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New release
author Adam Chlipala <adam@chlipala.net>
date Thu, 25 Aug 2011 14:55:38 -0400
parents 7b38729be069
children d5787b70cf48
rev   line source
adam@297 1 (* Copyright (c) 2008-2011, Adam Chlipala
adamc@142 2 *
adamc@142 3 * This work is licensed under a
adamc@142 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@142 5 * Unported License.
adamc@142 6 * The license text is available at:
adamc@142 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@142 8 *)
adamc@142 9
adamc@142 10 (* begin hide *)
adamc@142 11 Require Import List.
adamc@142 12
adamc@142 13 Require Import Tactics MoreSpecif.
adamc@142 14
adamc@142 15 Set Implicit Arguments.
adamc@142 16 (* end hide *)
adamc@142 17
adamc@142 18
adamc@142 19 (** %\chapter{Proof by Reflection}% *)
adamc@142 20
adamc@142 21 (** The last chapter highlighted a very heuristic approach to proving. In this chapter, we will study an alternative technique, %\textit{%#<i>#proof by reflection#</i>#%}%. We will write, in Gallina, decision procedures with proofs of correctness, and we will appeal to these procedures in writing very short proofs. Such a proof is checked by running the decision procedure. The term %\textit{%#<i>#reflection#</i>#%}% applies because we will need to translate Gallina propositions into values of inductive types representing syntax, so that Gallina programs may analyze them. *)
adamc@142 22
adamc@142 23
adamc@142 24 (** * Proving Evenness *)
adamc@142 25
adamc@142 26 (** Proving that particular natural number constants are even is certainly something we would rather have happen automatically. The Ltac-programming techniques that we learned in the last chapter make it easy to implement such a procedure. *)
adamc@142 27
adamc@142 28 Inductive isEven : nat -> Prop :=
adamc@144 29 | Even_O : isEven O
adamc@144 30 | Even_SS : forall n, isEven n -> isEven (S (S n)).
adamc@142 31
adamc@148 32 (* begin thide *)
adamc@142 33 Ltac prove_even := repeat constructor.
adamc@148 34 (* end thide *)
adamc@142 35
adamc@142 36 Theorem even_256 : isEven 256.
adamc@142 37 prove_even.
adamc@142 38 Qed.
adamc@142 39
adamc@142 40 Print even_256.
adamc@221 41 (** %\vspace{-.15in}% [[
adamc@142 42 even_256 =
adamc@142 43 Even_SS
adamc@142 44 (Even_SS
adamc@142 45 (Even_SS
adamc@142 46 (Even_SS
adamc@221 47
adamc@142 48 ]]
adamc@142 49
adam@289 50 %\noindent%...and so on. This procedure always works (at least on machines with infinite resources), but it has a serious drawback, which we see when we print the proof it generates that 256 is even. The final proof term has length super-linear in the input value. (Coq's implicit arguments mechanism is hiding the values given for parameter [n] of [Even_SS], which is why the proof term only appears linear here.) This seems like a shame, since we could write a trivial and trustworthy program to verify evenness of constants. The proof checker could simply call our program where needed.
adamc@142 51
adamc@142 52 It is also unfortunate not to have static typing guarantees that our tactic always behaves appropriately. Other invocations of similar tactics might fail with dynamic type errors, and we would not know about the bugs behind these errors until we happened to attempt to prove complex enough goals.
adamc@142 53
adamc@142 54 The techniques of proof by reflection address both complaints. We will be able to write proofs like this with constant size overhead beyond the size of the input, and we will do it with verified decision procedures written in Gallina.
adamc@142 55
adamc@221 56 For this example, we begin by using a type from the [MoreSpecif] module (included in the book source) to write a certified evenness checker. *)
adamc@142 57
adamc@142 58 Print partial.
adamc@221 59 (** %\vspace{-.15in}% [[
adamc@221 60 Inductive partial (P : Prop) : Set := Proved : P -> [P] | Uncertain : [P]
adamc@221 61
adamc@221 62 ]]
adamc@142 63
adamc@221 64 A [partial P] value is an optional proof of [P]. The notation [[P]] stands for [partial P]. *)
adamc@142 65
adamc@221 66 Local Open Scope partial_scope.
adamc@142 67
adamc@142 68 (** We bring into scope some notations for the [partial] type. These overlap with some of the notations we have seen previously for specification types, so they were placed in a separate scope that needs separate opening. *)
adamc@142 69
adamc@148 70 (* begin thide *)
adam@297 71 Definition check_even : forall n : nat, [isEven n].
adamc@142 72 Hint Constructors isEven.
adamc@142 73
adamc@142 74 refine (fix F (n : nat) : [isEven n] :=
adamc@221 75 match n with
adamc@142 76 | 0 => Yes
adamc@142 77 | 1 => No
adamc@142 78 | S (S n') => Reduce (F n')
adamc@142 79 end); auto.
adamc@142 80 Defined.
adamc@142 81
adamc@142 82 (** We can use dependent pattern-matching to write a function that performs a surprising feat. When given a [partial P], this function [partialOut] returns a proof of [P] if the [partial] value contains a proof, and it returns a (useless) proof of [True] otherwise. From the standpoint of ML and Haskell programming, it seems impossible to write such a type, but it is trivial with a [return] annotation. *)
adamc@142 83
adamc@142 84 Definition partialOut (P : Prop) (x : [P]) :=
adamc@142 85 match x return (match x with
adamc@142 86 | Proved _ => P
adamc@142 87 | Uncertain => True
adamc@142 88 end) with
adamc@142 89 | Proved pf => pf
adamc@142 90 | Uncertain => I
adamc@142 91 end.
adamc@142 92
adam@289 93 (** It may seem strange to define a function like this. However, it turns out to be very useful in writing a reflective version of our earlier [prove_even] tactic: *)
adamc@142 94
adamc@142 95 Ltac prove_even_reflective :=
adamc@142 96 match goal with
adamc@142 97 | [ |- isEven ?N] => exact (partialOut (check_even N))
adamc@142 98 end.
adamc@148 99 (* end thide *)
adamc@142 100
adam@289 101 (** We identify which natural number we are considering, and we %``%#"#prove#"#%''% its evenness by pulling the proof out of the appropriate [check_even] call. *)
adamc@142 102
adamc@142 103 Theorem even_256' : isEven 256.
adamc@142 104 prove_even_reflective.
adamc@142 105 Qed.
adamc@142 106
adamc@142 107 Print even_256'.
adamc@221 108 (** %\vspace{-.15in}% [[
adamc@142 109 even_256' = partialOut (check_even 256)
adamc@142 110 : isEven 256
adamc@221 111
adamc@142 112 ]]
adamc@142 113
adam@289 114 We can see a constant wrapper around the object of the proof. For any even number, this form of proof will suffice. The size of the proof term is now linear in the number being checked, containing two repetitions of the unary form of that number, one of which is hidden above within the implicit argument to [partialOut].
adam@289 115
adam@289 116 What happens if we try the tactic with an odd number? *)
adamc@142 117
adamc@142 118 Theorem even_255 : isEven 255.
adamc@142 119 (** [[
adamc@142 120 prove_even_reflective.
adamc@142 121
adamc@142 122 User error: No matching clauses for match goal
adamc@221 123
adamc@142 124 ]]
adamc@142 125
adamc@142 126 Thankfully, the tactic fails. To see more precisely what goes wrong, we can run manually the body of the [match].
adamc@142 127
adamc@142 128 [[
adamc@142 129 exact (partialOut (check_even 255)).
adamc@142 130
adamc@142 131 Error: The term "partialOut (check_even 255)" has type
adamc@142 132 "match check_even 255 with
adamc@142 133 | Yes => isEven 255
adamc@142 134 | No => True
adamc@142 135 end" while it is expected to have type "isEven 255"
adamc@221 136
adamc@142 137 ]]
adamc@142 138
adamc@142 139 As usual, the type-checker performs no reductions to simplify error messages. If we reduced the first term ourselves, we would see that [check_even 255] reduces to a [No], so that the first term is equivalent to [True], which certainly does not unify with [isEven 255]. *)
adamc@221 140
adamc@142 141 Abort.
adamc@143 142
adamc@143 143
adamc@143 144 (** * Reflecting the Syntax of a Trivial Tautology Language *)
adamc@143 145
adamc@143 146 (** We might also like to have reflective proofs of trivial tautologies like this one: *)
adamc@143 147
adamc@143 148 Theorem true_galore : (True /\ True) -> (True \/ (True /\ (True -> True))).
adamc@143 149 tauto.
adamc@143 150 Qed.
adamc@143 151
adamc@143 152 Print true_galore.
adamc@221 153 (** %\vspace{-.15in}% [[
adamc@143 154 true_galore =
adamc@143 155 fun H : True /\ True =>
adamc@143 156 and_ind (fun _ _ : True => or_introl (True /\ (True -> True)) I) H
adamc@143 157 : True /\ True -> True \/ True /\ (True -> True)
adamc@221 158
adamc@143 159 ]]
adamc@143 160
adamc@143 161 As we might expect, the proof that [tauto] builds contains explicit applications of natural deduction rules. For large formulas, this can add a linear amount of proof size overhead, beyond the size of the input.
adamc@143 162
adam@289 163 To write a reflective procedure for this class of goals, we will need to get into the actual %``%#"#reflection#"#%''% part of %``%#"#proof by reflection.#"#%''% It is impossible to case-analyze a [Prop] in any way in Gallina. We must %\textit{%#<i>#reflect#</i>#%}% [Prop] into some type that we %\textit{%#<i>#can#</i>#%}% analyze. This inductive type is a good candidate: *)
adamc@143 164
adamc@148 165 (* begin thide *)
adamc@143 166 Inductive taut : Set :=
adamc@143 167 | TautTrue : taut
adamc@143 168 | TautAnd : taut -> taut -> taut
adamc@143 169 | TautOr : taut -> taut -> taut
adamc@143 170 | TautImp : taut -> taut -> taut.
adamc@143 171
adam@289 172 (** We write a recursive function to %``%#"#unreflect#"#%''% this syntax back to [Prop]. *)
adamc@143 173
adamc@143 174 Fixpoint tautDenote (t : taut) : Prop :=
adamc@143 175 match t with
adamc@143 176 | TautTrue => True
adamc@143 177 | TautAnd t1 t2 => tautDenote t1 /\ tautDenote t2
adamc@143 178 | TautOr t1 t2 => tautDenote t1 \/ tautDenote t2
adamc@143 179 | TautImp t1 t2 => tautDenote t1 -> tautDenote t2
adamc@143 180 end.
adamc@143 181
adamc@143 182 (** It is easy to prove that every formula in the range of [tautDenote] is true. *)
adamc@143 183
adamc@143 184 Theorem tautTrue : forall t, tautDenote t.
adamc@143 185 induction t; crush.
adamc@143 186 Qed.
adamc@143 187
adamc@143 188 (** To use [tautTrue] to prove particular formulas, we need to implement the syntax reflection process. A recursive Ltac function does the job. *)
adamc@143 189
adamc@143 190 Ltac tautReflect P :=
adamc@143 191 match P with
adamc@143 192 | True => TautTrue
adamc@143 193 | ?P1 /\ ?P2 =>
adamc@143 194 let t1 := tautReflect P1 in
adamc@143 195 let t2 := tautReflect P2 in
adamc@143 196 constr:(TautAnd t1 t2)
adamc@143 197 | ?P1 \/ ?P2 =>
adamc@143 198 let t1 := tautReflect P1 in
adamc@143 199 let t2 := tautReflect P2 in
adamc@143 200 constr:(TautOr t1 t2)
adamc@143 201 | ?P1 -> ?P2 =>
adamc@143 202 let t1 := tautReflect P1 in
adamc@143 203 let t2 := tautReflect P2 in
adamc@143 204 constr:(TautImp t1 t2)
adamc@143 205 end.
adamc@143 206
adamc@143 207 (** With [tautReflect] available, it is easy to finish our reflective tactic. We look at the goal formula, reflect it, and apply [tautTrue] to the reflected formula. *)
adamc@143 208
adamc@143 209 Ltac obvious :=
adamc@143 210 match goal with
adamc@143 211 | [ |- ?P ] =>
adamc@143 212 let t := tautReflect P in
adamc@143 213 exact (tautTrue t)
adamc@143 214 end.
adamc@143 215
adamc@143 216 (** We can verify that [obvious] solves our original example, with a proof term that does not mention details of the proof. *)
adamc@148 217 (* end thide *)
adamc@143 218
adamc@143 219 Theorem true_galore' : (True /\ True) -> (True \/ (True /\ (True -> True))).
adamc@143 220 obvious.
adamc@143 221 Qed.
adamc@143 222
adamc@143 223 Print true_galore'.
adamc@143 224
adamc@221 225 (** %\vspace{-.15in}% [[
adamc@143 226 true_galore' =
adamc@143 227 tautTrue
adamc@143 228 (TautImp (TautAnd TautTrue TautTrue)
adamc@143 229 (TautOr TautTrue (TautAnd TautTrue (TautImp TautTrue TautTrue))))
adamc@143 230 : True /\ True -> True \/ True /\ (True -> True)
adamc@221 231
adamc@143 232 ]]
adamc@143 233
adam@289 234 It is worth considering how the reflective tactic improves on a pure-Ltac implementation. The formula reflection process is just as ad-hoc as before, so we gain little there. In general, proofs will be more complicated than formula translation, and the %``%#"#generic proof rule#"#%''% that we apply here %\textit{%#<i>#is#</i>#%}% on much better formal footing than a recursive Ltac function. The dependent type of the proof guarantees that it %``%#"#works#"#%''% on any input formula. This is all in addition to the proof-size improvement that we have already seen. *)
adamc@144 235
adamc@144 236
adamc@145 237 (** * A Monoid Expression Simplifier *)
adamc@145 238
adam@289 239 (** Proof by reflection does not require encoding of all of the syntax in a goal. We can insert %``%#"#variables#"#%''% in our syntax types to allow injection of arbitrary pieces, even if we cannot apply specialized reasoning to them. In this section, we explore that possibility by writing a tactic for normalizing monoid equations. *)
adamc@146 240
adamc@145 241 Section monoid.
adamc@145 242 Variable A : Set.
adamc@145 243 Variable e : A.
adamc@145 244 Variable f : A -> A -> A.
adamc@145 245
adamc@145 246 Infix "+" := f.
adamc@145 247
adamc@145 248 Hypothesis assoc : forall a b c, (a + b) + c = a + (b + c).
adamc@145 249 Hypothesis identl : forall a, e + a = a.
adamc@145 250 Hypothesis identr : forall a, a + e = a.
adamc@145 251
adamc@146 252 (** We add variables and hypotheses characterizing an arbitrary instance of the algebraic structure of monoids. We have an associative binary operator and an identity element for it.
adamc@146 253
adam@289 254 It is easy to define an expression tree type for monoid expressions. A [Var] constructor is a %``%#"#catch-all#"#%''% case for subexpressions that we cannot model. These subexpressions could be actual Gallina variables, or they could just use functions that our tactic is unable to understand. *)
adamc@146 255
adamc@148 256 (* begin thide *)
adamc@145 257 Inductive mexp : Set :=
adamc@145 258 | Ident : mexp
adamc@145 259 | Var : A -> mexp
adamc@145 260 | Op : mexp -> mexp -> mexp.
adamc@145 261
adam@289 262 (** Next, we write an %``%#"#un-reflect#"#%''% function. *)
adamc@146 263
adamc@145 264 Fixpoint mdenote (me : mexp) : A :=
adamc@145 265 match me with
adamc@145 266 | Ident => e
adamc@145 267 | Var v => v
adamc@145 268 | Op me1 me2 => mdenote me1 + mdenote me2
adamc@145 269 end.
adamc@145 270
adamc@146 271 (** We will normalize expressions by flattening them into lists, via associativity, so it is helpful to have a denotation function for lists of monoid values. *)
adamc@146 272
adamc@145 273 Fixpoint mldenote (ls : list A) : A :=
adamc@145 274 match ls with
adamc@145 275 | nil => e
adamc@145 276 | x :: ls' => x + mldenote ls'
adamc@145 277 end.
adamc@145 278
adamc@146 279 (** The flattening function itself is easy to implement. *)
adamc@146 280
adamc@145 281 Fixpoint flatten (me : mexp) : list A :=
adamc@145 282 match me with
adamc@145 283 | Ident => nil
adamc@145 284 | Var x => x :: nil
adamc@145 285 | Op me1 me2 => flatten me1 ++ flatten me2
adamc@145 286 end.
adamc@145 287
adamc@146 288 (** [flatten] has a straightforward correctness proof in terms of our [denote] functions. *)
adamc@146 289
adamc@146 290 Lemma flatten_correct' : forall ml2 ml1,
adamc@146 291 mldenote ml1 + mldenote ml2 = mldenote (ml1 ++ ml2).
adamc@145 292 induction ml1; crush.
adamc@145 293 Qed.
adamc@145 294
adamc@145 295 Theorem flatten_correct : forall me, mdenote me = mldenote (flatten me).
adamc@145 296 Hint Resolve flatten_correct'.
adamc@145 297
adamc@145 298 induction me; crush.
adamc@145 299 Qed.
adamc@145 300
adamc@146 301 (** Now it is easy to prove a theorem that will be the main tool behind our simplification tactic. *)
adamc@146 302
adamc@146 303 Theorem monoid_reflect : forall me1 me2,
adamc@146 304 mldenote (flatten me1) = mldenote (flatten me2)
adamc@146 305 -> mdenote me1 = mdenote me2.
adamc@145 306 intros; repeat rewrite flatten_correct; assumption.
adamc@145 307 Qed.
adamc@145 308
adamc@146 309 (** We implement reflection into the [mexp] type. *)
adamc@146 310
adamc@146 311 Ltac reflect me :=
adamc@146 312 match me with
adamc@145 313 | e => Ident
adamc@146 314 | ?me1 + ?me2 =>
adamc@146 315 let r1 := reflect me1 in
adamc@146 316 let r2 := reflect me2 in
adamc@145 317 constr:(Op r1 r2)
adamc@146 318 | _ => constr:(Var me)
adamc@145 319 end.
adamc@145 320
adamc@146 321 (** The final [monoid] tactic works on goals that equate two monoid terms. We reflect each and change the goal to refer to the reflected versions, finishing off by applying [monoid_reflect] and simplifying uses of [mldenote]. *)
adamc@146 322
adamc@145 323 Ltac monoid :=
adamc@145 324 match goal with
adamc@146 325 | [ |- ?me1 = ?me2 ] =>
adamc@146 326 let r1 := reflect me1 in
adamc@146 327 let r2 := reflect me2 in
adamc@145 328 change (mdenote r1 = mdenote r2);
adamc@145 329 apply monoid_reflect; simpl mldenote
adamc@145 330 end.
adamc@145 331
adamc@146 332 (** We can make short work of theorems like this one: *)
adamc@146 333
adamc@148 334 (* end thide *)
adamc@148 335
adamc@145 336 Theorem t1 : forall a b c d, a + b + c + d = a + (b + c) + d.
adamc@146 337 intros; monoid.
adamc@146 338 (** [[
adamc@146 339 ============================
adamc@146 340 a + (b + (c + (d + e))) = a + (b + (c + (d + e)))
adamc@221 341
adamc@146 342 ]]
adamc@146 343
adamc@146 344 [monoid] has canonicalized both sides of the equality, such that we can finish the proof by reflexivity. *)
adamc@146 345
adamc@145 346 reflexivity.
adamc@145 347 Qed.
adamc@146 348
adamc@146 349 (** It is interesting to look at the form of the proof. *)
adamc@146 350
adamc@146 351 Print t1.
adamc@221 352 (** %\vspace{-.15in}% [[
adamc@146 353 t1 =
adamc@146 354 fun a b c d : A =>
adamc@146 355 monoid_reflect (Op (Op (Op (Var a) (Var b)) (Var c)) (Var d))
adamc@146 356 (Op (Op (Var a) (Op (Var b) (Var c))) (Var d))
adamc@146 357 (refl_equal (a + (b + (c + (d + e)))))
adamc@146 358 : forall a b c d : A, a + b + c + d = a + (b + c) + d
adamc@221 359
adamc@146 360 ]]
adamc@146 361
adamc@146 362 The proof term contains only restatements of the equality operands in reflected form, followed by a use of reflexivity on the shared canonical form. *)
adamc@221 363
adamc@145 364 End monoid.
adamc@145 365
adamc@146 366 (** Extensions of this basic approach are used in the implementations of the [ring] and [field] tactics that come packaged with Coq. *)
adamc@146 367
adamc@145 368
adamc@144 369 (** * A Smarter Tautology Solver *)
adamc@144 370
adamc@221 371 (** Now we are ready to revisit our earlier tautology solver example. We want to broaden the scope of the tactic to include formulas whose truth is not syntactically apparent. We will want to allow injection of arbitrary formulas, like we allowed arbitrary monoid expressions in the last example. Since we are working in a richer theory, it is important to be able to use equalities between different injected formulas. For instance, we cannot prove [P -> P] by translating the formula into a value like [Imp (Var P) (Var P)], because a Gallina function has no way of comparing the two [P]s for equality.
adamc@147 372
adamc@147 373 To arrive at a nice implementation satisfying these criteria, we introduce the [quote] tactic and its associated library. *)
adamc@147 374
adamc@144 375 Require Import Quote.
adamc@144 376
adamc@148 377 (* begin thide *)
adamc@144 378 Inductive formula : Set :=
adamc@144 379 | Atomic : index -> formula
adamc@144 380 | Truth : formula
adamc@144 381 | Falsehood : formula
adamc@144 382 | And : formula -> formula -> formula
adamc@144 383 | Or : formula -> formula -> formula
adamc@144 384 | Imp : formula -> formula -> formula.
adamc@144 385
adamc@147 386 (** The type [index] comes from the [Quote] library and represents a countable variable type. The rest of [formula]'s definition should be old hat by now.
adamc@147 387
adamc@147 388 The [quote] tactic will implement injection from [Prop] into [formula] for us, but it is not quite as smart as we might like. In particular, it interprets implications incorrectly, so we will need to declare a wrapper definition for implication, as we did in the last chapter. *)
adamc@144 389
adamc@144 390 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@144 391 Infix "-->" := imp (no associativity, at level 95).
adamc@144 392
adamc@147 393 (** Now we can define our denotation function. *)
adamc@147 394
adamc@147 395 Definition asgn := varmap Prop.
adamc@147 396
adamc@144 397 Fixpoint formulaDenote (atomics : asgn) (f : formula) : Prop :=
adamc@144 398 match f with
adamc@144 399 | Atomic v => varmap_find False v atomics
adamc@144 400 | Truth => True
adamc@144 401 | Falsehood => False
adamc@144 402 | And f1 f2 => formulaDenote atomics f1 /\ formulaDenote atomics f2
adamc@144 403 | Or f1 f2 => formulaDenote atomics f1 \/ formulaDenote atomics f2
adamc@144 404 | Imp f1 f2 => formulaDenote atomics f1 --> formulaDenote atomics f2
adamc@144 405 end.
adamc@144 406
adamc@147 407 (** The [varmap] type family implements maps from [index] values. In this case, we define an assignment as a map from variables to [Prop]s. [formulaDenote] works with an assignment, and we use the [varmap_find] function to consult the assignment in the [Atomic] case. The first argument to [varmap_find] is a default value, in case the variable is not found. *)
adamc@147 408
adamc@144 409 Section my_tauto.
adamc@144 410 Variable atomics : asgn.
adamc@144 411
adamc@144 412 Definition holds (v : index) := varmap_find False v atomics.
adamc@144 413
adamc@147 414 (** We define some shorthand for a particular variable being true, and now we are ready to define some helpful functions based on the [ListSet] module of the standard library, which (unsurprisingly) presents a view of lists as sets. *)
adamc@147 415
adamc@144 416 Require Import ListSet.
adamc@144 417
adamc@144 418 Definition index_eq : forall x y : index, {x = y} + {x <> y}.
adamc@144 419 decide equality.
adamc@144 420 Defined.
adamc@144 421
adamc@144 422 Definition add (s : set index) (v : index) := set_add index_eq v s.
adamc@147 423
adamc@221 424 Definition In_dec : forall v (s : set index), {In v s} + {~ In v s}.
adamc@221 425 Local Open Scope specif_scope.
adamc@144 426
adamc@221 427 intro; refine (fix F (s : set index) : {In v s} + {~ In v s} :=
adamc@221 428 match s with
adamc@144 429 | nil => No
adamc@144 430 | v' :: s' => index_eq v' v || F s'
adamc@144 431 end); crush.
adamc@144 432 Defined.
adamc@144 433
adamc@147 434 (** We define what it means for all members of an index set to represent true propositions, and we prove some lemmas about this notion. *)
adamc@147 435
adamc@144 436 Fixpoint allTrue (s : set index) : Prop :=
adamc@144 437 match s with
adamc@144 438 | nil => True
adamc@144 439 | v :: s' => holds v /\ allTrue s'
adamc@144 440 end.
adamc@144 441
adamc@144 442 Theorem allTrue_add : forall v s,
adamc@144 443 allTrue s
adamc@144 444 -> holds v
adamc@144 445 -> allTrue (add s v).
adamc@144 446 induction s; crush;
adamc@144 447 match goal with
adamc@144 448 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@144 449 end; crush.
adamc@144 450 Qed.
adamc@144 451
adamc@144 452 Theorem allTrue_In : forall v s,
adamc@144 453 allTrue s
adamc@144 454 -> set_In v s
adamc@144 455 -> varmap_find False v atomics.
adamc@144 456 induction s; crush.
adamc@144 457 Qed.
adamc@144 458
adamc@144 459 Hint Resolve allTrue_add allTrue_In.
adamc@144 460
adamc@221 461 Local Open Scope partial_scope.
adamc@144 462
adamc@147 463 (** Now we can write a function [forward] which implements deconstruction of hypotheses. It has a dependent type, in the style of Chapter 6, guaranteeing correctness. The arguments to [forward] are a goal formula [f], a set [known] of atomic formulas that we may assume are true, a hypothesis formula [hyp], and a success continuation [cont] that we call when we have extended [known] to hold new truths implied by [hyp]. *)
adamc@147 464
adam@297 465 Definition forward : forall (f : formula) (known : set index) (hyp : formula)
adam@297 466 (cont : forall known', [allTrue known' -> formulaDenote atomics f]),
adam@297 467 [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f].
adamc@144 468 refine (fix F (f : formula) (known : set index) (hyp : formula)
adamc@221 469 (cont : forall known', [allTrue known' -> formulaDenote atomics f])
adamc@144 470 : [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f] :=
adamc@221 471 match hyp with
adamc@144 472 | Atomic v => Reduce (cont (add known v))
adamc@144 473 | Truth => Reduce (cont known)
adamc@144 474 | Falsehood => Yes
adamc@144 475 | And h1 h2 =>
adamc@144 476 Reduce (F (Imp h2 f) known h1 (fun known' =>
adamc@144 477 Reduce (F f known' h2 cont)))
adamc@144 478 | Or h1 h2 => F f known h1 cont && F f known h2 cont
adamc@144 479 | Imp _ _ => Reduce (cont known)
adamc@144 480 end); crush.
adamc@144 481 Defined.
adamc@144 482
adamc@147 483 (** A [backward] function implements analysis of the final goal. It calls [forward] to handle implications. *)
adamc@147 484
adam@297 485 Definition backward : forall (known : set index) (f : formula),
adam@297 486 [allTrue known -> formulaDenote atomics f].
adamc@221 487 refine (fix F (known : set index) (f : formula)
adamc@221 488 : [allTrue known -> formulaDenote atomics f] :=
adamc@221 489 match f with
adamc@144 490 | Atomic v => Reduce (In_dec v known)
adamc@144 491 | Truth => Yes
adamc@144 492 | Falsehood => No
adamc@144 493 | And f1 f2 => F known f1 && F known f2
adamc@144 494 | Or f1 f2 => F known f1 || F known f2
adamc@144 495 | Imp f1 f2 => forward f2 known f1 (fun known' => F known' f2)
adamc@144 496 end); crush; eauto.
adamc@144 497 Defined.
adamc@144 498
adamc@147 499 (** A simple wrapper around [backward] gives us the usual type of a partial decision procedure. *)
adamc@147 500
adam@297 501 Definition my_tauto : forall f : formula, [formulaDenote atomics f].
adamc@144 502 intro; refine (Reduce (backward nil f)); crush.
adamc@144 503 Defined.
adamc@144 504 End my_tauto.
adamc@144 505
adamc@147 506 (** Our final tactic implementation is now fairly straightforward. First, we [intro] all quantifiers that do not bind [Prop]s. Then we call the [quote] tactic, which implements the reflection for us. Finally, we are able to construct an exact proof via [partialOut] and the [my_tauto] Gallina function. *)
adamc@147 507
adamc@144 508 Ltac my_tauto :=
adamc@144 509 repeat match goal with
adamc@144 510 | [ |- forall x : ?P, _ ] =>
adamc@144 511 match type of P with
adamc@144 512 | Prop => fail 1
adamc@144 513 | _ => intro
adamc@144 514 end
adamc@144 515 end;
adamc@144 516 quote formulaDenote;
adamc@144 517 match goal with
adamc@144 518 | [ |- formulaDenote ?m ?f ] => exact (partialOut (my_tauto m f))
adamc@144 519 end.
adamc@148 520 (* end thide *)
adamc@144 521
adamc@147 522 (** A few examples demonstrate how the tactic works. *)
adamc@147 523
adamc@144 524 Theorem mt1 : True.
adamc@144 525 my_tauto.
adamc@144 526 Qed.
adamc@144 527
adamc@144 528 Print mt1.
adamc@221 529 (** %\vspace{-.15in}% [[
adamc@147 530 mt1 = partialOut (my_tauto (Empty_vm Prop) Truth)
adamc@147 531 : True
adamc@221 532
adamc@147 533 ]]
adamc@147 534
adamc@147 535 We see [my_tauto] applied with an empty [varmap], since every subformula is handled by [formulaDenote]. *)
adamc@144 536
adamc@144 537 Theorem mt2 : forall x y : nat, x = y --> x = y.
adamc@144 538 my_tauto.
adamc@144 539 Qed.
adamc@144 540
adamc@144 541 Print mt2.
adamc@221 542 (** %\vspace{-.15in}% [[
adamc@147 543 mt2 =
adamc@147 544 fun x y : nat =>
adamc@147 545 partialOut
adamc@147 546 (my_tauto (Node_vm (x = y) (Empty_vm Prop) (Empty_vm Prop))
adamc@147 547 (Imp (Atomic End_idx) (Atomic End_idx)))
adamc@147 548 : forall x y : nat, x = y --> x = y
adamc@221 549
adamc@147 550 ]]
adamc@147 551
adamc@147 552 Crucially, both instances of [x = y] are represented with the same index, [End_idx]. The value of this index only needs to appear once in the [varmap], whose form reveals that [varmap]s are represented as binary trees, where [index] values denote paths from tree roots to leaves. *)
adamc@144 553
adamc@144 554 Theorem mt3 : forall x y z,
adamc@144 555 (x < y /\ y > z) \/ (y > z /\ x < S y)
adamc@144 556 --> y > z /\ (x < y \/ x < S y).
adamc@144 557 my_tauto.
adamc@144 558 Qed.
adamc@144 559
adamc@144 560 Print mt3.
adamc@221 561 (** %\vspace{-.15in}% [[
adamc@147 562 fun x y z : nat =>
adamc@147 563 partialOut
adamc@147 564 (my_tauto
adamc@147 565 (Node_vm (x < S y) (Node_vm (x < y) (Empty_vm Prop) (Empty_vm Prop))
adamc@147 566 (Node_vm (y > z) (Empty_vm Prop) (Empty_vm Prop)))
adamc@147 567 (Imp
adamc@147 568 (Or (And (Atomic (Left_idx End_idx)) (Atomic (Right_idx End_idx)))
adamc@147 569 (And (Atomic (Right_idx End_idx)) (Atomic End_idx)))
adamc@147 570 (And (Atomic (Right_idx End_idx))
adamc@147 571 (Or (Atomic (Left_idx End_idx)) (Atomic End_idx)))))
adamc@147 572 : forall x y z : nat,
adamc@147 573 x < y /\ y > z \/ y > z /\ x < S y --> y > z /\ (x < y \/ x < S y)
adamc@221 574
adamc@147 575 ]]
adamc@147 576
adamc@147 577 Our goal contained three distinct atomic formulas, and we see that a three-element [varmap] is generated.
adamc@147 578
adamc@147 579 It can be interesting to observe differences between the level of repetition in proof terms generated by [my_tauto] and [tauto] for especially trivial theorems. *)
adamc@144 580
adamc@144 581 Theorem mt4 : True /\ True /\ True /\ True /\ True /\ True /\ False --> False.
adamc@144 582 my_tauto.
adamc@144 583 Qed.
adamc@144 584
adamc@144 585 Print mt4.
adamc@221 586 (** %\vspace{-.15in}% [[
adamc@147 587 mt4 =
adamc@147 588 partialOut
adamc@147 589 (my_tauto (Empty_vm Prop)
adamc@147 590 (Imp
adamc@147 591 (And Truth
adamc@147 592 (And Truth
adamc@147 593 (And Truth (And Truth (And Truth (And Truth Falsehood))))))
adamc@147 594 Falsehood))
adamc@147 595 : True /\ True /\ True /\ True /\ True /\ True /\ False --> False
adam@302 596 ]]
adam@302 597 *)
adamc@144 598
adamc@144 599 Theorem mt4' : True /\ True /\ True /\ True /\ True /\ True /\ False -> False.
adamc@144 600 tauto.
adamc@144 601 Qed.
adamc@144 602
adamc@144 603 Print mt4'.
adamc@221 604 (** %\vspace{-.15in}% [[
adamc@147 605 mt4' =
adamc@147 606 fun H : True /\ True /\ True /\ True /\ True /\ True /\ False =>
adamc@147 607 and_ind
adamc@147 608 (fun (_ : True) (H1 : True /\ True /\ True /\ True /\ True /\ False) =>
adamc@147 609 and_ind
adamc@147 610 (fun (_ : True) (H3 : True /\ True /\ True /\ True /\ False) =>
adamc@147 611 and_ind
adamc@147 612 (fun (_ : True) (H5 : True /\ True /\ True /\ False) =>
adamc@147 613 and_ind
adamc@147 614 (fun (_ : True) (H7 : True /\ True /\ False) =>
adamc@147 615 and_ind
adamc@147 616 (fun (_ : True) (H9 : True /\ False) =>
adamc@147 617 and_ind (fun (_ : True) (H11 : False) => False_ind False H11)
adamc@147 618 H9) H7) H5) H3) H1) H
adamc@147 619 : True /\ True /\ True /\ True /\ True /\ True /\ False -> False
adam@302 620 ]]
adam@302 621 *)
adamc@147 622
adamc@149 623
adamc@149 624 (** * Exercises *)
adamc@149 625
adamc@221 626 (** remove printing * *)
adamc@221 627
adamc@149 628 (** %\begin{enumerate}%#<ol>#
adamc@149 629
adamc@149 630 %\item%#<li># Implement a reflective procedure for normalizing systems of linear equations over rational numbers. In particular, the tactic should identify all hypotheses that are linear equations over rationals where the equation righthand sides are constants. It should normalize each hypothesis to have a lefthand side that is a sum of products of constants and variables, with no variable appearing multiple times. Then, your tactic should add together all of these equations to form a single new equation, possibly clearing the original equations. Some coefficients may cancel in the addition, reducing the number of variables that appear.
adamc@149 631
adamc@221 632 To work with rational numbers, import module [QArith] and use [Local Open Scope Q_scope]. All of the usual arithmetic operator notations will then work with rationals, and there are shorthands for constants 0 and 1. Other rationals must be written as [num # den] for numerator [num] and denominator [den]. Use the infix operator [==] in place of [=], to deal with different ways of expressing the same number as a fraction. For instance, a theorem and proof like this one should work with your tactic:
adamc@149 633
adamc@149 634 [[
adamc@149 635 Theorem t2 : forall x y z, (2 # 1) * (x - (3 # 2) * y) == 15 # 1
adamc@149 636 -> z + (8 # 1) * x == 20 # 1
adamc@149 637 -> (-6 # 2) * y + (10 # 1) * x + z == 35 # 1.
adamc@149 638 intros; reflectContext; assumption.
adamc@149 639 Qed.
adamc@221 640
adamc@205 641 ]]
adamc@205 642
adamc@149 643 Your solution can work in any way that involves reflecting syntax and doing most calculation with a Gallina function. These hints outline a particular possible solution. Throughout, the [ring] tactic will be helpful for proving many simple facts about rationals, and tactics like [rewrite] are correctly overloaded to work with rational equality [==].
adamc@149 644
adamc@149 645 %\begin{enumerate}%#<ol>#
adamc@221 646 %\item%#<li># Define an inductive type [exp] of expressions over rationals (which inhabit the Coq type [Q]). Include variables (represented as natural numbers), constants, addition, subtraction, and multiplication.#</li>#
adamc@149 647 %\item%#<li># Define a function [lookup] for reading an element out of a list of rationals, by its position in the list.#</li>#
adamc@149 648 %\item%#<li># Define a function [expDenote] that translates [exp]s, along with lists of rationals representing variable values, to [Q].#</li>#
adamc@149 649 %\item%#<li># Define a recursive function [eqsDenote] over [list (exp * Q)], characterizing when all of the equations are true.#</li>#
adamc@149 650 %\item%#<li># Fix a representation [lhs] of flattened expressions. Where [len] is the number of variables, represent a flattened equation as [ilist Q len]. Each position of the list gives the coefficient of the corresponding variable.#</li>#
adamc@151 651 %\item%#<li># Write a recursive function [linearize] that takes a constant [k] and an expression [e] and optionally returns an [lhs] equivalent to [k * e]. This function returns [None] when it discovers that the input expression is not linear. The parameter [len] of [lhs] should be a parameter of [linearize], too. The functions [singleton], [everywhere], and [map2] from [DepList] will probably be helpful. It is also helpful to know that [Qplus] is the identifier for rational addition.#</li>#
adamc@149 652 %\item%#<li># Write a recursive function [linearizeEqs : list (exp * Q) -> option (lhs * Q)]. This function linearizes all of the equations in the list in turn, building up the sum of the equations. It returns [None] if the linearization of any constituent equation fails.#</li>#
adamc@149 653 %\item%#<li># Define a denotation function for [lhs].#</li>#
adamc@149 654 %\item%#<li># Prove that, when [exp] linearization succeeds on constant [k] and expression [e], the linearized version has the same meaning as [k * e].#</li>#
adamc@149 655 %\item%#<li># Prove that, when [linearizeEqs] succeeds on an equation list [eqs], then the final summed-up equation is true whenever the original equation list is true.#</li>#
adamc@149 656 %\item%#<li># Write a tactic [findVarsHyps] to search through all equalities on rationals in the context, recursing through addition, subtraction, and multiplication to find the list of expressions that should be treated as variables. This list should be suitable as an argument to [expDenote] and [eqsDenote], associating a [Q] value to each natural number that stands for a variable.#</li>#
adamc@149 657 %\item%#<li># Write a tactic [reflect] to reflect a [Q] expression into [exp], with respect to a given list of variable values.#</li>#
adamc@149 658 %\item%#<li># Write a tactic [reflectEqs] to reflect a formula that begins with a sequence of implications from linear equalities whose lefthand sides are expressed with [expDenote]. This tactic should build a [list (exp * Q)] representing the equations. Remember to give an explicit type annotation when returning a nil list, as in [constr:(@nil (exp * Q))].#</li>#
adamc@149 659 %\item%#<li># Now this final tactic should do the job:
adamc@221 660
adamc@149 661 [[
adamc@149 662 Ltac reflectContext :=
adamc@149 663 let ls := findVarsHyps in
adamc@149 664 repeat match goal with
adamc@149 665 | [ H : ?e == ?num # ?den |- _ ] =>
adamc@149 666 let r := reflect ls e in
adamc@149 667 change (expDenote ls r == num # den) in H;
adamc@149 668 generalize H
adamc@149 669 end;
adamc@149 670 match goal with
adamc@149 671 | [ |- ?g ] => let re := reflectEqs g in
adamc@149 672 intros;
adamc@149 673 let H := fresh "H" in
adamc@149 674 assert (H : eqsDenote ls re); [ simpl in *; tauto
adamc@149 675 | repeat match goal with
adamc@149 676 | [ H : expDenote _ _ == _ |- _ ] => clear H
adamc@149 677 end;
adamc@149 678 generalize (linearizeEqsCorrect ls re H); clear H; simpl;
adamc@149 679 match goal with
adamc@149 680 | [ |- ?X == ?Y -> _ ] =>
adamc@149 681 ring_simplify X Y; intro
adamc@149 682 end ]
adamc@149 683 end.
adamc@149 684
adamc@205 685 ]]
adamc@205 686
adamc@149 687 #</ol>#%\end{enumerate}%
adamc@149 688 #</li>#
adamc@149 689
adamc@149 690 #</ol>#%\end{enumerate}% *)