588 (a0, |

adamc@119 | 589 fhapp (ls1:=ls1) (ls2:=ls2 ++ ls3) b |

adamc@119 | 590 (fhapp (ls1:=ls2) (ls2:=ls3) hls2 hls3)) = |

adamc@119 | 591 match pf in (_ = ls) return (fhlist B ls) with |

adam@426 | 592 | eq_refl => |

adamc@119 | 593 (a0, |

adamc@119 | 594 fhapp (ls1:=ls1 ++ ls2) (ls2:=ls3) |

adamc@119 | 595 (fhapp (ls1:=ls1) (ls2:=ls2) b hls2) hls3) |

adamc@119 | 596 end |

adamc@119 | 597 ]] |

adamc@119 | 598 |

adamc@119 | 599 Now we can rewrite using the inductive hypothesis. *) |

adamc@119 | 600 |

adamc@119 | 601 rewrite (IHls1 _ _ pf'). |

adamc@119 | 602 (** [[ |

adamc@119 | 603 ============================ |

adamc@119 | 604 (a0, |

adamc@119 | 605 match pf' in (_ = ls) return (fhlist B ls) with |

adam@426 | 606 | eq_refl => |

adamc@119 | 607 fhapp (ls1:=ls1 ++ ls2) (ls2:=ls3) |

adamc@119 | 608 (fhapp (ls1:=ls1) (ls2:=ls2) b hls2) hls3 |

adamc@119 | 609 end) = |

adamc@119 | 610 match pf in (_ = ls) return (fhlist B ls) with |

adam@426 | 611 | eq_refl => |

adamc@119 | 612 (a0, |

adamc@119 | 613 fhapp (ls1:=ls1 ++ ls2) (ls2:=ls3) |

adamc@119 | 614 (fhapp (ls1:=ls1) (ls2:=ls2) b hls2) hls3) |

adamc@119 | 615 end |

adamc@119 | 616 ]] |

adamc@119 | 617 |

adam@398 | 618 We have made an important bit of progress, as now only a single call to [fhapp] appears in the conclusion, repeated twice. Trying case analysis on our proofs still will not work, but there is a move we can make to enable it. Not only does just one call to [fhapp] matter to us now, but it also _does not matter what the result of the call is_. In other words, the subgoal should remain true if we replace this [fhapp] call with a fresh variable. The %\index{tactics!generalize}%[generalize] tactic helps us do exactly that. *) |

adamc@119 | 619 |

adamc@119 | 620 generalize (fhapp (fhapp b hls2) hls3). |

adamc@119 | 621 (** [[ |

adamc@119 | 622 forall f : fhlist B ((ls1 ++ ls2) ++ ls3), |

adamc@119 | 623 (a0, |

adamc@119 | 624 match pf' in (_ = ls) return (fhlist B ls) with |

adam@426 | 625 | eq_refl => f |

adamc@119 | 626 end) = |

adamc@119 | 627 match pf in (_ = ls) return (fhlist B ls) with |

adam@426 | 628 | eq_refl => (a0, f) |

adamc@119 | 629 end |

adamc@119 | 630 ]] |

adamc@119 | 631 |

adam@502 | 632 The conclusion has gotten markedly simpler. It seems counterintuitive that we can have an easier time of proving a more general theorem, but such a phenomenon applies to the case here and to many other proofs that use dependent types heavily. Speaking informally, the reason why this kind of activity helps is that [match] annotations contain some positions where only variables are allowed. By reducing more elements of a goal to variables, built-in tactics can have more success building [match] terms under the hood. |

adamc@119 | 633 |

adamc@119 | 634 In this case, it is helpful to generalize over our two proofs as well. *) |

adamc@119 | 635 |

adamc@119 | 636 generalize pf pf'. |

adamc@119 | 637 (** [[ |

adamc@119 | 638 forall (pf0 : a :: (ls1 ++ ls2) ++ ls3 = a :: ls1 ++ ls2 ++ ls3) |

adamc@119 | 639 (pf'0 : (ls1 ++ ls2) ++ ls3 = ls1 ++ ls2 ++ ls3) |

adamc@119 | 640 (f : fhlist B ((ls1 ++ ls2) ++ ls3)), |

adamc@119 | 641 (a0, |

adamc@119 | 642 match pf'0 in (_ = ls) return (fhlist B ls) with |

adam@426 | 643 | eq_refl => f |

adamc@119 | 644 end) = |

adamc@119 | 645 match pf0 in (_ = ls) return (fhlist B ls) with |

adam@426 | 646 | eq_refl => (a0, f) |

adamc@119 | 647 end |

adamc@119 | 648 ]] |

adamc@119 | 649 |

adam@502 | 650 To an experienced dependent types hacker, the appearance of this goal term calls for a celebration. The formula has a critical property that indicates that our problems are over. To get our proofs into the right form to apply [UIP_refl], we need to use associativity of list append to rewrite their types. We could not do so before because other parts of the goal require the proofs to retain their original types. In particular, the call to [fhapp] that we generalized must have type [(ls1 ++ ls2) ++ ls3], for some values of the list variables. If we rewrite the type of the proof used to type-cast this value to something like [ls1 ++ ls2 ++ ls3 = ls1 ++ ls2 ++ ls3], then the lefthand side of the equality would no longer match the type of the term we are trying to cast. |

adamc@119 | 651 |

adam@398 | 652 However, now that we have generalized over the [fhapp] call, the type of the term being type-cast appears explicitly in the goal and _may be rewritten as well_. In particular, the final masterstroke is rewriting everywhere in our goal using associativity of list append. *) |

adamc@119 | 653 |

adam@479 | 654 rewrite app_assoc. |

adamc@119 | 655 (** [[ |

adamc@119 | 656 ============================ |

adamc@119 | 657 forall (pf0 : a :: ls1 ++ ls2 ++ ls3 = a :: ls1 ++ ls2 ++ ls3) |

adamc@119 | 658 (pf'0 : ls1 ++ ls2 ++ ls3 = ls1 ++ ls2 ++ ls3) |

adamc@119 | 659 (f : fhlist B (ls1 ++ ls2 ++ ls3)), |

adamc@119 | 660 (a0, |

adamc@119 | 661 match pf'0 in (_ = ls) return (fhlist B ls) with |

adam@426 | 662 | eq_refl => f |

adamc@119 | 663 end) = |

adamc@119 | 664 match pf0 in (_ = ls) return (fhlist B ls) with |

adam@426 | 665 | eq_refl => (a0, f) |

adamc@119 | 666 end |

adamc@119 | 667 ]] |

adamc@119 | 668 |

adamc@119 | 669 We can see that we have achieved the crucial property: the type of each generalized equality proof has syntactically equal operands. This makes it easy to finish the proof with [UIP_refl]. *) |

adamc@119 | 670 |

adamc@119 | 671 intros. |

adamc@119 | 672 rewrite (UIP_refl _ _ pf0). |

adamc@119 | 673 rewrite (UIP_refl _ _ pf'0). |

adamc@119 | 674 reflexivity. |

adamc@119 | 675 Qed. |

adamc@124 | 676 (* end thide *) |

adamc@119 | 677 End fhapp. |

adamc@120 | 678 |

adamc@120 | 679 Implicit Arguments fhapp [A B ls1 ls2]. |

adamc@120 | 680 |

adam@364 | 681 (** This proof strategy was cumbersome and unorthodox, from the perspective of mainstream mathematics. The next section explores an alternative that leads to simpler developments in some cases. *) |

adam@364 | 682 |

adamc@120 | 683 |

adamc@120 | 684 (** * Heterogeneous Equality *) |

adamc@120 | 685 |

adam@407 | 686 (** There is another equality predicate, defined in the %\index{Gallina terms!JMeq}%[JMeq] module of the standard library, implementing%\index{heterogeneous equality}% _heterogeneous equality_. *) |

adamc@120 | 687 |

adamc@120 | 688 Print JMeq. |

adamc@218 | 689 (** %\vspace{-.15in}% [[ |

adamc@120 | 690 Inductive JMeq (A : Type) (x : A) : forall B : Type, B -> Prop := |

adamc@120 | 691 JMeq_refl : JMeq x x |

adamc@120 | 692 ]] |

adamc@120 | 693 |

adam@480 | 694 The identifier [JMeq] stands for %\index{John Major equality}%"John Major equality," a name coined by Conor McBride%~\cite{JMeq}% as an inside joke about British politics. The definition [JMeq] starts out looking a lot like the definition of [eq]. The crucial difference is that we may use [JMeq] _on arguments of different types_. For instance, a lemma that we failed to establish before is trivial with [JMeq]. It makes for prettier theorem statements to define some syntactic shorthand first. *) |

adamc@120 | 695 |

adamc@120 | 696 Infix "==" := JMeq (at level 70, no associativity). |

adamc@120 | 697 |

adam@426 | 698 (* EX: Prove UIP_refl' : forall (A : Type) (x : A) (pf : x = x), pf == eq_refl x *) |

adamc@124 | 699 (* begin thide *) |

adam@426 | 700 Definition UIP_refl' (A : Type) (x : A) (pf : x = x) : pf == eq_refl x := |

adam@426 | 701 match pf return (pf == eq_refl _) with |

adam@426 | 702 | eq_refl => JMeq_refl _ |

adamc@120 | 703 end. |

adamc@124 | 704 (* end thide *) |

adamc@120 | 705 |

adamc@120 | 706 (** There is no quick way to write such a proof by tactics, but the underlying proof term that we want is trivial. |

adamc@120 | 707 |

adamc@271 | 708 Suppose that we want to use [UIP_refl'] to establish another lemma of the kind we have run into several times so far. *) |

adamc@120 | 709 |

adamc@120 | 710 Lemma lemma4 : forall (A : Type) (x : A) (pf : x = x), |

adam@426 | 711 O = match pf with eq_refl => O end. |

adamc@124 | 712 (* begin thide *) |

adamc@121 | 713 intros; rewrite (UIP_refl' pf); reflexivity. |

adamc@120 | 714 Qed. |

adamc@124 | 715 (* end thide *) |

adamc@120 | 716 |

adamc@120 | 717 (** All in all, refreshingly straightforward, but there really is no such thing as a free lunch. The use of [rewrite] is implemented in terms of an axiom: *) |

adamc@120 | 718 |

adamc@120 | 719 Check JMeq_eq. |

adamc@218 | 720 (** %\vspace{-.15in}% [[ |

adamc@120 | 721 JMeq_eq |

adamc@120 | 722 : forall (A : Type) (x y : A), x == y -> x = y |

adamc@218 | 723 ]] |

adamc@120 | 724 |

adam@480 | 725 It may be surprising that we cannot prove that heterogeneous equality implies normal equality. The difficulties are the same kind we have seen so far, based on limitations of [match] annotations. The [JMeq_eq] axiom has been proved on paper to be consistent, but asserting it may still be considered to complicate the logic we work in, so there is some motivation for avoiding it. |

adamc@120 | 726 |

adamc@120 | 727 We can redo our [fhapp] associativity proof based around [JMeq]. *) |

adamc@120 | 728 |

adamc@120 | 729 Section fhapp'. |

adamc@120 | 730 Variable A : Type. |

adamc@120 | 731 Variable B : A -> Type. |

adamc@120 | 732 |

adam@484 | 733 (** This time, the %\%naive%{}% theorem statement type-checks. *) |

adamc@120 | 734 |

adamc@124 | 735 (* EX: Prove [fhapp] associativity using [JMeq]. *) |

adamc@124 | 736 |

adamc@124 | 737 (* begin thide *) |

adam@479 | 738 Theorem fhapp_assoc' : forall ls1 ls2 ls3 (hls1 : fhlist B ls1) (hls2 : fhlist B ls2) |

adam@364 | 739 (hls3 : fhlist B ls3), |

adamc@120 | 740 fhapp hls1 (fhapp hls2 hls3) == fhapp (fhapp hls1 hls2) hls3. |

adamc@120 | 741 induction ls1; crush. |

adamc@120 | 742 |

adamc@120 | 743 (** Even better, [crush] discharges the first subgoal automatically. The second subgoal is: |

adamc@120 | 744 [[ |

adamc@120 | 745 ============================ |

adam@297 | 746 (a0, fhapp b (fhapp hls2 hls3)) == (a0, fhapp (fhapp b hls2) hls3) |

adamc@120 | 747 ]] |

adamc@120 | 748 |

adam@297 | 749 It looks like one rewrite with the inductive hypothesis should be enough to make the goal trivial. Here is what happens when we try that in Coq 8.2: |

adamc@120 | 750 [[ |

adamc@120 | 751 rewrite IHls1. |

adam@364 | 752 ]] |

adamc@120 | 753 |

adam@364 | 754 << |

adamc@120 | 755 Error: Impossible to unify "fhlist B ((ls1 ++ ?1572) ++ ?1573)" with |

adamc@120 | 756 "fhlist B (ls1 ++ ?1572 ++ ?1573)" |

adam@364 | 757 >> |

adamc@120 | 758 |

adam@407 | 759 Coq 8.4 currently gives an error message about an uncaught exception. Perhaps that will be fixed soon. In any case, it is educational to consider a more explicit approach. |

adam@297 | 760 |

adamc@120 | 761 We see that [JMeq] is not a silver bullet. We can use it to simplify the statements of equality facts, but the Coq type-checker uses non-trivial heterogeneous equality facts no more readily than it uses standard equality facts. Here, the problem is that the form [(e1, e2)] is syntactic sugar for an explicit application of a constructor of an inductive type. That application mentions the type of each tuple element explicitly, and our [rewrite] tries to change one of those elements without updating the corresponding type argument. |

adamc@120 | 762 |

adamc@120 | 763 We can get around this problem by another multiple use of [generalize]. We want to bring into the goal the proper instance of the inductive hypothesis, and we also want to generalize the two relevant uses of [fhapp]. *) |

adamc@120 | 764 |

adamc@120 | 765 generalize (fhapp b (fhapp hls2 hls3)) |

adamc@120 | 766 (fhapp (fhapp b hls2) hls3) |

adamc@120 | 767 (IHls1 _ _ b hls2 hls3). |

adam@364 | 768 (** %\vspace{-.15in}%[[ |

adamc@120 | 769 ============================ |

adamc@120 | 770 forall (f : fhlist B (ls1 ++ ls2 ++ ls3)) |

adamc@120 | 771 (f0 : fhlist B ((ls1 ++ ls2) ++ ls3)), f == f0 -> (a0, f) == (a0, f0) |

adamc@120 | 772 ]] |

adamc@120 | 773 |

adamc@120 | 774 Now we can rewrite with append associativity, as before. *) |

adamc@120 | 775 |

adam@479 | 776 rewrite app_assoc. |

adam@364 | 777 (** %\vspace{-.15in}%[[ |

adamc@120 | 778 ============================ |

adamc@120 | 779 forall f f0 : fhlist B (ls1 ++ ls2 ++ ls3), f == f0 -> (a0, f) == (a0, f0) |

adamc@120 | 780 ]] |

adamc@120 | 781 |

adamc@120 | 782 From this point, the goal is trivial. *) |

adamc@120 | 783 |

adamc@120 | 784 intros f f0 H; rewrite H; reflexivity. |

adamc@120 | 785 Qed. |

adamc@124 | 786 (* end thide *) |

adamc@121 | 787 |

adam@385 | 788 End fhapp'. |

adam@385 | 789 |

adam@385 | 790 (** This example illustrates a general pattern: heterogeneous equality often simplifies theorem statements, but we still need to do some work to line up some dependent pattern matches that tactics will generate for us. |

adam@385 | 791 |

adam@385 | 792 The proof we have found relies on the [JMeq_eq] axiom, which we can verify with a command%\index{Vernacular commands!Print Assumptions}% that we will discuss more in two chapters. *) |

adam@385 | 793 |

adam@479 | 794 Print Assumptions fhapp_assoc'. |

adam@385 | 795 (** %\vspace{-.15in}%[[ |

adam@385 | 796 Axioms: |

adam@385 | 797 JMeq_eq : forall (A : Type) (x y : A), x == y -> x = y |

adam@385 | 798 ]] |

adam@385 | 799 |

adam@385 | 800 It was the [rewrite H] tactic that implicitly appealed to the axiom. By restructuring the proof, we can avoid axiom dependence. A general lemma about pairs provides the key element. (Our use of [generalize] above can be thought of as reducing the proof to another, more complex and specialized lemma.) *) |

adam@385 | 801 |

adam@385 | 802 Lemma pair_cong : forall A1 A2 B1 B2 (x1 : A1) (x2 : A2) (y1 : B1) (y2 : B2), |

adam@385 | 803 x1 == x2 |

adam@385 | 804 -> y1 == y2 |

adam@385 | 805 -> (x1, y1) == (x2, y2). |

adam@385 | 806 intros until y2; intros Hx Hy; rewrite Hx; rewrite Hy; reflexivity. |

adam@385 | 807 Qed. |

adam@385 | 808 |

adam@385 | 809 Hint Resolve pair_cong. |

adam@385 | 810 |

adam@385 | 811 Section fhapp''. |

adam@385 | 812 Variable A : Type. |

adam@385 | 813 Variable B : A -> Type. |

adam@385 | 814 |

adam@479 | 815 Theorem fhapp_assoc'' : forall ls1 ls2 ls3 (hls1 : fhlist B ls1) (hls2 : fhlist B ls2) |

adam@385 | 816 (hls3 : fhlist B ls3), |

adam@385 | 817 fhapp hls1 (fhapp hls2 hls3) == fhapp (fhapp hls1 hls2) hls3. |

adam@385 | 818 induction ls1; crush. |

adam@385 | 819 Qed. |

adam@385 | 820 End fhapp''. |

adam@385 | 821 |

adam@479 | 822 Print Assumptions fhapp_assoc''. |

adam@385 | 823 (** << |

adam@385 | 824 Closed under the global context |

adam@385 | 825 >> |

adam@385 | 826 |

adam@479 | 827 One might wonder exactly which elements of a proof involving [JMeq] imply that [JMeq_eq] must be used. For instance, above we noticed that [rewrite] had brought [JMeq_eq] into the proof of [fhapp_assoc'], yet here we have also used [rewrite] with [JMeq] hypotheses while avoiding axioms! One illuminating exercise is comparing the types of the lemmas that [rewrite] uses under the hood to implement the rewrites. Here is the normal lemma for [eq] rewriting:%\index{Gallina terms!eq\_ind\_r}% *) |

adam@385 | 828 |

adam@385 | 829 Check eq_ind_r. |

adam@385 | 830 (** %\vspace{-.15in}%[[ |

adam@385 | 831 eq_ind_r |

adam@385 | 832 : forall (A : Type) (x : A) (P : A -> Prop), |

adam@385 | 833 P x -> forall y : A, y = x -> P y |

adam@385 | 834 ]] |

adam@385 | 835 |

adam@536 | 836 The corresponding lemma used for [JMeq] in the proof of [pair_cong] is defined internally by [rewrite] as needed, but its type happens to be the following. *) |

adam@385 | 837 |

adam@385 | 838 (** %\vspace{-.15in}%[[ |

adam@398 | 839 internal_JMeq_rew_r |

adam@385 | 840 : forall (A : Type) (x : A) (B : Type) (b : B) |

adam@385 | 841 (P : forall B0 : Type, B0 -> Type), P B b -> x == b -> P A x |

adam@385 | 842 ]] |

adam@385 | 843 |

adam@534 | 844 The key difference is that, where the [eq] lemma is parameterized on a predicate of type [A -> Prop], the [JMeq] lemma is parameterized on a predicate of type more like [forall A : Type, A -> Prop]. To apply [eq_ind_r] with a proof of [x = y], it is only necessary to rearrange the goal into an application of a [fun] abstraction to [y]. In contrast, to apply the alternative principle, it is necessary to rearrange the goal to an application of a [fun] abstraction to both [y] and _its type_. In other words, the predicate must be _polymorphic_ in [y]'s type; any type must make sense, from a type-checking standpoint. There may be cases where the former rearrangement is easy to do in a type-correct way, but the second rearrangement done %\%naive%{}%ly leads to a type error. |

adam@385 | 845 |

adam@536 | 846 When [rewrite] cannot figure out how to apply the alternative principle for [x == y] where [x] and [y] have the same type, the tactic can instead use a different theorem, which is easy to prove as a composition of [eq_ind_r] and [JMeq_eq]. *) |

adam@385 | 847 |

adam@385 | 848 Check JMeq_ind_r. |

adam@385 | 849 (** %\vspace{-.15in}%[[ |

adam@385 | 850 JMeq_ind_r |

adam@385 | 851 : forall (A : Type) (x : A) (P : A -> Prop), |

adam@385 | 852 P x -> forall y : A, y == x -> P y |

adam@385 | 853 ]] |

adam@385 | 854 |

adam@534 | 855 Ironically, where in the proof of [fhapp_assoc'] we used [rewrite app_assoc] to make it clear that a use of [JMeq] was actually homogeneously typed, we created a situation where [rewrite] applied the axiom-based [JMeq_ind_r] instead of the axiom-free principle! |

adam@385 | 856 |

adam@385 | 857 For another simple example, consider this theorem that applies a heterogeneous equality to prove a congruence fact. *) |

adam@385 | 858 |

adam@385 | 859 Theorem out_of_luck : forall n m : nat, |

adam@385 | 860 n == m |

adam@385 | 861 -> S n == S m. |

adam@385 | 862 intros n m H. |

adam@385 | 863 |

adam@480 | 864 (** Applying [JMeq_ind_r] is easy, as the %\index{tactics!pattern}%[pattern] tactic will transform the goal into an application of an appropriate [fun] to a term that we want to abstract. (In general, [pattern] abstracts over a term by introducing a new anonymous function taking that term as argument.) *) |

adam@385 | 865 |

adam@385 | 866 pattern n. |

adam@385 | 867 (** %\vspace{-.15in}%[[ |

adam@385 | 868 n : nat |

adam@385 | 869 m : nat |

adam@385 | 870 H : n == m |

adam@385 | 871 ============================ |

adam@385 | 872 (fun n0 : nat => S n0 == S m) n |

adam@385 | 873 ]] |

adam@385 | 874 *) |

adam@385 | 875 apply JMeq_ind_r with (x := m); auto. |

adam@385 | 876 |

adam@534 | 877 (** However, we run into trouble trying to get the goal into a form compatible with the alternative principle. *) |

adam@427 | 878 |

adam@385 | 879 Undo 2. |

adam@385 | 880 (** %\vspace{-.15in}%[[ |

adam@385 | 881 pattern nat, n. |

adam@385 | 882 ]] |

adam@385 | 883 << |

adam@385 | 884 Error: The abstracted term "fun (P : Set) (n0 : P) => S n0 == S m" |

adam@385 | 885 is not well typed. |

adam@385 | 886 Illegal application (Type Error): |

adam@385 | 887 The term "S" of type "nat -> nat" |

adam@385 | 888 cannot be applied to the term |

adam@385 | 889 "n0" : "P" |

adam@385 | 890 This term has type "P" which should be coercible to |

adam@385 | 891 "nat". |

adam@385 | 892 >> |

adam@385 | 893 |

adam@385 | 894 In other words, the successor function [S] is insufficiently polymorphic. If we try to generalize over the type of [n], we find that [S] is no longer legal to apply to [n]. *) |

adam@385 | 895 |

adam@385 | 896 Abort. |

adam@385 | 897 |

adam@479 | 898 (** Why did we not run into this problem in our proof of [fhapp_assoc'']? The reason is that the pair constructor is polymorphic in the types of the pair components, while functions like [S] are not polymorphic at all. Use of such non-polymorphic functions with [JMeq] tends to push toward use of axioms. The example with [nat] here is a bit unrealistic; more likely cases would involve functions that have _some_ polymorphism, but not enough to allow abstractions of the sort we attempted above with [pattern]. For instance, we might have an equality between two lists, where the goal only type-checks when the terms involved really are lists, though everything is polymorphic in the types of list data elements. The {{http://www.mpi-sws.org/~gil/Heq/}Heq} library builds up a slightly different foundation to help avoid such problems. *) |

adam@364 | 899 |

adamc@121 | 900 |

adamc@121 | 901 (** * Equivalence of Equality Axioms *) |

adamc@121 | 902 |

adamc@124 | 903 (* EX: Show that the approaches based on K and JMeq are equivalent logically. *) |

adamc@124 | 904 |

adamc@124 | 905 (* begin thide *) |

adam@502 | 906 (** Assuming axioms (like axiom K and [JMeq_eq]) is a hazardous business. The due diligence associated with it is necessarily global in scope, since two axioms may be consistent alone but inconsistent together. It turns out that all of the major axioms proposed for reasoning about equality in Coq are logically equivalent, so that we only need to pick one to assert without proof. In this section, we demonstrate by showing how each of the previous two sections' approaches reduces to the other logically. |

adamc@121 | 907 |

adamc@121 | 908 To show that [JMeq] and its axiom let us prove [UIP_refl], we start from the lemma [UIP_refl'] from the previous section. The rest of the proof is trivial. *) |

adamc@121 | 909 |

adam@426 | 910 Lemma UIP_refl'' : forall (A : Type) (x : A) (pf : x = x), pf = eq_refl x. |

adamc@121 | 911 intros; rewrite (UIP_refl' pf); reflexivity. |

adamc@121 | 912 Qed. |

adamc@121 | 913 |

adamc@121 | 914 (** The other direction is perhaps more interesting. Assume that we only have the axiom of the [Eqdep] module available. We can define [JMeq] in a way that satisfies the same interface as the combination of the [JMeq] module's inductive definition and axiom. *) |

adamc@121 | 915 |

adamc@121 | 916 Definition JMeq' (A : Type) (x : A) (B : Type) (y : B) : Prop := |

adam@426 | 917 exists pf : B = A, x = match pf with eq_refl => y end. |

adamc@121 | 918 |

adamc@121 | 919 Infix "===" := JMeq' (at level 70, no associativity). |

adamc@121 | 920 |

adam@427 | 921 (** remove printing exists *) |

adam@427 | 922 |

adamc@121 | 923 (** We say that, by definition, [x] and [y] are equal if and only if there exists a proof [pf] that their types are equal, such that [x] equals the result of casting [y] with [pf]. This statement can look strange from the standpoint of classical math, where we almost never mention proofs explicitly with quantifiers in formulas, but it is perfectly legal Coq code. |

adamc@121 | 924 |

adamc@121 | 925 We can easily prove a theorem with the same type as that of the [JMeq_refl] constructor of [JMeq]. *) |

adamc@121 | 926 |

adamc@121 | 927 Theorem JMeq_refl' : forall (A : Type) (x : A), x === x. |

adam@426 | 928 intros; unfold JMeq'; exists (eq_refl A); reflexivity. |

adamc@121 | 929 Qed. |

adamc@121 | 930 |

adamc@121 | 931 (** printing exists $\exists$ *) |

adamc@121 | 932 |

adamc@121 | 933 (** The proof of an analogue to [JMeq_eq] is a little more interesting, but most of the action is in appealing to [UIP_refl]. *) |

adamc@121 | 934 |

adamc@121 | 935 Theorem JMeq_eq' : forall (A : Type) (x y : A), |

adamc@121 | 936 x === y -> x = y. |

adamc@121 | 937 unfold JMeq'; intros. |

adamc@121 | 938 (** [[ |

adamc@121 | 939 H : exists pf : A = A, |

adamc@121 | 940 x = match pf in (_ = T) return T with |

adam@426 | 941 | eq_refl => y |

adamc@121 | 942 end |

adamc@121 | 943 ============================ |

adamc@121 | 944 x = y |

adam@302 | 945 ]] |

adam@302 | 946 *) |

adamc@121 | 947 |

adamc@121 | 948 destruct H. |

adamc@121 | 949 (** [[ |

adamc@121 | 950 x0 : A = A |

adamc@121 | 951 H : x = match x0 in (_ = T) return T with |

adam@426 | 952 | eq_refl => y |

adamc@121 | 953 end |

adamc@121 | 954 ============================ |

adamc@121 | 955 x = y |

adam@302 | 956 ]] |

adam@302 | 957 *) |

adamc@121 | 958 |

adamc@121 | 959 rewrite H. |

adamc@121 | 960 (** [[ |

adamc@121 | 961 x0 : A = A |

adamc@121 | 962 ============================ |

adamc@121 | 963 match x0 in (_ = T) return T with |

adam@426 | 964 | eq_refl => y |

adamc@121 | 965 end = y |

adam@302 | 966 ]] |

adam@302 | 967 *) |

adamc@121 | 968 |

adamc@121 | 969 rewrite (UIP_refl _ _ x0); reflexivity. |

adamc@121 | 970 Qed. |

adamc@121 | 971 |

adam@427 | 972 (** We see that, in a very formal sense, we are free to switch back and forth between the two styles of proofs about equality proofs. One style may be more convenient than the other for some proofs, but we can always interconvert between our results. The style that does not use heterogeneous equality may be preferable in cases where many results do not require the tricks of this chapter, since then the use of axioms is avoided altogether for the simple cases, and a wider audience will be able to follow those "simple" proofs. On the other hand, heterogeneous equality often makes for shorter and more readable theorem statements. *) |

adamc@123 | 973 |

adamc@124 | 974 (* end thide *) |

adamc@123 | 975 |

adamc@123 | 976 |

adamc@123 | 977 (** * Equality of Functions *) |

adamc@123 | 978 |

adam@444 | 979 (** The following seems like a reasonable theorem to want to hold, and it does hold in set theory. |

adam@444 | 980 %\vspace{-.15in}%[[ |

adam@456 | 981 Theorem two_funs : (fun n => n) = (fun n => n + 0). |

adamc@205 | 982 ]] |

adam@444 | 983 %\vspace{-.15in}%Unfortunately, this theorem is not provable in CIC without additional axioms. None of the definitional equality rules force function equality to be%\index{extensionality of function equality}% _extensional_. That is, the fact that two functions return equal results on equal inputs does not imply that the functions are equal. We _can_ assert function extensionality as an axiom, and indeed the standard library already contains that axiom. *) |

adamc@123 | 984 |

adam@407 | 985 Require Import FunctionalExtensionality. |

adam@407 | 986 About functional_extensionality. |

adam@407 | 987 (** %\vspace{-.15in}%[[ |

adam@407 | 988 functional_extensionality : |

adam@407 | 989 forall (A B : Type) (f g : A -> B), (forall x : A, f x = g x) -> f = g |

adam@407 | 990 ]] |

adam@407 | 991 *) |

adamc@123 | 992 |

adam@456 | 993 (** This axiom has been verified metatheoretically to be consistent with CIC and the two equality axioms we considered previously. With it, the proof of [two_funs] is trivial. *) |

adamc@123 | 994 |

adam@456 | 995 Theorem two_funs : (fun n => n) = (fun n => n + 0). |

adamc@124 | 996 (* begin thide *) |

adam@407 | 997 apply functional_extensionality; crush. |

adamc@123 | 998 Qed. |

adamc@124 | 999 (* end thide *) |

adamc@123 | 1000 |

adam@427 | 1001 (** The same axiom can help us prove equality of types, where we need to "reason under quantifiers." *) |

adamc@123 | 1002 |

adamc@123 | 1003 Theorem forall_eq : (forall x : nat, match x with |

adamc@123 | 1004 | O => True |

adamc@123 | 1005 | S _ => True |

adamc@123 | 1006 end) |

adamc@123 | 1007 = (forall _ : nat, True). |

adamc@123 | 1008 |

adam@456 | 1009 (** There are no immediate opportunities to apply [functional_extensionality], but we can use %\index{tactics!change}%[change] to fix that problem. *) |

adamc@123 | 1010 |

adamc@124 | 1011 (* begin thide *) |

adamc@123 | 1012 change ((forall x : nat, (fun x => match x with |

adamc@123 | 1013 | 0 => True |

adamc@123 | 1014 | S _ => True |

adamc@123 | 1015 end) x) = (nat -> True)). |

adam@407 | 1016 rewrite (functional_extensionality (fun x => match x with |

adam@407 | 1017 | 0 => True |

adam@407 | 1018 | S _ => True |

adam@407 | 1019 end) (fun _ => True)). |

adamc@123 | 1020 (** [[ |

adamc@123 | 1021 2 subgoals |

adamc@123 | 1022 |

adamc@123 | 1023 ============================ |

adamc@123 | 1024 (nat -> True) = (nat -> True) |

adamc@123 | 1025 |

adamc@123 | 1026 subgoal 2 is: |

adamc@123 | 1027 forall x : nat, match x with |

adamc@123 | 1028 | 0 => True |

adamc@123 | 1029 | S _ => True |

adamc@123 | 1030 end = True |

adam@302 | 1031 ]] |

adam@302 | 1032 *) |

adamc@123 | 1033 |

adamc@123 | 1034 reflexivity. |

adamc@123 | 1035 |

adamc@123 | 1036 destruct x; constructor. |

adamc@123 | 1037 Qed. |

adamc@124 | 1038 (* end thide *) |

adamc@127 | 1039 |

adam@407 | 1040 (** Unlike in the case of [eq_rect_eq], we have no way of deriving this axiom of%\index{functional extensionality}% _functional extensionality_ for types with decidable equality. To allow equality reasoning without axioms, it may be worth rewriting a development to replace functions with alternate representations, such as finite map types for which extensionality is derivable in CIC. *) |