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Start of Reflection
author Adam Chlipala <adamc@hcoop.net>
date Tue, 28 Oct 2008 11:29:14 -0400
parents
children f3e018e167f8
rev   line source
adamc@142 1 (* Copyright (c) 2008, 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@142 29 | Even_O : isEven O
adamc@142 30 | Even_SS : forall n, isEven n -> isEven (S (S n)).
adamc@142 31
adamc@142 32 Ltac prove_even := repeat constructor.
adamc@142 33
adamc@142 34 Theorem even_256 : isEven 256.
adamc@142 35 prove_even.
adamc@142 36 Qed.
adamc@142 37
adamc@142 38 Print even_256.
adamc@142 39 (** [[
adamc@142 40
adamc@142 41 even_256 =
adamc@142 42 Even_SS
adamc@142 43 (Even_SS
adamc@142 44 (Even_SS
adamc@142 45 (Even_SS
adamc@142 46 ]]
adamc@142 47
adamc@142 48 ...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 49
adamc@142 50 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 51
adamc@142 52 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 53
adamc@142 54 For this example, we begin by using a type from the [MoreSpecif] module to write a certified evenness checker. *)
adamc@142 55
adamc@142 56 Print partial.
adamc@142 57 (** [[
adamc@142 58
adamc@142 59 Inductive partial (P : Prop) : Set := Proved : P -> [P] | Uncertain : [P]
adamc@142 60 ]] *)
adamc@142 61
adamc@142 62 (** A [partial P] value is an optional proof of [P]. The notation [[P]] stands for [partial P]. *)
adamc@142 63
adamc@142 64 Open Local Scope partial_scope.
adamc@142 65
adamc@142 66 (** 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 67
adamc@142 68 Definition check_even (n : nat) : [isEven n].
adamc@142 69 Hint Constructors isEven.
adamc@142 70
adamc@142 71 refine (fix F (n : nat) : [isEven n] :=
adamc@142 72 match n return [isEven n] with
adamc@142 73 | 0 => Yes
adamc@142 74 | 1 => No
adamc@142 75 | S (S n') => Reduce (F n')
adamc@142 76 end); auto.
adamc@142 77 Defined.
adamc@142 78
adamc@142 79 (** 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 80
adamc@142 81 Definition partialOut (P : Prop) (x : [P]) :=
adamc@142 82 match x return (match x with
adamc@142 83 | Proved _ => P
adamc@142 84 | Uncertain => True
adamc@142 85 end) with
adamc@142 86 | Proved pf => pf
adamc@142 87 | Uncertain => I
adamc@142 88 end.
adamc@142 89
adamc@142 90 (** 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 91
adamc@142 92 Ltac prove_even_reflective :=
adamc@142 93 match goal with
adamc@142 94 | [ |- isEven ?N] => exact (partialOut (check_even N))
adamc@142 95 end.
adamc@142 96
adamc@142 97 (** 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 98
adamc@142 99 Theorem even_256' : isEven 256.
adamc@142 100 prove_even_reflective.
adamc@142 101 Qed.
adamc@142 102
adamc@142 103 Print even_256'.
adamc@142 104 (** [[
adamc@142 105
adamc@142 106 even_256' = partialOut (check_even 256)
adamc@142 107 : isEven 256
adamc@142 108 ]]
adamc@142 109
adamc@142 110 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@142 111
adamc@142 112 Theorem even_255 : isEven 255.
adamc@142 113 (** [[
adamc@142 114
adamc@142 115 prove_even_reflective.
adamc@142 116
adamc@142 117 [[
adamc@142 118
adamc@142 119 User error: No matching clauses for match goal
adamc@142 120 ]]
adamc@142 121
adamc@142 122 Thankfully, the tactic fails. To see more precisely what goes wrong, we can run manually the body of the [match].
adamc@142 123
adamc@142 124 [[
adamc@142 125
adamc@142 126 exact (partialOut (check_even 255)).
adamc@142 127
adamc@142 128 [[
adamc@142 129
adamc@142 130 Error: The term "partialOut (check_even 255)" has type
adamc@142 131 "match check_even 255 with
adamc@142 132 | Yes => isEven 255
adamc@142 133 | No => True
adamc@142 134 end" while it is expected to have type "isEven 255"
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@142 138 Abort.