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author Adam Chlipala <adamc@hcoop.net>
date Mon, 28 Dec 2009 12:35:44 -0500
parents 54e8ecb5ec88
children 4662b6f099b0
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adamc@221 1 (* Copyright (c) 2008-2009, 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
adamc@221 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 linear in the input value. 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 *)
adamc@142 71 Definition check_even (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
adamc@142 93 (** It may seem strange to define a function like this. However, it turns out to be very useful in writing a reflective verison 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
adamc@142 101 (** 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
adamc@142 114 We can see a constant wrapper around the object of the proof. For any even number, this form of proof will suffice. What happens if we try the tactic with an odd number? *)
adamc@142 115
adamc@142 116 Theorem even_255 : isEven 255.
adamc@142 117 (** [[
adamc@142 118 prove_even_reflective.
adamc@142 119
adamc@142 120 User error: No matching clauses for match goal
adamc@221 121
adamc@142 122 ]]
adamc@142 123
adamc@142 124 Thankfully, the tactic fails. To see more precisely what goes wrong, we can run manually the body of the [match].
adamc@142 125
adamc@142 126 [[
adamc@142 127 exact (partialOut (check_even 255)).
adamc@142 128
adamc@142 129 Error: The term "partialOut (check_even 255)" has type
adamc@142 130 "match check_even 255 with
adamc@142 131 | Yes => isEven 255
adamc@142 132 | No => True
adamc@142 133 end" while it is expected to have type "isEven 255"
adamc@221 134
adamc@142 135 ]]
adamc@142 136
adamc@142 137 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@221 138
adamc@142 139 Abort.
adamc@143 140
adamc@143 141
adamc@143 142 (** * Reflecting the Syntax of a Trivial Tautology Language *)
adamc@143 143
adamc@143 144 (** We might also like to have reflective proofs of trivial tautologies like this one: *)
adamc@143 145
adamc@143 146 Theorem true_galore : (True /\ True) -> (True \/ (True /\ (True -> True))).
adamc@143 147 tauto.
adamc@143 148 Qed.
adamc@143 149
adamc@143 150 Print true_galore.
adamc@221 151 (** %\vspace{-.15in}% [[
adamc@143 152 true_galore =
adamc@143 153 fun H : True /\ True =>
adamc@143 154 and_ind (fun _ _ : True => or_introl (True /\ (True -> True)) I) H
adamc@143 155 : True /\ True -> True \/ True /\ (True -> True)
adamc@221 156
adamc@143 157 ]]
adamc@143 158
adamc@143 159 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@143 160
adamc@143 161 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@143 162
adamc@148 163 (* begin thide *)
adamc@143 164 Inductive taut : Set :=
adamc@143 165 | TautTrue : taut
adamc@143 166 | TautAnd : taut -> taut -> taut
adamc@143 167 | TautOr : taut -> taut -> taut
adamc@143 168 | TautImp : taut -> taut -> taut.
adamc@143 169
adamc@143 170 (** We write a recursive function to "unreflect" this syntax back to [Prop]. *)
adamc@143 171
adamc@143 172 Fixpoint tautDenote (t : taut) : Prop :=
adamc@143 173 match t with
adamc@143 174 | TautTrue => True
adamc@143 175 | TautAnd t1 t2 => tautDenote t1 /\ tautDenote t2
adamc@143 176 | TautOr t1 t2 => tautDenote t1 \/ tautDenote t2
adamc@143 177 | TautImp t1 t2 => tautDenote t1 -> tautDenote t2
adamc@143 178 end.
adamc@143 179
adamc@143 180 (** It is easy to prove that every formula in the range of [tautDenote] is true. *)
adamc@143 181
adamc@143 182 Theorem tautTrue : forall t, tautDenote t.
adamc@143 183 induction t; crush.
adamc@143 184 Qed.
adamc@143 185
adamc@143 186 (** To use [tautTrue] to prove particular formulas, we need to implement the syntax reflection process. A recursive Ltac function does the job. *)
adamc@143 187
adamc@143 188 Ltac tautReflect P :=
adamc@143 189 match P with
adamc@143 190 | True => TautTrue
adamc@143 191 | ?P1 /\ ?P2 =>
adamc@143 192 let t1 := tautReflect P1 in
adamc@143 193 let t2 := tautReflect P2 in
adamc@143 194 constr:(TautAnd t1 t2)
adamc@143 195 | ?P1 \/ ?P2 =>
adamc@143 196 let t1 := tautReflect P1 in
adamc@143 197 let t2 := tautReflect P2 in
adamc@143 198 constr:(TautOr t1 t2)
adamc@143 199 | ?P1 -> ?P2 =>
adamc@143 200 let t1 := tautReflect P1 in
adamc@143 201 let t2 := tautReflect P2 in
adamc@143 202 constr:(TautImp t1 t2)
adamc@143 203 end.
adamc@143 204
adamc@143 205 (** 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@143 206
adamc@143 207 Ltac obvious :=
adamc@143 208 match goal with
adamc@143 209 | [ |- ?P ] =>
adamc@143 210 let t := tautReflect P in
adamc@143 211 exact (tautTrue t)
adamc@143 212 end.
adamc@143 213
adamc@143 214 (** We can verify that [obvious] solves our original example, with a proof term that does not mention details of the proof. *)
adamc@148 215 (* end thide *)
adamc@143 216
adamc@143 217 Theorem true_galore' : (True /\ True) -> (True \/ (True /\ (True -> True))).
adamc@143 218 obvious.
adamc@143 219 Qed.
adamc@143 220
adamc@143 221 Print true_galore'.
adamc@143 222
adamc@221 223 (** %\vspace{-.15in}% [[
adamc@143 224 true_galore' =
adamc@143 225 tautTrue
adamc@143 226 (TautImp (TautAnd TautTrue TautTrue)
adamc@143 227 (TautOr TautTrue (TautAnd TautTrue (TautImp TautTrue TautTrue))))
adamc@143 228 : True /\ True -> True \/ True /\ (True -> True)
adamc@221 229
adamc@143 230 ]]
adamc@143 231
adamc@143 232 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@144 233
adamc@144 234
adamc@145 235 (** * A Monoid Expression Simplifier *)
adamc@145 236
adamc@146 237 (** 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@146 238
adamc@145 239 Section monoid.
adamc@145 240 Variable A : Set.
adamc@145 241 Variable e : A.
adamc@145 242 Variable f : A -> A -> A.
adamc@145 243
adamc@145 244 Infix "+" := f.
adamc@145 245
adamc@145 246 Hypothesis assoc : forall a b c, (a + b) + c = a + (b + c).
adamc@145 247 Hypothesis identl : forall a, e + a = a.
adamc@145 248 Hypothesis identr : forall a, a + e = a.
adamc@145 249
adamc@146 250 (** 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@146 251
adamc@146 252 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@146 253
adamc@148 254 (* begin thide *)
adamc@145 255 Inductive mexp : Set :=
adamc@145 256 | Ident : mexp
adamc@145 257 | Var : A -> mexp
adamc@145 258 | Op : mexp -> mexp -> mexp.
adamc@145 259
adamc@146 260 (** Next, we write an "un-reflect" function. *)
adamc@146 261
adamc@145 262 Fixpoint mdenote (me : mexp) : A :=
adamc@145 263 match me with
adamc@145 264 | Ident => e
adamc@145 265 | Var v => v
adamc@145 266 | Op me1 me2 => mdenote me1 + mdenote me2
adamc@145 267 end.
adamc@145 268
adamc@146 269 (** 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@146 270
adamc@145 271 Fixpoint mldenote (ls : list A) : A :=
adamc@145 272 match ls with
adamc@145 273 | nil => e
adamc@145 274 | x :: ls' => x + mldenote ls'
adamc@145 275 end.
adamc@145 276
adamc@146 277 (** The flattening function itself is easy to implement. *)
adamc@146 278
adamc@145 279 Fixpoint flatten (me : mexp) : list A :=
adamc@145 280 match me with
adamc@145 281 | Ident => nil
adamc@145 282 | Var x => x :: nil
adamc@145 283 | Op me1 me2 => flatten me1 ++ flatten me2
adamc@145 284 end.
adamc@145 285
adamc@146 286 (** [flatten] has a straightforward correctness proof in terms of our [denote] functions. *)
adamc@146 287
adamc@146 288 Lemma flatten_correct' : forall ml2 ml1,
adamc@146 289 mldenote ml1 + mldenote ml2 = mldenote (ml1 ++ ml2).
adamc@145 290 induction ml1; crush.
adamc@145 291 Qed.
adamc@145 292
adamc@145 293 Theorem flatten_correct : forall me, mdenote me = mldenote (flatten me).
adamc@145 294 Hint Resolve flatten_correct'.
adamc@145 295
adamc@145 296 induction me; crush.
adamc@145 297 Qed.
adamc@145 298
adamc@146 299 (** Now it is easy to prove a theorem that will be the main tool behind our simplification tactic. *)
adamc@146 300
adamc@146 301 Theorem monoid_reflect : forall me1 me2,
adamc@146 302 mldenote (flatten me1) = mldenote (flatten me2)
adamc@146 303 -> mdenote me1 = mdenote me2.
adamc@145 304 intros; repeat rewrite flatten_correct; assumption.
adamc@145 305 Qed.
adamc@145 306
adamc@146 307 (** We implement reflection into the [mexp] type. *)
adamc@146 308
adamc@146 309 Ltac reflect me :=
adamc@146 310 match me with
adamc@145 311 | e => Ident
adamc@146 312 | ?me1 + ?me2 =>
adamc@146 313 let r1 := reflect me1 in
adamc@146 314 let r2 := reflect me2 in
adamc@145 315 constr:(Op r1 r2)
adamc@146 316 | _ => constr:(Var me)
adamc@145 317 end.
adamc@145 318
adamc@146 319 (** 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@146 320
adamc@145 321 Ltac monoid :=
adamc@145 322 match goal with
adamc@146 323 | [ |- ?me1 = ?me2 ] =>
adamc@146 324 let r1 := reflect me1 in
adamc@146 325 let r2 := reflect me2 in
adamc@145 326 change (mdenote r1 = mdenote r2);
adamc@145 327 apply monoid_reflect; simpl mldenote
adamc@145 328 end.
adamc@145 329
adamc@146 330 (** We can make short work of theorems like this one: *)
adamc@146 331
adamc@148 332 (* end thide *)
adamc@148 333
adamc@145 334 Theorem t1 : forall a b c d, a + b + c + d = a + (b + c) + d.
adamc@146 335 intros; monoid.
adamc@146 336 (** [[
adamc@146 337 ============================
adamc@146 338 a + (b + (c + (d + e))) = a + (b + (c + (d + e)))
adamc@221 339
adamc@146 340 ]]
adamc@146 341
adamc@146 342 [monoid] has canonicalized both sides of the equality, such that we can finish the proof by reflexivity. *)
adamc@146 343
adamc@145 344 reflexivity.
adamc@145 345 Qed.
adamc@146 346
adamc@146 347 (** It is interesting to look at the form of the proof. *)
adamc@146 348
adamc@146 349 Print t1.
adamc@221 350 (** %\vspace{-.15in}% [[
adamc@146 351 t1 =
adamc@146 352 fun a b c d : A =>
adamc@146 353 monoid_reflect (Op (Op (Op (Var a) (Var b)) (Var c)) (Var d))
adamc@146 354 (Op (Op (Var a) (Op (Var b) (Var c))) (Var d))
adamc@146 355 (refl_equal (a + (b + (c + (d + e)))))
adamc@146 356 : forall a b c d : A, a + b + c + d = a + (b + c) + d
adamc@221 357
adamc@146 358 ]]
adamc@146 359
adamc@146 360 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@221 361
adamc@145 362 End monoid.
adamc@145 363
adamc@146 364 (** Extensions of this basic approach are used in the implementations of the [ring] and [field] tactics that come packaged with Coq. *)
adamc@146 365
adamc@145 366
adamc@144 367 (** * A Smarter Tautology Solver *)
adamc@144 368
adamc@221 369 (** 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@147 370
adamc@147 371 To arrive at a nice implementation satisfying these criteria, we introduce the [quote] tactic and its associated library. *)
adamc@147 372
adamc@144 373 Require Import Quote.
adamc@144 374
adamc@148 375 (* begin thide *)
adamc@144 376 Inductive formula : Set :=
adamc@144 377 | Atomic : index -> formula
adamc@144 378 | Truth : formula
adamc@144 379 | Falsehood : formula
adamc@144 380 | And : formula -> formula -> formula
adamc@144 381 | Or : formula -> formula -> formula
adamc@144 382 | Imp : formula -> formula -> formula.
adamc@144 383
adamc@147 384 (** 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@147 385
adamc@147 386 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@144 387
adamc@144 388 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@144 389 Infix "-->" := imp (no associativity, at level 95).
adamc@144 390
adamc@147 391 (** Now we can define our denotation function. *)
adamc@147 392
adamc@147 393 Definition asgn := varmap Prop.
adamc@147 394
adamc@144 395 Fixpoint formulaDenote (atomics : asgn) (f : formula) : Prop :=
adamc@144 396 match f with
adamc@144 397 | Atomic v => varmap_find False v atomics
adamc@144 398 | Truth => True
adamc@144 399 | Falsehood => False
adamc@144 400 | And f1 f2 => formulaDenote atomics f1 /\ formulaDenote atomics f2
adamc@144 401 | Or f1 f2 => formulaDenote atomics f1 \/ formulaDenote atomics f2
adamc@144 402 | Imp f1 f2 => formulaDenote atomics f1 --> formulaDenote atomics f2
adamc@144 403 end.
adamc@144 404
adamc@147 405 (** 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@147 406
adamc@144 407 Section my_tauto.
adamc@144 408 Variable atomics : asgn.
adamc@144 409
adamc@144 410 Definition holds (v : index) := varmap_find False v atomics.
adamc@144 411
adamc@147 412 (** 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@147 413
adamc@144 414 Require Import ListSet.
adamc@144 415
adamc@144 416 Definition index_eq : forall x y : index, {x = y} + {x <> y}.
adamc@144 417 decide equality.
adamc@144 418 Defined.
adamc@144 419
adamc@144 420 Definition add (s : set index) (v : index) := set_add index_eq v s.
adamc@147 421
adamc@221 422 Definition In_dec : forall v (s : set index), {In v s} + {~ In v s}.
adamc@221 423 Local Open Scope specif_scope.
adamc@144 424
adamc@221 425 intro; refine (fix F (s : set index) : {In v s} + {~ In v s} :=
adamc@221 426 match s with
adamc@144 427 | nil => No
adamc@144 428 | v' :: s' => index_eq v' v || F s'
adamc@144 429 end); crush.
adamc@144 430 Defined.
adamc@144 431
adamc@147 432 (** 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@147 433
adamc@144 434 Fixpoint allTrue (s : set index) : Prop :=
adamc@144 435 match s with
adamc@144 436 | nil => True
adamc@144 437 | v :: s' => holds v /\ allTrue s'
adamc@144 438 end.
adamc@144 439
adamc@144 440 Theorem allTrue_add : forall v s,
adamc@144 441 allTrue s
adamc@144 442 -> holds v
adamc@144 443 -> allTrue (add s v).
adamc@144 444 induction s; crush;
adamc@144 445 match goal with
adamc@144 446 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@144 447 end; crush.
adamc@144 448 Qed.
adamc@144 449
adamc@144 450 Theorem allTrue_In : forall v s,
adamc@144 451 allTrue s
adamc@144 452 -> set_In v s
adamc@144 453 -> varmap_find False v atomics.
adamc@144 454 induction s; crush.
adamc@144 455 Qed.
adamc@144 456
adamc@144 457 Hint Resolve allTrue_add allTrue_In.
adamc@144 458
adamc@221 459 Local Open Scope partial_scope.
adamc@144 460
adamc@147 461 (** 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@147 462
adamc@144 463 Definition forward (f : formula) (known : set index) (hyp : formula)
adamc@144 464 (cont : forall known', [allTrue known' -> formulaDenote atomics f])
adamc@144 465 : [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f].
adamc@144 466 refine (fix F (f : formula) (known : set index) (hyp : formula)
adamc@221 467 (cont : forall known', [allTrue known' -> formulaDenote atomics f])
adamc@144 468 : [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f] :=
adamc@221 469 match hyp with
adamc@144 470 | Atomic v => Reduce (cont (add known v))
adamc@144 471 | Truth => Reduce (cont known)
adamc@144 472 | Falsehood => Yes
adamc@144 473 | And h1 h2 =>
adamc@144 474 Reduce (F (Imp h2 f) known h1 (fun known' =>
adamc@144 475 Reduce (F f known' h2 cont)))
adamc@144 476 | Or h1 h2 => F f known h1 cont && F f known h2 cont
adamc@144 477 | Imp _ _ => Reduce (cont known)
adamc@144 478 end); crush.
adamc@144 479 Defined.
adamc@144 480
adamc@147 481 (** A [backward] function implements analysis of the final goal. It calls [forward] to handle implications. *)
adamc@147 482
adamc@221 483 Definition backward (known : set index) (f : formula)
adamc@221 484 : [allTrue known -> formulaDenote atomics f].
adamc@221 485 refine (fix F (known : set index) (f : formula)
adamc@221 486 : [allTrue known -> formulaDenote atomics f] :=
adamc@221 487 match f with
adamc@144 488 | Atomic v => Reduce (In_dec v known)
adamc@144 489 | Truth => Yes
adamc@144 490 | Falsehood => No
adamc@144 491 | And f1 f2 => F known f1 && F known f2
adamc@144 492 | Or f1 f2 => F known f1 || F known f2
adamc@144 493 | Imp f1 f2 => forward f2 known f1 (fun known' => F known' f2)
adamc@144 494 end); crush; eauto.
adamc@144 495 Defined.
adamc@144 496
adamc@147 497 (** A simple wrapper around [backward] gives us the usual type of a partial decision procedure. *)
adamc@147 498
adamc@144 499 Definition my_tauto (f : formula) : [formulaDenote atomics f].
adamc@144 500 intro; refine (Reduce (backward nil f)); crush.
adamc@144 501 Defined.
adamc@144 502 End my_tauto.
adamc@144 503
adamc@147 504 (** 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@147 505
adamc@144 506 Ltac my_tauto :=
adamc@144 507 repeat match goal with
adamc@144 508 | [ |- forall x : ?P, _ ] =>
adamc@144 509 match type of P with
adamc@144 510 | Prop => fail 1
adamc@144 511 | _ => intro
adamc@144 512 end
adamc@144 513 end;
adamc@144 514 quote formulaDenote;
adamc@144 515 match goal with
adamc@144 516 | [ |- formulaDenote ?m ?f ] => exact (partialOut (my_tauto m f))
adamc@144 517 end.
adamc@148 518 (* end thide *)
adamc@144 519
adamc@147 520 (** A few examples demonstrate how the tactic works. *)
adamc@147 521
adamc@144 522 Theorem mt1 : True.
adamc@144 523 my_tauto.
adamc@144 524 Qed.
adamc@144 525
adamc@144 526 Print mt1.
adamc@221 527 (** %\vspace{-.15in}% [[
adamc@147 528 mt1 = partialOut (my_tauto (Empty_vm Prop) Truth)
adamc@147 529 : True
adamc@221 530
adamc@147 531 ]]
adamc@147 532
adamc@147 533 We see [my_tauto] applied with an empty [varmap], since every subformula is handled by [formulaDenote]. *)
adamc@144 534
adamc@144 535 Theorem mt2 : forall x y : nat, x = y --> x = y.
adamc@144 536 my_tauto.
adamc@144 537 Qed.
adamc@144 538
adamc@144 539 Print mt2.
adamc@221 540 (** %\vspace{-.15in}% [[
adamc@147 541 mt2 =
adamc@147 542 fun x y : nat =>
adamc@147 543 partialOut
adamc@147 544 (my_tauto (Node_vm (x = y) (Empty_vm Prop) (Empty_vm Prop))
adamc@147 545 (Imp (Atomic End_idx) (Atomic End_idx)))
adamc@147 546 : forall x y : nat, x = y --> x = y
adamc@221 547
adamc@147 548 ]]
adamc@147 549
adamc@147 550 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@144 551
adamc@144 552 Theorem mt3 : forall x y z,
adamc@144 553 (x < y /\ y > z) \/ (y > z /\ x < S y)
adamc@144 554 --> y > z /\ (x < y \/ x < S y).
adamc@144 555 my_tauto.
adamc@144 556 Qed.
adamc@144 557
adamc@144 558 Print mt3.
adamc@221 559 (** %\vspace{-.15in}% [[
adamc@147 560 fun x y z : nat =>
adamc@147 561 partialOut
adamc@147 562 (my_tauto
adamc@147 563 (Node_vm (x < S y) (Node_vm (x < y) (Empty_vm Prop) (Empty_vm Prop))
adamc@147 564 (Node_vm (y > z) (Empty_vm Prop) (Empty_vm Prop)))
adamc@147 565 (Imp
adamc@147 566 (Or (And (Atomic (Left_idx End_idx)) (Atomic (Right_idx End_idx)))
adamc@147 567 (And (Atomic (Right_idx End_idx)) (Atomic End_idx)))
adamc@147 568 (And (Atomic (Right_idx End_idx))
adamc@147 569 (Or (Atomic (Left_idx End_idx)) (Atomic End_idx)))))
adamc@147 570 : forall x y z : nat,
adamc@147 571 x < y /\ y > z \/ y > z /\ x < S y --> y > z /\ (x < y \/ x < S y)
adamc@221 572
adamc@147 573 ]]
adamc@147 574
adamc@147 575 Our goal contained three distinct atomic formulas, and we see that a three-element [varmap] is generated.
adamc@147 576
adamc@147 577 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@144 578
adamc@144 579 Theorem mt4 : True /\ True /\ True /\ True /\ True /\ True /\ False --> False.
adamc@144 580 my_tauto.
adamc@144 581 Qed.
adamc@144 582
adamc@144 583 Print mt4.
adamc@221 584 (** %\vspace{-.15in}% [[
adamc@147 585 mt4 =
adamc@147 586 partialOut
adamc@147 587 (my_tauto (Empty_vm Prop)
adamc@147 588 (Imp
adamc@147 589 (And Truth
adamc@147 590 (And Truth
adamc@147 591 (And Truth (And Truth (And Truth (And Truth Falsehood))))))
adamc@147 592 Falsehood))
adamc@147 593 : True /\ True /\ True /\ True /\ True /\ True /\ False --> False
adamc@147 594 ]] *)
adamc@144 595
adamc@144 596 Theorem mt4' : True /\ True /\ True /\ True /\ True /\ True /\ False -> False.
adamc@144 597 tauto.
adamc@144 598 Qed.
adamc@144 599
adamc@144 600 Print mt4'.
adamc@221 601 (** %\vspace{-.15in}% [[
adamc@147 602 mt4' =
adamc@147 603 fun H : True /\ True /\ True /\ True /\ True /\ True /\ False =>
adamc@147 604 and_ind
adamc@147 605 (fun (_ : True) (H1 : True /\ True /\ True /\ True /\ True /\ False) =>
adamc@147 606 and_ind
adamc@147 607 (fun (_ : True) (H3 : True /\ True /\ True /\ True /\ False) =>
adamc@147 608 and_ind
adamc@147 609 (fun (_ : True) (H5 : True /\ True /\ True /\ False) =>
adamc@147 610 and_ind
adamc@147 611 (fun (_ : True) (H7 : True /\ True /\ False) =>
adamc@147 612 and_ind
adamc@147 613 (fun (_ : True) (H9 : True /\ False) =>
adamc@147 614 and_ind (fun (_ : True) (H11 : False) => False_ind False H11)
adamc@147 615 H9) H7) H5) H3) H1) H
adamc@147 616 : True /\ True /\ True /\ True /\ True /\ True /\ False -> False
adamc@147 617 ]] *)
adamc@147 618
adamc@149 619
adamc@149 620 (** * Exercises *)
adamc@149 621
adamc@221 622 (** remove printing * *)
adamc@221 623
adamc@149 624 (** %\begin{enumerate}%#<ol>#
adamc@149 625
adamc@149 626 %\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@149 627
adamc@221 628 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@149 629
adamc@149 630 [[
adamc@149 631 Theorem t2 : forall x y z, (2 # 1) * (x - (3 # 2) * y) == 15 # 1
adamc@149 632 -> z + (8 # 1) * x == 20 # 1
adamc@149 633 -> (-6 # 2) * y + (10 # 1) * x + z == 35 # 1.
adamc@149 634 intros; reflectContext; assumption.
adamc@149 635 Qed.
adamc@221 636
adamc@205 637 ]]
adamc@205 638
adamc@149 639 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@149 640
adamc@149 641 %\begin{enumerate}%#<ol>#
adamc@221 642 %\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@149 643 %\item%#<li># Define a function [lookup] for reading an element out of a list of rationals, by its position in the list.#</li>#
adamc@149 644 %\item%#<li># Define a function [expDenote] that translates [exp]s, along with lists of rationals representing variable values, to [Q].#</li>#
adamc@149 645 %\item%#<li># Define a recursive function [eqsDenote] over [list (exp * Q)], characterizing when all of the equations are true.#</li>#
adamc@149 646 %\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@151 647 %\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@149 648 %\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@149 649 %\item%#<li># Define a denotation function for [lhs].#</li>#
adamc@149 650 %\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@149 651 %\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@149 652 %\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@149 653 %\item%#<li># Write a tactic [reflect] to reflect a [Q] expression into [exp], with respect to a given list of variable values.#</li>#
adamc@149 654 %\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@149 655 %\item%#<li># Now this final tactic should do the job:
adamc@221 656
adamc@149 657 [[
adamc@149 658 Ltac reflectContext :=
adamc@149 659 let ls := findVarsHyps in
adamc@149 660 repeat match goal with
adamc@149 661 | [ H : ?e == ?num # ?den |- _ ] =>
adamc@149 662 let r := reflect ls e in
adamc@149 663 change (expDenote ls r == num # den) in H;
adamc@149 664 generalize H
adamc@149 665 end;
adamc@149 666 match goal with
adamc@149 667 | [ |- ?g ] => let re := reflectEqs g in
adamc@149 668 intros;
adamc@149 669 let H := fresh "H" in
adamc@149 670 assert (H : eqsDenote ls re); [ simpl in *; tauto
adamc@149 671 | repeat match goal with
adamc@149 672 | [ H : expDenote _ _ == _ |- _ ] => clear H
adamc@149 673 end;
adamc@149 674 generalize (linearizeEqsCorrect ls re H); clear H; simpl;
adamc@149 675 match goal with
adamc@149 676 | [ |- ?X == ?Y -> _ ] =>
adamc@149 677 ring_simplify X Y; intro
adamc@149 678 end ]
adamc@149 679 end.
adamc@149 680
adamc@205 681 ]]
adamc@205 682
adamc@149 683 #</ol>#%\end{enumerate}%
adamc@149 684 #</li>#
adamc@149 685
adamc@149 686 #</ol>#%\end{enumerate}% *)