adamc@142
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1 (* Copyright (c) 2008, Adam Chlipala
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2 *
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3 * This work is licensed under a
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4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
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5 * Unported License.
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6 * The license text is available at:
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7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
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8 *)
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9
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10 (* begin hide *)
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11 Require Import List.
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12
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13 Require Import Tactics MoreSpecif.
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14
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15 Set Implicit Arguments.
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16 (* end hide *)
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17
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18
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19 (** %\chapter{Proof by Reflection}% *)
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20
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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. *)
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22
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23
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24 (** * Proving Evenness *)
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25
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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. *)
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27
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28 Inductive isEven : nat -> Prop :=
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29 | Even_O : isEven O
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30 | Even_SS : forall n, isEven n -> isEven (S (S n)).
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31
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32 Ltac prove_even := repeat constructor.
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33
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34 Theorem even_256 : isEven 256.
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35 prove_even.
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36 Qed.
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37
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38 Print even_256.
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39 (** [[
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40
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41 even_256 =
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42 Even_SS
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43 (Even_SS
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44 (Even_SS
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45 (Even_SS
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46 ]]
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47
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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.
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49
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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.
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51
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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.
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53
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54 For this example, we begin by using a type from the [MoreSpecif] module to write a certified evenness checker. *)
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55
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56 Print partial.
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57 (** [[
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58
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59 Inductive partial (P : Prop) : Set := Proved : P -> [P] | Uncertain : [P]
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60 ]] *)
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61
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62 (** A [partial P] value is an optional proof of [P]. The notation [[P]] stands for [partial P]. *)
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63
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64 Open Local Scope partial_scope.
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65
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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. *)
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67
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68 Definition check_even (n : nat) : [isEven n].
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69 Hint Constructors isEven.
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70
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71 refine (fix F (n : nat) : [isEven n] :=
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72 match n return [isEven n] with
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73 | 0 => Yes
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74 | 1 => No
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75 | S (S n') => Reduce (F n')
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76 end); auto.
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77 Defined.
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78
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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. *)
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80
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81 Definition partialOut (P : Prop) (x : [P]) :=
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82 match x return (match x with
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83 | Proved _ => P
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84 | Uncertain => True
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85 end) with
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86 | Proved pf => pf
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87 | Uncertain => I
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88 end.
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89
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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: *)
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91
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92 Ltac prove_even_reflective :=
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93 match goal with
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94 | [ |- isEven ?N] => exact (partialOut (check_even N))
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95 end.
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96
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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. *)
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98
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99 Theorem even_256' : isEven 256.
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100 prove_even_reflective.
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101 Qed.
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102
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103 Print even_256'.
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104 (** [[
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105
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106 even_256' = partialOut (check_even 256)
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107 : isEven 256
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108 ]]
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109
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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? *)
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111
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112 Theorem even_255 : isEven 255.
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113 (** [[
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114
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115 prove_even_reflective.
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116
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117 [[
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118
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119 User error: No matching clauses for match goal
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120 ]]
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121
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122 Thankfully, the tactic fails. To see more precisely what goes wrong, we can run manually the body of the [match].
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123
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124 [[
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125
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126 exact (partialOut (check_even 255)).
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127
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128 [[
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129
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130 Error: The term "partialOut (check_even 255)" has type
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131 "match check_even 255 with
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132 | Yes => isEven 255
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133 | No => True
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134 end" while it is expected to have type "isEven 255"
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135 ]]
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136
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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]. *)
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138 Abort.
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139
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140
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141 (** * Reflecting the Syntax of a Trivial Tautology Language *)
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142
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143 (** We might also like to have reflective proofs of trivial tautologies like this one: *)
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144
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145 Theorem true_galore : (True /\ True) -> (True \/ (True /\ (True -> True))).
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146 tauto.
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147 Qed.
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148
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149 Print true_galore.
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150
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151 (** [[
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152
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153 true_galore =
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154 fun H : True /\ True =>
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155 and_ind (fun _ _ : True => or_introl (True /\ (True -> True)) I) H
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156 : True /\ True -> True \/ True /\ (True -> True)
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157 ]]
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158
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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.
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160
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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: *)
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162
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163 Inductive taut : Set :=
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164 | TautTrue : taut
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165 | TautAnd : taut -> taut -> taut
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166 | TautOr : taut -> taut -> taut
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167 | TautImp : taut -> taut -> taut.
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168
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169 (** We write a recursive function to "unreflect" this syntax back to [Prop]. *)
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170
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171 Fixpoint tautDenote (t : taut) : Prop :=
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172 match t with
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173 | TautTrue => True
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174 | TautAnd t1 t2 => tautDenote t1 /\ tautDenote t2
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175 | TautOr t1 t2 => tautDenote t1 \/ tautDenote t2
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176 | TautImp t1 t2 => tautDenote t1 -> tautDenote t2
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177 end.
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178
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179 (** It is easy to prove that every formula in the range of [tautDenote] is true. *)
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180
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181 Theorem tautTrue : forall t, tautDenote t.
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182 induction t; crush.
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183 Qed.
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184
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185 (** To use [tautTrue] to prove particular formulas, we need to implement the syntax reflection process. A recursive Ltac function does the job. *)
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186
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187 Ltac tautReflect P :=
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188 match P with
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189 | True => TautTrue
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190 | ?P1 /\ ?P2 =>
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191 let t1 := tautReflect P1 in
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192 let t2 := tautReflect P2 in
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193 constr:(TautAnd t1 t2)
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194 | ?P1 \/ ?P2 =>
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195 let t1 := tautReflect P1 in
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196 let t2 := tautReflect P2 in
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197 constr:(TautOr t1 t2)
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198 | ?P1 -> ?P2 =>
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199 let t1 := tautReflect P1 in
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200 let t2 := tautReflect P2 in
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201 constr:(TautImp t1 t2)
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202 end.
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203
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204 (** 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. *)
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205
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206 Ltac obvious :=
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207 match goal with
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208 | [ |- ?P ] =>
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209 let t := tautReflect P in
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210 exact (tautTrue t)
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211 end.
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212
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213 (** We can verify that [obvious] solves our original example, with a proof term that does not mention details of the proof. *)
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214
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215 Theorem true_galore' : (True /\ True) -> (True \/ (True /\ (True -> True))).
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216 obvious.
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217 Qed.
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218
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219 Print true_galore'.
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220
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221 (** [[
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222
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223 true_galore' =
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224 tautTrue
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225 (TautImp (TautAnd TautTrue TautTrue)
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226 (TautOr TautTrue (TautAnd TautTrue (TautImp TautTrue TautTrue))))
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227 : True /\ True -> True \/ True /\ (True -> True)
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228
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229 ]]
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230
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231 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. *)
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232
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233
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234 (** * A Smarter Tautology Solver *)
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235
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236 Require Import Quote.
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237
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238 Inductive formula : Set :=
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239 | Atomic : index -> formula
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240 | Truth : formula
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241 | Falsehood : formula
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242 | And : formula -> formula -> formula
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243 | Or : formula -> formula -> formula
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244 | Imp : formula -> formula -> formula.
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245
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246 Definition asgn := varmap Prop.
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247
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248 Definition imp (P1 P2 : Prop) := P1 -> P2.
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249 Infix "-->" := imp (no associativity, at level 95).
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250
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251 Fixpoint formulaDenote (atomics : asgn) (f : formula) : Prop :=
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252 match f with
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253 | Atomic v => varmap_find False v atomics
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254 | Truth => True
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255 | Falsehood => False
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256 | And f1 f2 => formulaDenote atomics f1 /\ formulaDenote atomics f2
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257 | Or f1 f2 => formulaDenote atomics f1 \/ formulaDenote atomics f2
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258 | Imp f1 f2 => formulaDenote atomics f1 --> formulaDenote atomics f2
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259 end.
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260
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261 Section my_tauto.
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262 Variable atomics : asgn.
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263
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264 Definition holds (v : index) := varmap_find False v atomics.
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265
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266 Require Import ListSet.
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267
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268 Definition index_eq : forall x y : index, {x = y} + {x <> y}.
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269 decide equality.
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270 Defined.
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271
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272 Definition add (s : set index) (v : index) := set_add index_eq v s.
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273 Definition In_dec : forall v (s : set index), {In v s} + {~In v s}.
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274 Open Local Scope specif_scope.
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275
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276 intro; refine (fix F (s : set index) : {In v s} + {~In v s} :=
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277 match s return {In v s} + {~In v s} with
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278 | nil => No
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279 | v' :: s' => index_eq v' v || F s'
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280 end); crush.
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281 Defined.
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282
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283 Fixpoint allTrue (s : set index) : Prop :=
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284 match s with
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285 | nil => True
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286 | v :: s' => holds v /\ allTrue s'
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287 end.
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288
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289 Theorem allTrue_add : forall v s,
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290 allTrue s
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291 -> holds v
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292 -> allTrue (add s v).
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293 induction s; crush;
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294 match goal with
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295 | [ |- context[if ?E then _ else _] ] => destruct E
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296 end; crush.
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297 Qed.
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298
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299 Theorem allTrue_In : forall v s,
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300 allTrue s
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301 -> set_In v s
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302 -> varmap_find False v atomics.
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303 induction s; crush.
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304 Qed.
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305
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306 Hint Resolve allTrue_add allTrue_In.
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307
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308 Open Local Scope partial_scope.
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309
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310 Definition forward (f : formula) (known : set index) (hyp : formula)
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311 (cont : forall known', [allTrue known' -> formulaDenote atomics f])
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312 : [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f].
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313 refine (fix F (f : formula) (known : set index) (hyp : formula)
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314 (cont : forall known', [allTrue known' -> formulaDenote atomics f]){struct hyp}
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315 : [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f] :=
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316 match hyp return [allTrue known -> formulaDenote atomics hyp -> formulaDenote atomics f] with
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317 | Atomic v => Reduce (cont (add known v))
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318 | Truth => Reduce (cont known)
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319 | Falsehood => Yes
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320 | And h1 h2 =>
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321 Reduce (F (Imp h2 f) known h1 (fun known' =>
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322 Reduce (F f known' h2 cont)))
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323 | Or h1 h2 => F f known h1 cont && F f known h2 cont
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324 | Imp _ _ => Reduce (cont known)
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325 end); crush.
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326 Defined.
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327
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328 Definition backward (known : set index) (f : formula) : [allTrue known -> formulaDenote atomics f].
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329 refine (fix F (known : set index) (f : formula) : [allTrue known -> formulaDenote atomics f] :=
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330 match f return [allTrue known -> formulaDenote atomics f] with
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331 | Atomic v => Reduce (In_dec v known)
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332 | Truth => Yes
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333 | Falsehood => No
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334 | And f1 f2 => F known f1 && F known f2
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335 | Or f1 f2 => F known f1 || F known f2
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adamc@144
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336 | Imp f1 f2 => forward f2 known f1 (fun known' => F known' f2)
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adamc@144
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337 end); crush; eauto.
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adamc@144
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338 Defined.
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adamc@144
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339
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adamc@144
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340 Definition my_tauto (f : formula) : [formulaDenote atomics f].
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adamc@144
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341 intro; refine (Reduce (backward nil f)); crush.
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adamc@144
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342 Defined.
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adamc@144
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343 End my_tauto.
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adamc@144
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344
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adamc@144
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345 Ltac my_tauto :=
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adamc@144
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346 repeat match goal with
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adamc@144
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347 | [ |- forall x : ?P, _ ] =>
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adamc@144
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348 match type of P with
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adamc@144
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349 | Prop => fail 1
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adamc@144
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350 | _ => intro
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adamc@144
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351 end
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adamc@144
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352 end;
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adamc@144
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353 quote formulaDenote;
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adamc@144
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354 match goal with
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adamc@144
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355 | [ |- formulaDenote ?m ?f ] => exact (partialOut (my_tauto m f))
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adamc@144
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356 end.
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adamc@144
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357
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adamc@144
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358 Theorem mt1 : True.
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adamc@144
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359 my_tauto.
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adamc@144
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360 Qed.
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adamc@144
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361
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adamc@144
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362 Print mt1.
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adamc@144
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363
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adamc@144
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364 Theorem mt2 : forall x y : nat, x = y --> x = y.
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adamc@144
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365 my_tauto.
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adamc@144
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366 Qed.
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adamc@144
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367
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adamc@144
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368 Print mt2.
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adamc@144
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369
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adamc@144
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370 Theorem mt3 : forall x y z,
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adamc@144
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371 (x < y /\ y > z) \/ (y > z /\ x < S y)
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adamc@144
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372 --> y > z /\ (x < y \/ x < S y).
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adamc@144
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373 my_tauto.
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adamc@144
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374 Qed.
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adamc@144
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375
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adamc@144
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376 Print mt3.
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adamc@144
|
377
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adamc@144
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378 Theorem mt4 : True /\ True /\ True /\ True /\ True /\ True /\ False --> False.
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adamc@144
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379 my_tauto.
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adamc@144
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380 Qed.
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adamc@144
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381
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adamc@144
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382 Print mt4.
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adamc@144
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383
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adamc@144
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384 Theorem mt4' : True /\ True /\ True /\ True /\ True /\ True /\ False -> False.
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adamc@144
|
385 tauto.
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adamc@144
|
386 Qed.
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adamc@144
|
387
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adamc@144
|
388 Print mt4'.
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