adamc@132
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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.
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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 (** %\part{Proof Engineering}
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20
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21 \chapter{Proof Search in Ltac}% *)
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22
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23 (** We have seen many examples of proof automation so far. This chapter aims to give a principled presentation of the features of Ltac, focusing in particular on the Ltac [match] construct, which supports a novel approach to backtracking search. First, though, we will run through some useful automation tactics that are built into Coq. They are described in detail in the manual, so we only outline what is possible. *)
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24
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25 (** * Some Built-In Automation Tactics *)
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26
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27 (** A number of tactics are called repeatedly by [crush]. [intuition] simplifies propositional structure of goals. [congruence] applies the rules of equality and congruence closure, plus properties of constructors of inductive types. The [omega] tactic provides a complete decision procedure for a theory that is called quantifier-free linear arithmetic or Presburger arithmetic, depending on whom you ask. That is, [omega] proves any goal that follows from looking only at parts of that goal that can be interpreted as propositional formulas whose atomic formulas are basic comparison operations on natural numbers or integers.
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28
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29 The [ring] tactic solves goals by appealing to the axioms of rings or semi-rings (as in algebra), depending on the type involved. Coq developments may declare new types to be parts of rings and semi-rings by proving the associated axioms. There is a simlar tactic [field] for simplifying values in fields by conversion to fractions over rings. Both [ring] and [field] can only solve goals that are equalities. The [fourier] tactic uses Fourier's method to prove inequalities over real numbers, which are axiomatized in the Coq standard library.
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30
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31 The %\textit{%#<i>#setoid#</i>#%}% facility makes it possible to register new equivalence relations to be understood by tactics like [rewrite]. For instance, [Prop] is registered as a setoid with the equivalence relation "if and only if." The ability to register new setoids can be very useful in proofs of a kind common in math, where all reasoning is done after "modding out by a relation." *)
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32
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33
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34 (** * Hint Databases *)
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35
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36 (** Another class of built-in tactics includes [auto], [eauto], and [autorewrite]. These are based on %\textit{%#<i>#hint databases#</i>#%}%, which we have seen extended in many examples so far. These tactics are important, because, in Ltac programming, we cannot create "global variables" whose values can be extended seamlessly by different modules in different source files. We have seen the advantages of hints so far, where [crush] can be defined once and for all, while still automatically applying the hints we add throughout developments.
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37
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38 The basic hints for [auto] and [eauto] are [Hint Immediate lemma], asking to try solving a goal immediately by applying the premise-free lemma; [Resolve lemma], which does the same but may add new premises that are themselves to be subjects of proof search; [Constructor type], which acts like [Resolve] applied to every constructor of an inductive type; and [Unfold ident], which tries unfolding [ident] when it appears at the head of a proof goal. Each of these [Hint] commands may be used with a suffix, as in [Hint Resolve lemma : my_db]. This adds the hint only to the specified database, so that it would only be used by, for instance, [auto with my_db]. An additional argument to [auto] specifies the maximum depth of proof trees to search in depth-first order, as in [auto 8] or [auto 8 with my_db]. The default depth is 5.
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39
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40 All of these [Hint] commands can be issued alternatively with a more primitive hint kind, [Extern]. A few examples should do best to explain how [Hint Extern] works. *)
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41
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42 Theorem bool_neq : true <> false.
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43 (* begin thide *)
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44 auto.
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45 (** [crush] would have discharged this goal, but the default hint database for [auto] contains no hint that applies. *)
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46 Abort.
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47
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48 (** It is hard to come up with a [bool]-specific hint that is not just a restatement of the theorem we mean to prove. Luckily, a simpler form suffices. *)
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49
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50 Hint Extern 1 (_ <> _) => congruence.
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51
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52 Theorem bool_neq : true <> false.
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53 auto.
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54 Qed.
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55 (* end thide *)
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56
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57 (** Our hint says: "whenever the conclusion matches the pattern [_ <> _], try applying [congruence]." The [1] is a cost for this rule. During proof search, whenever multiple rules apply, rules are tried in increasing cost order, so it pays to assign high costs to relatively expensive [Extern] hints.
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58
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59 [Extern] hints may be implemented with the full Ltac language. This example shows a case where a hint uses a [match]. *)
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60
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61 Section forall_and.
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62 Variable A : Set.
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63 Variables P Q : A -> Prop.
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64
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65 Hypothesis both : forall x, P x /\ Q x.
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66
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67 Theorem forall_and : forall z, P z.
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68 (* begin thide *)
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69 crush.
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70 (** [crush] makes no progress beyond what [intros] would have accomplished. [auto] will not apply the hypothesis [both] to prove the goal, because the conclusion of [both] does not unify with the conclusion of the goal. However, we can teach [auto] to handle this kind of goal. *)
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71
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72 Hint Extern 1 (P ?X) =>
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73 match goal with
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74 | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
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75 end.
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76
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77 auto.
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78 Qed.
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79 (* end thide *)
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80
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81 (** We see that an [Extern] pattern may bind unification variables that we use in the associated tactic. [proj1] is a function from the standard library for extracting a proof of [R] from a proof of [R /\ S]. *)
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82 End forall_and.
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83
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84 (** After our success on this example, we might get more ambitious and seek to generalize the hint to all possible predicates [P].
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85
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86 [[
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87 Hint Extern 1 (?P ?X) =>
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88 match goal with
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89 | [ H : forall x, ?P x /\ _ |- _ ] => apply (proj1 (H X))
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90 end.
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91
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92 ]]
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93
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94 [[
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95 User error: Bound head variable
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96 ]]
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97
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98 Coq's [auto] hint databases work as tables mapping %\textit{%#<i>#head symbols#</i>#%}% to lists of tactics to try. Because of this, the constant head of an [Extern] pattern must be determinable statically. In our first [Extern] hint, the head symbol was [not], since [x <> y] desugars to [not (eq x y)]; and, in the second example, the head symbol was [P].
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99
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100 This restriction on [Extern] hints is the main limitation of the [auto] mechanism, preventing us from using it for general context simplifications that are not keyed off of the form of the conclusion. This is perhaps just as well, since we can often code more efficient tactics with specialized Ltac programs, and we will see how in later sections of the chapter.
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101
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102 We have used [Hint Rewrite] in many examples so far. [crush] uses these hints by calling [autorewrite]. Our rewrite hints have taken the form [Hint Rewrite lemma : cpdt], adding them to the [cpdt] rewrite database. This is because, in contrast to [auto], [autorewrite] has no default database. Thus, we set the convention that [crush] uses the [cpdt] database.
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103
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104 This example shows a direct use of [autorewrite]. *)
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105
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106 Section autorewrite.
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107 Variable A : Set.
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108 Variable f : A -> A.
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109
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110 Hypothesis f_f : forall x, f (f x) = f x.
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111
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112 Hint Rewrite f_f : my_db.
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113
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114 Lemma f_f_f : forall x, f (f (f x)) = f x.
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115 intros; autorewrite with my_db; reflexivity.
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116 Qed.
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117
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118 (** There are a few ways in which [autorewrite] can lead to trouble when insufficient care is taken in choosing hints. First, the set of hints may define a nonterminating rewrite system, in which case invocations to [autorewrite] may not terminate. Second, we may add hints that "lead [autorewrite] down the wrong path." For instance: *)
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119
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120 Section garden_path.
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121 Variable g : A -> A.
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122 Hypothesis f_g : forall x, f x = g x.
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123 Hint Rewrite f_g : my_db.
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124
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125 Lemma f_f_f' : forall x, f (f (f x)) = f x.
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126 intros; autorewrite with my_db.
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127 (** [[
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128
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129 ============================
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130 g (g (g x)) = g x
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131 ]] *)
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132 Abort.
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133
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134 (** Our new hint was used to rewrite the goal into a form where the old hint could no longer be applied. This "non-monotonicity" of rewrite hints contrasts with the situation for [auto], where new hints may slow down proof search but can never "break" old proofs. *)
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135
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136 Reset garden_path.
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137
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138 (** [autorewrite] works with quantified equalities that include additional premises, but we must be careful to avoid similar incorrect rewritings. *)
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139
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140 Section garden_path.
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141 Variable P : A -> Prop.
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142 Variable g : A -> A.
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143 Hypothesis f_g : forall x, P x -> f x = g x.
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144 Hint Rewrite f_g : my_db.
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145
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146 Lemma f_f_f' : forall x, f (f (f x)) = f x.
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147 intros; autorewrite with my_db.
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148 (** [[
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149
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150 ============================
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151 g (g (g x)) = g x
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152
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153 subgoal 2 is:
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154 P x
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155 subgoal 3 is:
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156 P (f x)
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157 subgoal 4 is:
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158 P (f x)
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159 ]] *)
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160 Abort.
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161
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162 (** The inappropriate rule fired the same three times as before, even though we know we will not be able to prove the premises. *)
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163
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164 Reset garden_path.
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165
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166 (** Our final, successful, attempt uses an extra argument to [Hint Rewrite] that specifies a tactic to apply to generated premises. *)
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167
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168 Section garden_path.
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169 Variable P : A -> Prop.
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170 Variable g : A -> A.
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171 Hypothesis f_g : forall x, P x -> f x = g x.
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172 (* begin thide *)
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173 Hint Rewrite f_g using assumption : my_db.
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174 (* end thide *)
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175
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176 Lemma f_f_f' : forall x, f (f (f x)) = f x.
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177 (* begin thide *)
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178 intros; autorewrite with my_db; reflexivity.
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179 Qed.
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180 (* end thide *)
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181
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182 (** [autorewrite] will still use [f_g] when the generated premise is among our assumptions. *)
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183
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184 Lemma f_f_f_g : forall x, P x -> f (f x) = g x.
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185 (* begin thide *)
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186 intros; autorewrite with my_db; reflexivity.
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187 (* end thide *)
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188 Qed.
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189 End garden_path.
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190
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191 (** It can also be useful to use the [autorewrite with db in *] form, which does rewriting in hypotheses, as well as in the conclusion. *)
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192
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193 Lemma in_star : forall x y, f (f (f (f x))) = f (f y)
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194 -> f x = f (f (f y)).
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195 (* begin thide *)
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196 intros; autorewrite with my_db in *; assumption.
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197 (* end thide *)
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198 Qed.
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199
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200 End autorewrite.
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201
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202
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203 (** * Ltac Programming Basics *)
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204
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205 (** We have already seen many examples of Ltac programs. In the rest of this chapter, we attempt to give a more principled introduction to the important features and design patterns.
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206
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207 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
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208
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209 (* begin thide *)
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210 Ltac find_if :=
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211 match goal with
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212 | [ |- if ?X then _ else _ ] => destruct X
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213 end.
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214 (* end thide *)
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215
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216 (** The tactic checks if the conclusion is an [if], [destruct]ing the test expression if so. Certain classes of theorem are trivial to prove automatically with such a tactic. *)
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217
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218 Theorem hmm : forall (a b c : bool),
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219 if a
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220 then if b
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221 then True
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222 else True
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223 else if c
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224 then True
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225 else True.
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226 (* begin thide *)
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227 intros; repeat find_if; constructor.
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228 Qed.
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229 (* end thide *)
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230
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231 (** The [repeat] that we use here is called a %\textit{%#<i>#tactical#</i>#%}%, or tactic combinator. The behavior of [repeat t] is to loop through running [t], running [t] on all generated subgoals, running [t] on %\textit{%#<i>#their#</i>#%}% generated subgoals, and so on. When [t] fails at any point in this search tree, that particular subgoal is left to be handled by later tactics. Thus, it is important never to use [repeat] with a tactic that always succeeds.
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232
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233 Another very useful Ltac building block is %\textit{%#<i>#context patterns#</i>#%}%. *)
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234
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235 (* begin thide *)
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236 Ltac find_if_inside :=
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237 match goal with
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238 | [ |- context[if ?X then _ else _] ] => destruct X
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239 end.
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240 (* end thide *)
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241
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242 (** The behavior of this tactic is to find any subterm of the conclusion that is an [if] and then [destruct] the test expression. This version subsumes [find_if]. *)
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243
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244 Theorem hmm' : forall (a b c : bool),
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245 if a
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246 then if b
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247 then True
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248 else True
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249 else if c
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250 then True
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251 else True.
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252 (* begin thide *)
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253 intros; repeat find_if_inside; constructor.
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254 Qed.
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255 (* end thide *)
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256
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257 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
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258
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259 Theorem hmm2 : forall (a b : bool),
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260 (if a then 42 else 42) = (if b then 42 else 42).
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261 (* begin thide *)
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262 intros; repeat find_if_inside; reflexivity.
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263 Qed.
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264 (* end thide *)
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265
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266 (** Many decision procedures can be coded in Ltac via "[repeat match] loops." For instance, we can implement a subset of the functionality of [tauto]. *)
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267
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268 (* begin thide *)
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269 Ltac my_tauto :=
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270 repeat match goal with
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271 | [ H : ?P |- ?P ] => exact H
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272
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273 | [ |- True ] => constructor
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274 | [ |- _ /\ _ ] => constructor
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275 | [ |- _ -> _ ] => intro
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276
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277 | [ H : False |- _ ] => destruct H
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278 | [ H : _ /\ _ |- _ ] => destruct H
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279 | [ H : _ \/ _ |- _ ] => destruct H
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280
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281 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] =>
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282 let H := fresh "H" in
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283 generalize (H1 H2); clear H1; intro H
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284 end.
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285 (* end thide *)
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286
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287 (** Since [match] patterns can share unification variables between hypothesis and conclusion patterns, it is easy to figure out when the conclusion matches a hypothesis. The [exact] tactic solves a goal completely when given a proof term of the proper type.
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288
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289 It is also trivial to implement the "introduction rules" for a few of the connectives. Implementing elimination rules is only a little more work, since we must bind a name for a hypothesis to [destruct].
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290
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291 The last rule implements modus ponens. The most interesting part is the use of the Ltac-level [let] with a [fresh] expression. [fresh] takes in a name base and returns a fresh hypothesis variable based on that name. We use the new name variable [H] as the name we assign to the result of modus ponens. The use of [generalize] changes our conclusion to be an implication from [Q]. We clear the original hypothesis and move [Q] into the context with name [H]. *)
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292
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293 Section propositional.
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294 Variables P Q R : Prop.
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295
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296 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
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297 (* begin thide *)
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298 my_tauto.
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299 Qed.
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300 (* end thide *)
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301 End propositional.
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302
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303 (** It was relatively easy to implement modus ponens, because we do not lose information by clearing every implication that we use. If we want to implement a similarly-complete procedure for quantifier instantiation, we need a way to ensure that a particular proposition is not already included among our hypotheses. To do that effectively, we first need to learn a bit more about the semantics of [match].
|
adamc@135
|
304
|
adamc@135
|
305 It is tempting to assume that [match] works like it does in ML. In fact, there are a few critical differences in its behavior. One is that we may include arbitrary expressions in patterns, instead of being restricted to variables and constructors. Another is that the same variable may appear multiple times, inducing an implicit equality constraint.
|
adamc@135
|
306
|
adamc@135
|
307 There is a related pair of two other differences that are much more important than the others. [match] has a %\textit{%#<i>#backtracking semantics for failure#</i>#%}%. In ML, pattern matching works by finding the first pattern to match and then executing its body. If the body raises an exception, then the overall match raises the same exception. In Coq, failures in case bodies instead trigger continued search through the list of cases.
|
adamc@135
|
308
|
adamc@135
|
309 For instance, this (unnecessarily verbose) proof script works: *)
|
adamc@135
|
310
|
adamc@135
|
311 Theorem m1 : True.
|
adamc@135
|
312 match goal with
|
adamc@135
|
313 | [ |- _ ] => intro
|
adamc@135
|
314 | [ |- True ] => constructor
|
adamc@135
|
315 end.
|
adamc@141
|
316 (* begin thide *)
|
adamc@135
|
317 Qed.
|
adamc@141
|
318 (* end thide *)
|
adamc@135
|
319
|
adamc@135
|
320 (** The first case matches trivially, but its body tactic fails, since the conclusion does not begin with a quantifier or implication. In a similar ML match, that would mean that the whole pattern-match fails. In Coq, we backtrack and try the next pattern, which also matches. Its body tactic succeeds, so the overall tactic succeeds as well.
|
adamc@135
|
321
|
adamc@135
|
322 The example shows how failure can move to a different pattern within a [match]. Failure can also trigger an attempt to find %\textit{%#<i>#a different way of matching a single pattern#</i>#%}%. Consider another example: *)
|
adamc@135
|
323
|
adamc@135
|
324 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
|
adamc@135
|
325 intros; match goal with
|
adamc@135
|
326 | [ H : _ |- _ ] => pose H
|
adamc@135
|
327 end.
|
adamc@135
|
328 (** [[
|
adamc@135
|
329
|
adamc@135
|
330 r := H1 : R
|
adamc@135
|
331 ============================
|
adamc@135
|
332 Q
|
adamc@135
|
333 ]]
|
adamc@135
|
334
|
adamc@135
|
335 By applying [pose], a convenient debugging tool for "leaking information out of [match]es," we see that this [match] first tries binding [H] to [H1], which cannot be used to prove [Q]. Nonetheless, the following variation on the tactic succeeds at proving the goal: *)
|
adamc@135
|
336
|
adamc@141
|
337 (* begin thide *)
|
adamc@135
|
338 match goal with
|
adamc@135
|
339 | [ H : _ |- _ ] => exact H
|
adamc@135
|
340 end.
|
adamc@135
|
341 Qed.
|
adamc@141
|
342 (* end thide *)
|
adamc@135
|
343
|
adamc@135
|
344 (** The tactic first unifies [H] with [H1], as before, but [exact H] fails in that case, so the tactic engine searches for more possible values of [H]. Eventually, it arrives at the correct value, so that [exact H] and the overall tactic succeed. *)
|
adamc@135
|
345
|
adamc@135
|
346 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *)
|
adamc@135
|
347
|
adamc@141
|
348 (* begin thide *)
|
adamc@135
|
349 Ltac notHyp P :=
|
adamc@135
|
350 match goal with
|
adamc@135
|
351 | [ _ : P |- _ ] => fail 1
|
adamc@135
|
352 | _ =>
|
adamc@135
|
353 match P with
|
adamc@135
|
354 | ?P1 /\ ?P2 => first [ notHyp P1 | notHyp P2 | fail 2 ]
|
adamc@135
|
355 | _ => idtac
|
adamc@135
|
356 end
|
adamc@135
|
357 end.
|
adamc@141
|
358 (* end thide *)
|
adamc@135
|
359
|
adamc@135
|
360 (** We use the equality checking that is built into pattern-matching to see if there is a hypothesis that matches the proposition exactly. If so, we use the [fail] tactic. Without arguments, [fail] signals normal tactic failure, as you might expect. When [fail] is passed an argument [n], [n] is used to count outwards through the enclosing cases of backtracking search. In this case, [fail 1] says "fail not just in this pattern-matching branch, but for the whole [match]." The second case will never be tried when the [fail 1] is reached.
|
adamc@135
|
361
|
adamc@135
|
362 This second case, used when [P] matches no hypothesis, checks if [P] is a conjunction. Other simplifications may have split conjunctions into their component formulas, so we need to check that at least one of those components is also not represented. To achieve this, we apply the [first] tactical, which takes a list of tactics and continues down the list until one of them does not fail. The [fail 2] at the end says to [fail] both the [first] and the [match] wrapped around it.
|
adamc@135
|
363
|
adamc@135
|
364 The body of the [?P1 /\ ?P2] case guarantees that, if it is reached, we either succeed completely or fail completely. Thus, if we reach the wildcard case, [P] is not a conjunction. We use [idtac], a tactic that would be silly to apply on its own, since its effect is to succeed at doing nothing. Nonetheless, [idtac] is a useful placeholder for cases like what we see here.
|
adamc@135
|
365
|
adamc@135
|
366 With the non-presence check implemented, it is easy to build a tactic that takes as input a proof term and adds its conclusion as a new hypothesis, only if that conclusion is not already present, failing otherwise. *)
|
adamc@135
|
367
|
adamc@141
|
368 (* begin thide *)
|
adamc@135
|
369 Ltac extend pf :=
|
adamc@135
|
370 let t := type of pf in
|
adamc@135
|
371 notHyp t; generalize pf; intro.
|
adamc@141
|
372 (* end thide *)
|
adamc@135
|
373
|
adamc@135
|
374 (** We see the useful [type of] operator of Ltac. This operator could not be implemented in Gallina, but it is easy to support in Ltac. We end up with [t] bound to the type of [pf]. We check that [t] is not already present. If so, we use a [generalize]/[intro] combo to add a new hypothesis proved by [pf].
|
adamc@135
|
375
|
adamc@135
|
376 With these tactics defined, we can write a tactic [completer] for adding to the context all consequences of a set of simple first-order formulas. *)
|
adamc@135
|
377
|
adamc@141
|
378 (* begin thide *)
|
adamc@135
|
379 Ltac completer :=
|
adamc@135
|
380 repeat match goal with
|
adamc@135
|
381 | [ |- _ /\ _ ] => constructor
|
adamc@135
|
382 | [ H : _ /\ _ |- _ ] => destruct H
|
adamc@135
|
383 | [ H : ?P -> ?Q, H' : ?P |- _ ] =>
|
adamc@135
|
384 generalize (H H'); clear H; intro H
|
adamc@135
|
385 | [ |- forall x, _ ] => intro
|
adamc@135
|
386
|
adamc@135
|
387 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
|
adamc@135
|
388 extend (H X H')
|
adamc@135
|
389 end.
|
adamc@141
|
390 (* end thide *)
|
adamc@135
|
391
|
adamc@135
|
392 (** We use the same kind of conjunction and implication handling as previously. Note that, since [->] is the special non-dependent case of [forall], the fourth rule handles [intro] for implications, too.
|
adamc@135
|
393
|
adamc@135
|
394 In the fifth rule, when we find a [forall] fact [H] with a premise matching one of our hypotheses, we add the appropriate instantiation of [H]'s conclusion, if we have not already added it.
|
adamc@135
|
395
|
adamc@135
|
396 We can check that [completer] is working properly: *)
|
adamc@135
|
397
|
adamc@135
|
398 Section firstorder.
|
adamc@135
|
399 Variable A : Set.
|
adamc@135
|
400 Variables P Q R S : A -> Prop.
|
adamc@135
|
401
|
adamc@135
|
402 Hypothesis H1 : forall x, P x -> Q x /\ R x.
|
adamc@135
|
403 Hypothesis H2 : forall x, R x -> S x.
|
adamc@135
|
404
|
adamc@135
|
405 Theorem fo : forall x, P x -> S x.
|
adamc@141
|
406 (* begin thide *)
|
adamc@135
|
407 completer.
|
adamc@135
|
408 (** [[
|
adamc@135
|
409
|
adamc@135
|
410 x : A
|
adamc@135
|
411 H : P x
|
adamc@135
|
412 H0 : Q x
|
adamc@135
|
413 H3 : R x
|
adamc@135
|
414 H4 : S x
|
adamc@135
|
415 ============================
|
adamc@135
|
416 S x
|
adamc@135
|
417 ]] *)
|
adamc@135
|
418
|
adamc@135
|
419 assumption.
|
adamc@135
|
420 Qed.
|
adamc@141
|
421 (* end thide *)
|
adamc@135
|
422 End firstorder.
|
adamc@135
|
423
|
adamc@135
|
424 (** We narrowly avoided a subtle pitfall in our definition of [completer]. Let us try another definition that even seems preferable to the original, to the untrained eye. *)
|
adamc@135
|
425
|
adamc@141
|
426 (* begin thide *)
|
adamc@135
|
427 Ltac completer' :=
|
adamc@135
|
428 repeat match goal with
|
adamc@135
|
429 | [ |- _ /\ _ ] => constructor
|
adamc@135
|
430 | [ H : _ /\ _ |- _ ] => destruct H
|
adamc@135
|
431 | [ H : ?P -> _, H' : ?P |- _ ] =>
|
adamc@135
|
432 generalize (H H'); clear H; intro H
|
adamc@135
|
433 | [ |- forall x, _ ] => intro
|
adamc@135
|
434
|
adamc@135
|
435 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
|
adamc@135
|
436 extend (H X H')
|
adamc@135
|
437 end.
|
adamc@141
|
438 (* end thide *)
|
adamc@135
|
439
|
adamc@135
|
440 (** The only difference is in the modus ponens rule, where we have replaced an unused unification variable [?Q] with a wildcard. Let us try our example again with this version: *)
|
adamc@135
|
441
|
adamc@135
|
442 Section firstorder'.
|
adamc@135
|
443 Variable A : Set.
|
adamc@135
|
444 Variables P Q R S : A -> Prop.
|
adamc@135
|
445
|
adamc@135
|
446 Hypothesis H1 : forall x, P x -> Q x /\ R x.
|
adamc@135
|
447 Hypothesis H2 : forall x, R x -> S x.
|
adamc@135
|
448
|
adamc@135
|
449 Theorem fo' : forall x, P x -> S x.
|
adamc@141
|
450 (* begin thide *)
|
adamc@135
|
451 (** [[
|
adamc@135
|
452
|
adamc@135
|
453 completer'.
|
adamc@135
|
454
|
adamc@205
|
455 ]]
|
adamc@205
|
456
|
adamc@135
|
457 Coq loops forever at this point. What went wrong? *)
|
adamc@135
|
458 Abort.
|
adamc@141
|
459 (* end thide *)
|
adamc@135
|
460 End firstorder'.
|
adamc@136
|
461
|
adamc@136
|
462 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
|
adamc@136
|
463
|
adamc@136
|
464 Theorem t1 : forall x : nat, x = x.
|
adamc@136
|
465 match goal with
|
adamc@136
|
466 | [ |- forall x, _ ] => trivial
|
adamc@136
|
467 end.
|
adamc@141
|
468 (* begin thide *)
|
adamc@136
|
469 Qed.
|
adamc@141
|
470 (* end thide *)
|
adamc@136
|
471
|
adamc@136
|
472 (** This one fails. *)
|
adamc@136
|
473
|
adamc@141
|
474 (* begin thide *)
|
adamc@136
|
475 Theorem t1' : forall x : nat, x = x.
|
adamc@136
|
476 (** [[
|
adamc@136
|
477
|
adamc@136
|
478 match goal with
|
adamc@136
|
479 | [ |- forall x, ?P ] => trivial
|
adamc@136
|
480 end.
|
adamc@136
|
481
|
adamc@205
|
482 ]]
|
adamc@205
|
483
|
adamc@136
|
484 [[
|
adamc@136
|
485 User error: No matching clauses for match goal
|
adamc@136
|
486 ]] *)
|
adamc@136
|
487 Abort.
|
adamc@141
|
488 (* end thide *)
|
adamc@136
|
489
|
adamc@136
|
490 (** The problem is that unification variables may not contain locally-bound variables. In this case, [?P] would need to be bound to [x = x], which contains the local quantified variable [x]. By using a wildcard in the earlier version, we avoided this restriction.
|
adamc@136
|
491
|
adamc@136
|
492 The Coq 8.2 release includes a special pattern form for a unification variable with an explicit set of free variables. That unification variable is then bound to a function from the free variables to the "real" value. In Coq 8.1 and earlier, there is no such workaround.
|
adamc@136
|
493
|
adamc@136
|
494 No matter which version you use, it is important to be aware of this restriction. As we have alluded to, the restriction is the culprit behind the infinite-looping behavior of [completer']. We unintentionally match quantified facts with the modus ponens rule, circumventing the "already present" check and leading to different behavior. *)
|
adamc@137
|
495
|
adamc@137
|
496
|
adamc@137
|
497 (** * Functional Programming in Ltac *)
|
adamc@137
|
498
|
adamc@141
|
499 (* EX: Write a list length function in Ltac. *)
|
adamc@141
|
500
|
adamc@137
|
501 (** Ltac supports quite convenient functional programming, with a Lisp-with-syntax kind of flavor. However, there are a few syntactic conventions involved in getting programs to be accepted. The Ltac syntax is optimized for tactic-writing, so one has to deal with some inconveniences in writing more standard functional programs.
|
adamc@137
|
502
|
adamc@137
|
503 To illustrate, let us try to write a simple list length function. We start out writing it just like in Gallina, simply replacing [Fixpoint] (and its annotations) with [Ltac].
|
adamc@137
|
504
|
adamc@137
|
505 [[
|
adamc@137
|
506 Ltac length ls :=
|
adamc@137
|
507 match ls with
|
adamc@137
|
508 | nil => O
|
adamc@137
|
509 | _ :: ls' => S (length ls')
|
adamc@137
|
510 end.
|
adamc@137
|
511
|
adamc@205
|
512 ]]
|
adamc@205
|
513
|
adamc@137
|
514 [[
|
adamc@137
|
515 Error: The reference ls' was not found in the current environment
|
adamc@137
|
516 ]]
|
adamc@137
|
517
|
adamc@137
|
518 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
|
adamc@137
|
519
|
adamc@137
|
520 [[
|
adamc@137
|
521 Ltac length ls :=
|
adamc@137
|
522 match ls with
|
adamc@137
|
523 | nil => O
|
adamc@137
|
524 | _ :: ?ls' => S (length ls')
|
adamc@137
|
525 end.
|
adamc@137
|
526
|
adamc@205
|
527 ]]
|
adamc@205
|
528
|
adamc@137
|
529 [[
|
adamc@137
|
530 Error: The reference S was not found in the current environment
|
adamc@137
|
531 ]]
|
adamc@137
|
532
|
adamc@137
|
533 The problem is that Ltac treats the expression [S (length ls')] as an invocation of a tactic [S] with argument [length ls']. We need to use a special annotation to "escape into" the Gallina parsing nonterminal. *)
|
adamc@137
|
534
|
adamc@141
|
535 (* begin thide *)
|
adamc@137
|
536 Ltac length ls :=
|
adamc@137
|
537 match ls with
|
adamc@137
|
538 | nil => O
|
adamc@137
|
539 | _ :: ?ls' => constr:(S (length ls'))
|
adamc@137
|
540 end.
|
adamc@137
|
541
|
adamc@137
|
542 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *)
|
adamc@137
|
543
|
adamc@137
|
544 Goal False.
|
adamc@137
|
545 let n := length (1 :: 2 :: 3 :: nil) in
|
adamc@137
|
546 pose n.
|
adamc@137
|
547 (** [[
|
adamc@137
|
548
|
adamc@137
|
549 n := S (length (2 :: 3 :: nil)) : nat
|
adamc@137
|
550 ============================
|
adamc@137
|
551 False
|
adamc@137
|
552 ]]
|
adamc@137
|
553
|
adamc@137
|
554 [n] only has the length calculation unrolled one step. What has happened here is that, by escaping into the [constr] nonterminal, we referred to the [length] function of Gallina, rather than the [length] Ltac function that we are defining. *)Abort.
|
adamc@137
|
555
|
adamc@137
|
556 Reset length.
|
adamc@137
|
557
|
adamc@137
|
558 (** The thing to remember is that Gallina terms built by tactics must be bound explicitly via [let] or a similar technique, rather than inserting Ltac calls directly in other Gallina terms. *)
|
adamc@137
|
559
|
adamc@137
|
560 Ltac length ls :=
|
adamc@137
|
561 match ls with
|
adamc@137
|
562 | nil => O
|
adamc@137
|
563 | _ :: ?ls' =>
|
adamc@137
|
564 let ls'' := length ls' in
|
adamc@137
|
565 constr:(S ls'')
|
adamc@137
|
566 end.
|
adamc@137
|
567
|
adamc@137
|
568 Goal False.
|
adamc@137
|
569 let n := length (1 :: 2 :: 3 :: nil) in
|
adamc@137
|
570 pose n.
|
adamc@137
|
571 (** [[
|
adamc@137
|
572
|
adamc@137
|
573 n := 3 : nat
|
adamc@137
|
574 ============================
|
adamc@137
|
575 False
|
adamc@137
|
576 ]] *)
|
adamc@137
|
577 Abort.
|
adamc@141
|
578 (* end thide *)
|
adamc@141
|
579
|
adamc@141
|
580 (* EX: Write a list map function in Ltac. *)
|
adamc@137
|
581
|
adamc@137
|
582 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
|
adamc@137
|
583
|
adamc@141
|
584 (* begin thide *)
|
adamc@137
|
585 Ltac map T f :=
|
adamc@137
|
586 let rec map' ls :=
|
adamc@137
|
587 match ls with
|
adamc@137
|
588 | nil => constr:(@nil T)
|
adamc@137
|
589 | ?x :: ?ls' =>
|
adamc@137
|
590 let x' := f x in
|
adamc@137
|
591 let ls'' := map' ls' in
|
adamc@137
|
592 constr:(x' :: ls'')
|
adamc@137
|
593 end in
|
adamc@137
|
594 map'.
|
adamc@137
|
595
|
adamc@137
|
596 (** Ltac functions can have no implicit arguments. It may seem surprising that we need to pass [T], the carried type of the output list, explicitly. We cannot just use [type of f], because [f] is an Ltac term, not a Gallina term, and Ltac programs are dynamically typed. [f] could use very syntactic methods to decide to return differently typed terms for different inputs. We also could not replace [constr:(@nil T)] with [constr:nil], because we have no strongly-typed context to use to infer the parameter to [nil]. Luckily, we do have sufficient context within [constr:(x' :: ls'')].
|
adamc@137
|
597
|
adamc@137
|
598 Sometimes we need to employ the opposite direction of "nonterminal escape," when we want to pass a complicated tactic expression as an argument to another tactic, as we might want to do in invoking [map]. *)
|
adamc@137
|
599
|
adamc@137
|
600 Goal False.
|
adamc@137
|
601 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in
|
adamc@137
|
602 pose ls.
|
adamc@137
|
603 (** [[
|
adamc@137
|
604
|
adamc@137
|
605 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
|
adamc@137
|
606 ============================
|
adamc@137
|
607 False
|
adamc@137
|
608 ]] *)
|
adamc@137
|
609 Abort.
|
adamc@141
|
610 (* end thide *)
|
adamc@137
|
611
|
adamc@138
|
612
|
adamc@139
|
613 (** * Recursive Proof Search *)
|
adamc@139
|
614
|
adamc@139
|
615 (** Deciding how to instantiate quantifiers is one of the hardest parts of automated first-order theorem proving. For a given problem, we can consider all possible bounded-length sequences of quantifier instantiations, applying only propositional reasoning at the end. This is probably a bad idea for almost all goals, but it makes for a nice example of recursive proof search procedures in Ltac.
|
adamc@139
|
616
|
adamc@139
|
617 We can consider the maximum "dependency chain" length for a first-order proof. We define the chain length for a hypothesis to be 0, and the chain length for an instantiation of a quantified fact to be one greater than the length for that fact. The tactic [inster n] is meant to try all possible proofs with chain length at most [n]. *)
|
adamc@139
|
618
|
adamc@141
|
619 (* begin thide *)
|
adamc@139
|
620 Ltac inster n :=
|
adamc@139
|
621 intuition;
|
adamc@139
|
622 match n with
|
adamc@139
|
623 | S ?n' =>
|
adamc@139
|
624 match goal with
|
adamc@139
|
625 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n'
|
adamc@139
|
626 end
|
adamc@139
|
627 end.
|
adamc@141
|
628 (* end thide *)
|
adamc@139
|
629
|
adamc@139
|
630 (** [inster] begins by applying propositional simplification. Next, it checks if any chain length remains. If so, it tries all possible ways of instantiating quantified hypotheses with properly-typed local variables. It is critical to realize that, if the recursive call [inster n'] fails, then the [match goal] just seeks out another way of unifying its pattern against proof state. Thus, this small amount of code provides an elegant demonstration of how backtracking [match] enables exhaustive search.
|
adamc@139
|
631
|
adamc@139
|
632 We can verify the efficacy of [inster] with two short examples. The built-in [firstorder] tactic (with no extra arguments) is able to prove the first but not the second. *)
|
adamc@139
|
633
|
adamc@139
|
634 Section test_inster.
|
adamc@139
|
635 Variable A : Set.
|
adamc@139
|
636 Variables P Q : A -> Prop.
|
adamc@139
|
637 Variable f : A -> A.
|
adamc@139
|
638 Variable g : A -> A -> A.
|
adamc@139
|
639
|
adamc@139
|
640 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
|
adamc@139
|
641
|
adamc@139
|
642 Theorem test_inster : forall x y, P (g x y) -> Q (f x).
|
adamc@139
|
643 intros; inster 2.
|
adamc@139
|
644 Qed.
|
adamc@139
|
645
|
adamc@139
|
646 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
|
adamc@139
|
647 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
|
adamc@139
|
648
|
adamc@139
|
649 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
|
adamc@139
|
650 intros; inster 3.
|
adamc@139
|
651 Qed.
|
adamc@139
|
652 End test_inster.
|
adamc@139
|
653
|
adamc@140
|
654 (** The style employed in the definition of [inster] can seem very counterintuitive to functional programmers. Usually, functional programs accumulate state changes in explicit arguments to recursive functions. In Ltac, the state of the current subgoal is always implicit. Nonetheless, in contrast to general imperative programming, it is easy to undo any changes to this state, and indeed such "undoing" happens automatically at failures within [match]es. In this way, Ltac programming is similar to programming in Haskell with a stateful failure monad that supports a composition operator along the lines of the [first] tactical.
|
adamc@140
|
655
|
adamc@140
|
656 Functional programming purists may react indignantly to the suggestion of programming this way. Nonetheless, as with other kinds of "monadic programming," many problems are much simpler to solve with Ltac than they would be with explicit, pure proof manipulation in ML or Haskell. To demonstrate, we will write a basic simplification procedure for logical implications.
|
adamc@140
|
657
|
adamc@140
|
658 This procedure is inspired by one for separation logic, where conjuncts in formulas are thought of as "resources," such that we lose no completeness by "crossing out" equal conjuncts on the two sides of an implication. This process is complicated by the fact that, for reasons of modularity, our formulas can have arbitrary nested tree structure (branching at conjunctions) and may include existential quantifiers. It is helpful for the matching process to "go under" quantifiers and in fact decide how to instantiate existential quantifiers in the conclusion.
|
adamc@140
|
659
|
adamc@140
|
660 To distinguish the implications that our tactic handles from the implications that will show up as "plumbing" in various lemmas, we define a wrapper definition, a notation, and a tactic. *)
|
adamc@138
|
661
|
adamc@138
|
662 Definition imp (P1 P2 : Prop) := P1 -> P2.
|
adamc@140
|
663 Infix "-->" := imp (no associativity, at level 95).
|
adamc@140
|
664 Ltac imp := unfold imp; firstorder.
|
adamc@138
|
665
|
adamc@140
|
666 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
|
adamc@138
|
667
|
adamc@138
|
668 Theorem and_True_prem : forall P Q,
|
adamc@138
|
669 (P /\ True --> Q)
|
adamc@138
|
670 -> (P --> Q).
|
adamc@138
|
671 imp.
|
adamc@138
|
672 Qed.
|
adamc@138
|
673
|
adamc@138
|
674 Theorem and_True_conc : forall P Q,
|
adamc@138
|
675 (P --> Q /\ True)
|
adamc@138
|
676 -> (P --> Q).
|
adamc@138
|
677 imp.
|
adamc@138
|
678 Qed.
|
adamc@138
|
679
|
adamc@138
|
680 Theorem assoc_prem1 : forall P Q R S,
|
adamc@138
|
681 (P /\ (Q /\ R) --> S)
|
adamc@138
|
682 -> ((P /\ Q) /\ R --> S).
|
adamc@138
|
683 imp.
|
adamc@138
|
684 Qed.
|
adamc@138
|
685
|
adamc@138
|
686 Theorem assoc_prem2 : forall P Q R S,
|
adamc@138
|
687 (Q /\ (P /\ R) --> S)
|
adamc@138
|
688 -> ((P /\ Q) /\ R --> S).
|
adamc@138
|
689 imp.
|
adamc@138
|
690 Qed.
|
adamc@138
|
691
|
adamc@138
|
692 Theorem comm_prem : forall P Q R,
|
adamc@138
|
693 (P /\ Q --> R)
|
adamc@138
|
694 -> (Q /\ P --> R).
|
adamc@138
|
695 imp.
|
adamc@138
|
696 Qed.
|
adamc@138
|
697
|
adamc@138
|
698 Theorem assoc_conc1 : forall P Q R S,
|
adamc@138
|
699 (S --> P /\ (Q /\ R))
|
adamc@138
|
700 -> (S --> (P /\ Q) /\ R).
|
adamc@138
|
701 imp.
|
adamc@138
|
702 Qed.
|
adamc@138
|
703
|
adamc@138
|
704 Theorem assoc_conc2 : forall P Q R S,
|
adamc@138
|
705 (S --> Q /\ (P /\ R))
|
adamc@138
|
706 -> (S --> (P /\ Q) /\ R).
|
adamc@138
|
707 imp.
|
adamc@138
|
708 Qed.
|
adamc@138
|
709
|
adamc@138
|
710 Theorem comm_conc : forall P Q R,
|
adamc@138
|
711 (R --> P /\ Q)
|
adamc@138
|
712 -> (R --> Q /\ P).
|
adamc@138
|
713 imp.
|
adamc@138
|
714 Qed.
|
adamc@138
|
715
|
adamc@140
|
716 (** The first order of business in crafting our [matcher] tactic will be auxiliary support for searching through formula trees. The [search_prem] tactic implements running its tactic argument [tac] on every subformula of an [imp] premise. As it traverses a tree, [search_prem] applies some of the above lemmas to rewrite the goal to bring different subformulas to the head of the goal. That is, for every subformula [P] of the implication premise, we want [P] to "have a turn," where the premise is rearranged into the form [P /\ Q] for some [Q]. The tactic [tac] should expect to see a goal in this form and focus its attention on the first conjunct of the premise. *)
|
adamc@140
|
717
|
adamc@141
|
718 (* begin thide *)
|
adamc@138
|
719 Ltac search_prem tac :=
|
adamc@138
|
720 let rec search P :=
|
adamc@138
|
721 tac
|
adamc@138
|
722 || (apply and_True_prem; tac)
|
adamc@138
|
723 || match P with
|
adamc@138
|
724 | ?P1 /\ ?P2 =>
|
adamc@138
|
725 (apply assoc_prem1; search P1)
|
adamc@138
|
726 || (apply assoc_prem2; search P2)
|
adamc@138
|
727 end
|
adamc@138
|
728 in match goal with
|
adamc@138
|
729 | [ |- ?P /\ _ --> _ ] => search P
|
adamc@138
|
730 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
|
adamc@138
|
731 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
|
adamc@138
|
732 end.
|
adamc@138
|
733
|
adamc@140
|
734 (** To understand how [search_prem] works, we turn first to the final [match]. If the premise begins with a conjunction, we call the [search] procedure on each of the conjuncts, or only the first conjunct, if that already yields a case where [tac] does not fail. [search P] expects and maintains the invariant that the premise is of the form [P /\ Q] for some [Q]. We pass [P] explicitly as a kind of decreasing induction measure, to avoid looping forever when [tac] always fails. The second [match] case calls a commutativity lemma to realize this invariant, before passing control to [search]. The final [match] case tries applying [tac] directly and then, if that fails, changes the form of the goal by adding an extraneous [True] conjunct and calls [tac] again.
|
adamc@140
|
735
|
adamc@140
|
736 [search] itself tries the same tricks as in the last case of the final [match]. Additionally, if neither works, it checks if [P] is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity lemmas to maintain the goal-form invariant.
|
adamc@140
|
737
|
adamc@140
|
738 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
|
adamc@140
|
739
|
adamc@138
|
740 Ltac search_conc tac :=
|
adamc@138
|
741 let rec search P :=
|
adamc@138
|
742 tac
|
adamc@138
|
743 || (apply and_True_conc; tac)
|
adamc@138
|
744 || match P with
|
adamc@138
|
745 | ?P1 /\ ?P2 =>
|
adamc@138
|
746 (apply assoc_conc1; search P1)
|
adamc@138
|
747 || (apply assoc_conc2; search P2)
|
adamc@138
|
748 end
|
adamc@138
|
749 in match goal with
|
adamc@138
|
750 | [ |- _ --> ?P /\ _ ] => search P
|
adamc@138
|
751 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
|
adamc@138
|
752 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
|
adamc@138
|
753 end.
|
adamc@138
|
754
|
adamc@140
|
755 (** Now we can prove a number of lemmas that are suitable for application by our search tactics. A lemma that is meant to handle a premise should have the form [P /\ Q --> R] for some interesting [P], and a lemma that is meant to handle a conclusion should have the form [P --> Q /\ R] for some interesting [Q]. *)
|
adamc@140
|
756
|
adamc@138
|
757 Theorem False_prem : forall P Q,
|
adamc@138
|
758 False /\ P --> Q.
|
adamc@138
|
759 imp.
|
adamc@138
|
760 Qed.
|
adamc@138
|
761
|
adamc@138
|
762 Theorem True_conc : forall P Q : Prop,
|
adamc@138
|
763 (P --> Q)
|
adamc@138
|
764 -> (P --> True /\ Q).
|
adamc@138
|
765 imp.
|
adamc@138
|
766 Qed.
|
adamc@138
|
767
|
adamc@138
|
768 Theorem Match : forall P Q R : Prop,
|
adamc@138
|
769 (Q --> R)
|
adamc@138
|
770 -> (P /\ Q --> P /\ R).
|
adamc@138
|
771 imp.
|
adamc@138
|
772 Qed.
|
adamc@138
|
773
|
adamc@138
|
774 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
|
adamc@138
|
775 (forall x, P x /\ Q --> R)
|
adamc@138
|
776 -> (ex P /\ Q --> R).
|
adamc@138
|
777 imp.
|
adamc@138
|
778 Qed.
|
adamc@138
|
779
|
adamc@138
|
780 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
|
adamc@138
|
781 (Q --> P x /\ R)
|
adamc@138
|
782 -> (Q --> ex P /\ R).
|
adamc@138
|
783 imp.
|
adamc@138
|
784 Qed.
|
adamc@138
|
785
|
adamc@140
|
786 (** We will also want a "base case" lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
|
adamc@140
|
787
|
adamc@138
|
788 Theorem imp_True : forall P,
|
adamc@138
|
789 P --> True.
|
adamc@138
|
790 imp.
|
adamc@138
|
791 Qed.
|
adamc@138
|
792
|
adamc@140
|
793 (** Our final [matcher] tactic is now straightforward. First, we [intros] all variables into scope. Then we attempt simple premise simplifications, finishing the proof upon finding [False] and eliminating any existential quantifiers that we find. After that, we search through the conclusion. We remove [True] conjuncts, remove existential quantifiers by introducing unification variables for their bound variables, and search for matching premises to cancel. Finally, when no more progress is made, we see if the goal has become trivial and can be solved by [imp_True]. *)
|
adamc@140
|
794
|
adamc@138
|
795 Ltac matcher :=
|
adamc@138
|
796 intros;
|
adamc@204
|
797 repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
|
adamc@204
|
798 repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
|
adamc@204
|
799 || search_prem ltac:(simple apply Match));
|
adamc@204
|
800 try simple apply imp_True.
|
adamc@141
|
801 (* end thide *)
|
adamc@140
|
802
|
adamc@140
|
803 (** Our tactic succeeds at proving a simple example. *)
|
adamc@138
|
804
|
adamc@138
|
805 Theorem t2 : forall P Q : Prop,
|
adamc@138
|
806 Q /\ (P /\ False) /\ P --> P /\ Q.
|
adamc@138
|
807 matcher.
|
adamc@138
|
808 Qed.
|
adamc@138
|
809
|
adamc@140
|
810 (** In the generated proof, we find a trace of the workings of the search tactics. *)
|
adamc@140
|
811
|
adamc@140
|
812 Print t2.
|
adamc@140
|
813 (** [[
|
adamc@140
|
814
|
adamc@140
|
815 t2 =
|
adamc@140
|
816 fun P Q : Prop =>
|
adamc@140
|
817 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
|
adamc@140
|
818 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
|
adamc@140
|
819 ]] *)
|
adamc@140
|
820
|
adamc@140
|
821 (** We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
|
adamc@140
|
822
|
adamc@138
|
823 Theorem t3 : forall P Q R : Prop,
|
adamc@138
|
824 P /\ Q --> Q /\ R /\ P.
|
adamc@138
|
825 matcher.
|
adamc@140
|
826 (** [[
|
adamc@140
|
827
|
adamc@140
|
828 ============================
|
adamc@140
|
829 True --> R
|
adamc@140
|
830 ]]
|
adamc@140
|
831
|
adamc@140
|
832 [matcher] canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
|
adamc@138
|
833 Abort.
|
adamc@138
|
834
|
adamc@140
|
835 (** [matcher] even succeeds at guessing quantifier instantiations. It is the unification that occurs in uses of the [Match] lemma that does the real work here. *)
|
adamc@140
|
836
|
adamc@138
|
837 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
|
adamc@138
|
838 matcher.
|
adamc@138
|
839 Qed.
|
adamc@138
|
840
|
adamc@140
|
841 Print t4.
|
adamc@140
|
842
|
adamc@140
|
843 (** [[
|
adamc@140
|
844
|
adamc@140
|
845 t4 =
|
adamc@140
|
846 fun (P : nat -> Prop) (Q : Prop) =>
|
adamc@140
|
847 and_True_prem
|
adamc@140
|
848 (ex_prem (P:=fun x : nat => P x /\ Q)
|
adamc@140
|
849 (fun x : nat =>
|
adamc@140
|
850 assoc_prem2
|
adamc@140
|
851 (Match (P:=Q)
|
adamc@140
|
852 (and_True_conc
|
adamc@140
|
853 (ex_conc (fun x0 : nat => P x0) x
|
adamc@140
|
854 (Match (P:=P x) (imp_True (P:=True))))))))
|
adamc@140
|
855 : forall (P : nat -> Prop) (Q : Prop),
|
adamc@140
|
856 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
|
adamc@140
|
857 ]] *)
|