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adamc@45
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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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adamc@45
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19 (** %\chapter{Inductive Predicates}% *)
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20
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21 (** The so-called "Curry-Howard Correspondence" states a formal connection between functional programs and mathematical proofs. In the last chapter, we snuck in a first introduction to this subject in Coq. Witness the close similarity between the types [unit] and [True] from the standard library: *)
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22
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23 Print unit.
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24 (** [[
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25
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26 Inductive unit : Set := tt : unit
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27 ]] *)
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28
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29 Print True.
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30 (** [[
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31
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32 Inductive True : Prop := I : True
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33 ]] *)
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34
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35 (** Recall that [unit] is the type with only one value, and [True] is the proposition that always holds. Despite this superficial difference between the two concepts, in both cases we can use the same inductive definition mechanism. The connection goes further than this. We see that we arrive at the definition of [True] by replacing [unit] by [True], [tt] by [I], and [Set] by [Prop]. The first two of these differences are superficial changes of names, while the third difference is the crucial one for separating programs from proofs. A term [T] of type [Set] is a type of programs, and a term of type [T] is a program. A term [T] of type [Prop] is a logical proposition, and its proofs are of type [T].
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36
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37 [unit] has one value, [tt]. [True] has one proof, [I]. Why distinguish between these two types? Many people who have read about Curry-Howard in an abstract context and not put it to use in proof engineering answer that the two types in fact %\textit{%#<i>#should not#</i>#%}% be distinguished. There is a certain aesthetic appeal to this point of view, but I want to argue that it is best to treat Curry-Howard very loosely in practical proving. There are Coq-specific reasons for preferring the distinction, involving efficient compilation and avoidance of paradoxes in the presence of classical math, but I will argue that there is a more general principle that should lead us to avoid conflating programming and proving.
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38
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39 The essence of the argument is roughly this: to an engineer, not all functions of type [A -> B] are created equal, but all proofs of a proposition [P -> Q] are. This idea is known as %\textit{%#<i>#proof irrelevance#</i>#%}%, and its formalizations in logics prevent us from distinguishing between alternate proofs of the same proposition. Proof irrelevance is compatible with, but not derivable in, Gallina. Apart from this theoretical concern, I will argue that it is most effective to do engineering with Coq by employing different techniques for programs versus proofs. Most of this book is organized around that distinction, describing how to program, by applying standard functional programming techniques in the presence of dependent types; and how to prove, by writing custom Ltac decision procedures.
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40
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41 With that perspective in mind, this chapter is sort of a mirror image of the last chapter, introducing how to define predicates with inductive definitions. We will point out similarities in places, but much of the effective Coq user's bag of tricks is disjoint for predicates versus "datatypes." This chapter is also a covert introduction to dependent types, which are the foundation on which interesting inductive predicates are built, though we will rely on tactics to build dependently-typed proof terms for us for now. A future chapter introduces more manual application of dependent types. *)
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42
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43
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adamc@45
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44 (** * Propositional logic *)
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45
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adamc@45
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46 (** Let us begin with a brief tour through the definitions of the connectives for propositional logic. We will work within a Coq section that provides us with a set of propositional variables. In Coq parlance, these are just terms of type [Prop.] *)
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47
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48 Section Propositional.
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49 Variables P Q R : Prop.
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50
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51 (** In Coq, the most basic propositional connective is implication, written [->], which we have already used in almost every proof. Rather than being defined inductively, implication is built into Coq as the function type constructor.
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52
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53 We have also already seen the definition of [True]. For a demonstration of a lower-level way of establishing proofs of inductive predicates, we turn to this trivial theorem. *)
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54
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55 Theorem obvious : True.
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56 apply I.
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57 Qed.
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58
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59 (** We may always use the [apply] tactic to take a proof step based on applying a particular constructor of the inductive predicate that we are trying to establish. Sometimes there is only one constructor that could possibly apply, in which case a shortcut is available: *)
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60
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61 Theorem obvious' : True.
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62 constructor.
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63 Qed.
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64
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65 (** There is also a predicate [False], which is the Curry-Howard mirror image of [Empty_set] from the last chapter. *)
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66
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67 Print False.
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68 (** [[
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69
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70 Inductive False : Prop :=
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71 ]] *)
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72
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73 (** We can conclude anything from [False], doing case analysis on a proof of [False] in the same way we might do case analysis on, say, a natural number. Since there are no cases to consider, any such case analysis succeeds immediately in proving the goal. *)
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74
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75 Theorem False_imp : False -> 2 + 2 = 5.
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76 destruct 1.
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77 Qed.
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78
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79 (** In a consistent context, we can never build a proof of [False]. In inconsistent contexts that appear in the courses of proofs, it is usually easiest to proceed by demonstrating that inconsistency with an explicit proof of [False]. *)
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80
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81 Theorem arith_neq : 2 + 2 = 5 -> 9 + 9 = 835.
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82 intro.
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83
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84 (** At this point, we have an inconsistent hypothesis [2 + 2 = 5], so the specific conclusion is not important. We use the [elimtype] tactic to state a proposition, telling Coq that we wish to construct a proof of the new proposition and then prove the original goal by case analysis on the structure of the new auxiliary proof. Since [False] has no constructors, [elimtype False] simply leaves us with the obligation to prove [False]. *)
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85
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86 elimtype False.
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87 (** [[
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88
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89 H : 2 + 2 = 5
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90 ============================
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91 False
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92 ]] *)
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93
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94 (** For now, we will leave the details of this proof about arithmetic to [crush]. *)
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95
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96 crush.
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97 Qed.
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98
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99 (** A related notion to [False] is logical negation. *)
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100
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101 Print not.
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102 (** [[
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103
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104 not = fun A : Prop => A -> False
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105 : Prop -> Prop
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106 ]] *)
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107
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108 (** We see that [not] is just shorthand for implication of [False]. We can use that fact explicitly in proofs. The syntax [~P] expands to [not P]. *)
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109
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110 Theorem arith_neq' : ~ (2 + 2 = 5).
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111 unfold not.
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112
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113 (** [[
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114
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115 ============================
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116 2 + 2 = 5 -> False
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117 ]] *)
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118
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119 crush.
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120 Qed.
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121
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122 (** We also have conjunction, which we introduced in the last chapter. *)
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123
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124 Print and.
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125 (** [[
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126
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127 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
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128 ]] *)
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129
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130 (** The interested reader can check that [and] has a Curry-Howard doppleganger called [prod], the type of pairs. However, it is generally most convenient to reason about conjunction using tactics. An explicit proof of commutativity of [and] illustrates the usual suspects for such tasks. [/\] is an infix shorthand for [and]. *)
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131
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132 Theorem and_comm : P /\ Q -> Q /\ P.
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133 (** We start by case analysis on the proof of [P /\ Q]. *)
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134
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135 destruct 1.
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136 (** [[
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137
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138 H : P
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139 H0 : Q
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140 ============================
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141 Q /\ P
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142 ]] *)
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143
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144 (** Every proof of a conjunction provides proofs for both conjuncts, so we get a single subgoal reflecting that. We can proceed by splitting this subgoal into a case for each conjunct of [Q /\ P]. *)
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145
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146 split.
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147 (** [[
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148 2 subgoals
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149
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150 H : P
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151 H0 : Q
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152 ============================
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153 Q
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154
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155 subgoal 2 is:
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156 P
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157 ]] *)
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158
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159 (** In each case, the conclusion is among our hypotheses, so the [assumption] tactic finishes the process. *)
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160
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161 assumption.
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162 assumption.
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163 Qed.
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164
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165 (** Coq disjunction is called [or] and abbreviated with the infix operator [\/]. *)
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166
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167 Print or.
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168 (** [[
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169
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170 Inductive or (A : Prop) (B : Prop) : Prop :=
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171 or_introl : A -> A \/ B | or_intror : B -> A \/ B
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172 ]] *)
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173
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174 (** We see that there are two ways to prove a disjunction: prove the first disjunct or prove the second. The Curry-Howard analogue of this is the Coq [sum] type. We can demonstrate the main tactics here with another proof of commutativity. *)
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175
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176 Theorem or_comm : P \/ Q -> Q \/ P.
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177 (** As in the proof for [and], we begin with case analysis, though this time we are met by two cases instead of one. *)
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178 destruct 1.
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179 (** [[
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180
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181 2 subgoals
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182
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183 H : P
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184 ============================
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185 Q \/ P
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186
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187 subgoal 2 is:
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188 Q \/ P
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189 ]] *)
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190
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191 (** We can see that, in the first subgoal, we want to prove the disjunction by proving its second disjunct. The [right] tactic telegraphs this intent. *)
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192
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193 right; assumption.
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194
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195 (** The second subgoal has a symmetric proof.
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196
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197 [[
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198
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199 1 subgoal
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200
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201 H : Q
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202 ============================
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203 Q \/ P
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204 ]] *)
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205
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206 left; assumption.
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207 Qed.
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208
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209
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210 (* begin hide *)
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211 (* In-class exercises *)
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212
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213 Theorem contra : P -> ~P -> R.
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214 Admitted.
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215
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216 Theorem and_assoc : (P /\ Q) /\ R -> P /\ (Q /\ R).
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217 Admitted.
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218
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219 Theorem or_assoc : (P \/ Q) \/ R -> P \/ (Q \/ R).
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220 Admitted.
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221
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222 (* end hide *)
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223
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224
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225 (** It would be a shame to have to plod manually through all proofs about propositional logic. Luckily, there is no need. One of the most basic Coq automation tactics is [tauto], which is a complete decision procedure for constructive propositional logic. (More on what "constructive" means in the next section.) We can use [tauto] to dispatch all of the purely propositional theorems we have proved so far. *)
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226
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227 Theorem or_comm' : P \/ Q -> Q \/ P.
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228 tauto.
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229 Qed.
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230
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231 (** Sometimes propositional reasoning forms important plumbing for the proof of a theorem, but we still need to apply some other smarts about, say, arithmetic. [intuition] is a generalization of [tauto] that proves everything it can using propositional reasoning. When some goals remain, it uses propositional laws to simplify them as far as possible. Consider this example, which uses the list concatenation operator [++] from the standard library. *)
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232
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233 Theorem arith_comm : forall ls1 ls2 : list nat,
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234 length ls1 = length ls2 \/ length ls1 + length ls2 = 6
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235 -> length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2.
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236 intuition.
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237
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238 (** A lot of the proof structure has been generated for us by [intuition], but the final proof depends on a fact about lists. The remaining subgoal hints at what cleverness we need to inject. *)
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239
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240 (** [[
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241
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242 ls1 : list nat
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243 ls2 : list nat
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244 H0 : length ls1 + length ls2 = 6
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245 ============================
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246 length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2
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247 ]] *)
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248
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249 (** We can see that we need a theorem about lengths of concatenated lists, which we proved last chapter and is also in the standard library. *)
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250
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251 rewrite app_length.
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252 (** [[
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253
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254 ls1 : list nat
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255 ls2 : list nat
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256 H0 : length ls1 + length ls2 = 6
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257 ============================
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258 length ls1 + length ls2 = 6 \/ length ls1 = length ls2
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259 ]] *)
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260
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261 (** Now the subgoal follows by purely propositional reasoning. That is, we could replace [length ls1 + length ls2 = 6] with [P] and [length ls1 = length ls2] with [Q] and arrive at a tautology of propositional logic. *)
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262
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263 tauto.
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264 Qed.
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265
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266 (** [intuition] is one of the main bits of glue in the implementation of [crush], so, with a little help, we can get a short automated proof of the theorem. *)
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267
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268 Theorem arith_comm' : forall ls1 ls2 : list nat,
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269 length ls1 = length ls2 \/ length ls1 + length ls2 = 6
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270 -> length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2.
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271 Hint Rewrite app_length : cpdt.
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272
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273 crush.
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274 Qed.
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275
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276 End Propositional.
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277
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278
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279
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280
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