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adam@322
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1 (* Copyright (c) 2008-2011, 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 CpdtTactics.
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14
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15 Set Implicit Arguments.
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16
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17 (* Extra definitions to get coqdoc to choose the right fonts. *)
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18
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19 (* begin thide *)
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20 Inductive unit := tt.
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21 Inductive Empty_set := .
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22 Inductive bool := true | false.
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23 Inductive sum := .
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24 Inductive prod := .
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25 Inductive and := conj.
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26 Inductive or := or_introl | or_intror.
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27 Inductive ex := ex_intro.
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28 Inductive eq := refl_equal.
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29 Reset unit.
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adam@323
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30 (* end thide *)
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adamc@45
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31 (* end hide *)
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32
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adamc@45
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33 (** %\chapter{Inductive Predicates}% *)
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34
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35 (** The so-called %\index{Curry-Howard correspondence}``%#"#Curry-Howard correspondence#"#%''~\cite{Curry,Howard}% 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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36
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37 (* begin hide *)
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38 Print unit.
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adam@322
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39 (* end hide *)
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40 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#unit#</tt>#%}%[.] *)
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41 (** [[
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42 Inductive unit : Set := tt : unit
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43 ]]
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44 *)
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45
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46 (* begin hide *)
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47 Print True.
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48 (* end hide *)
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49 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#True#</tt>#%}%[.] *)
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50 (** [[
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51 Inductive True : Prop := I : True
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52 ]]
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53 *)
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54
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55 (** 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]. Chapter 12 goes into more detail about the theoretical differences between [Prop] and [Set]. For now, we will simply follow common intuitions about what a proof is.
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56
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57 The type [unit] has one value, [tt]. The type [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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58
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59 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 %\index{proof irrelevance}\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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60
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61 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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62
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63
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64 (** * Propositional Logic *)
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65
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66 (** 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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67
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68 Section Propositional.
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69 Variables P Q R : Prop.
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70
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71 (** 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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72
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73 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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74
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75 Theorem obvious : True.
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76 (* begin thide *)
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77 apply I.
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78 (* end thide *)
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79 Qed.
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80
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81 (** 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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82
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83 (* begin thide *)
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84 Theorem obvious' : True.
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85 (* begin hide *)
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86 constructor.
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87 (* end hide *)
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88 (** %\hspace{.075in}\coqdockw{%#<tt>#constructor#</tt>#%}%.%\vspace{-.1in}% *)
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89
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90 Qed.
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91
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92 (* end thide *)
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93
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94 (** There is also a predicate [False], which is the Curry-Howard mirror image of [Empty_set] from the last chapter. *)
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95
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96 (* begin hide *)
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97 Print False.
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98 (* end hide *)
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99 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#False#</tt>#%}%[.] *)
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100 (** [[
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101 Inductive False : Prop :=
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102
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103 ]]
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104
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105 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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106
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107 Theorem False_imp : False -> 2 + 2 = 5.
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108 (* begin thide *)
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109 destruct 1.
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110 (* end thide *)
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111 Qed.
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112
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113 (** 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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114
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115 Theorem arith_neq : 2 + 2 = 5 -> 9 + 9 = 835.
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116 (* begin thide *)
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117 intro.
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118
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119 (** At this point, we have an inconsistent hypothesis [2 + 2 = 5], so the specific conclusion is not important. We use the %\index{tactics!elimtype}%[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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120
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121 elimtype False.
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122 (** [[
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123 H : 2 + 2 = 5
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124 ============================
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125 False
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126
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127 ]]
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128
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129 For now, we will leave the details of this proof about arithmetic to [crush]. *)
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130
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131 crush.
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132 (* end thide *)
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133 Qed.
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134
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135 (** A related notion to [False] is logical negation. *)
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136
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137 Print not.
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138 (** %\vspace{-.15in}% [[
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139 not = fun A : Prop => A -> False
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140 : Prop -> Prop
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141
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142 ]]
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143
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144 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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145
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146 Theorem arith_neq' : ~ (2 + 2 = 5).
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147 (* begin thide *)
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148 unfold not.
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149 (** [[
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150 ============================
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151 2 + 2 = 5 -> False
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152 ]]
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153 *)
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154
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155 crush.
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156 (* end thide *)
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157 Qed.
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158
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159 (** We also have conjunction, which we introduced in the last chapter. *)
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160
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161 (* begin hide *)
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162 Print and.
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163 (* end hide *)
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164 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#and#</tt>#%}%[.]
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165 [[
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166 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
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167
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168 ]]
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169
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170 The interested reader can check that [and] has a Curry-Howard equivalent called %\index{Gallina terms!prod}%[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. The operator [/\] is an infix shorthand for [and]. *)
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171
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172 Theorem and_comm : P /\ Q -> Q /\ P.
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173
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174 (* begin thide *)
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175 (** We start by case analysis on the proof of [P /\ Q]. *)
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176
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177 destruct 1.
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178 (** [[
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179 H : P
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180 H0 : Q
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181 ============================
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182 Q /\ P
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183
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184 ]]
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185
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186 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].%\index{tactics!split}% *)
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187
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188 split.
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189 (** %\vspace{.1in} \noindent 2 \coqdockw{subgoals}\vspace{-.1in}%#<tt>2 subgoals</tt>#
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190 [[
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191
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192 H : P
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193 H0 : Q
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194 ============================
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195 Q
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196 ]]
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197 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
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198 [[
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199 P
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200
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201 ]]
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202
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203 In each case, the conclusion is among our hypotheses, so the %\index{tactics!assumption}%[assumption] tactic finishes the process. *)
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204
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205 assumption.
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206 assumption.
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207 (* end thide *)
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208 Qed.
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209
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210 (** Coq disjunction is called %\index{Gallina terms!or}%[or] and abbreviated with the infix operator [\/]. *)
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211
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212 (* begin hide *)
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213 Print or.
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214 (* end hide *)
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215 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#or#</tt>#%}%[.]
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216 [[
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217 Inductive or (A : Prop) (B : Prop) : Prop :=
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218 or_introl : A -> A \/ B | or_intror : B -> A \/ B
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219
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220 ]]
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221
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222 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 %\index{Gallina terms!sum}%[sum] type. We can demonstrate the main tactics here with another proof of commutativity. *)
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223
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224 Theorem or_comm : P \/ Q -> Q \/ P.
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225
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226 (* begin thide *)
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227 (** 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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228
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229 destruct 1.
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230 (** %\vspace{.1in} \noindent 2 \coqdockw{subgoals}\vspace{-.1in}%#<tt>2 subgoals</tt>#
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231 [[
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232
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233 H : P
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234 ============================
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235 Q \/ P
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236 ]]
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237 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
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238 [[
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239 Q \/ P
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240
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241 ]]
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242
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243 We can see that, in the first subgoal, we want to prove the disjunction by proving its second disjunct. The %\index{tactics!right}\coqdockw{%#<tt>#right#</tt>#%}% tactic telegraphs this intent. *)
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244
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245 (* begin hide *)
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246 right; assumption.
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247 (* end hide *)
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248 (** %\hspace{.075in}\coqdockw{%#<tt>#right#</tt>#%}%[; assumption.] *)
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249
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250 (** The second subgoal has a symmetric proof.%\index{tactics!left}%
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251
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252 [[
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253 1 subgoal
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254
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255 H : Q
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256 ============================
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257 Q \/ P
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258 ]]
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259 *)
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260
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261 (* begin hide *)
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262 left; assumption.
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263 (* end hide *)
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adam@322
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264 (** %\hspace{.075in}\coqdockw{%#<tt>#left#</tt>#%}%[; assumption.] *)
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265
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266 (* end thide *)
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267 Qed.
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268
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269
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adamc@46
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270 (* begin hide *)
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adamc@46
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271 (* In-class exercises *)
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272
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273 Theorem contra : P -> ~P -> R.
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adamc@52
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274 (* begin thide *)
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275 unfold not.
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adamc@52
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276 intros.
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adamc@52
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277 elimtype False.
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278 apply H0.
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279 assumption.
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adamc@52
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280 (* end thide *)
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281 Admitted.
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282
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283 Theorem and_assoc : (P /\ Q) /\ R -> P /\ (Q /\ R).
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adamc@52
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284 (* begin thide *)
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adamc@52
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285 intros.
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adamc@52
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286 destruct H.
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adamc@52
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287 destruct H.
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288 split.
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289 assumption.
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adamc@52
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290 split.
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291 assumption.
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292 assumption.
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adamc@52
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293 (* end thide *)
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294 Admitted.
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295
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adamc@46
|
296 Theorem or_assoc : (P \/ Q) \/ R -> P \/ (Q \/ R).
|
|
adamc@52
|
297 (* begin thide *)
|
|
adamc@52
|
298 intros.
|
|
adamc@52
|
299 destruct H.
|
|
adamc@52
|
300 destruct H.
|
|
adamc@52
|
301 left.
|
|
adamc@52
|
302 assumption.
|
|
adamc@52
|
303 right.
|
|
adamc@52
|
304 left.
|
|
adamc@52
|
305 assumption.
|
|
adamc@52
|
306 right.
|
|
adamc@52
|
307 right.
|
|
adamc@52
|
308 assumption.
|
|
adamc@52
|
309 (* end thide *)
|
|
adamc@46
|
310 Admitted.
|
|
adamc@46
|
311
|
|
adamc@46
|
312 (* end hide *)
|
|
adamc@46
|
313
|
|
adamc@46
|
314
|
|
adam@322
|
315 (** 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 %\index{tactics!tauto}%[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. *)
|
|
adamc@46
|
316
|
|
adamc@46
|
317 Theorem or_comm' : P \/ Q -> Q \/ P.
|
|
adamc@55
|
318 (* begin thide *)
|
|
adamc@46
|
319 tauto.
|
|
adamc@55
|
320 (* end thide *)
|
|
adamc@46
|
321 Qed.
|
|
adamc@46
|
322
|
|
adam@322
|
323 (** Sometimes propositional reasoning forms important plumbing for the proof of a theorem, but we still need to apply some other smarts about, say, arithmetic. %\index{tactics!intuition}%[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. *)
|
|
adamc@46
|
324
|
|
adamc@46
|
325 Theorem arith_comm : forall ls1 ls2 : list nat,
|
|
adamc@46
|
326 length ls1 = length ls2 \/ length ls1 + length ls2 = 6
|
|
adamc@46
|
327 -> length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2.
|
|
adamc@55
|
328 (* begin thide *)
|
|
adamc@46
|
329 intuition.
|
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adamc@46
|
330
|
|
adamc@46
|
331 (** 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. *)
|
|
adamc@46
|
332
|
|
adamc@46
|
333 (** [[
|
|
adamc@46
|
334 ls1 : list nat
|
|
adamc@46
|
335 ls2 : list nat
|
|
adamc@46
|
336 H0 : length ls1 + length ls2 = 6
|
|
adamc@46
|
337 ============================
|
|
adamc@46
|
338 length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2
|
|
adamc@209
|
339
|
|
adamc@209
|
340 ]]
|
|
adamc@46
|
341
|
|
adamc@209
|
342 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. *)
|
|
adamc@46
|
343
|
|
adamc@46
|
344 rewrite app_length.
|
|
adamc@46
|
345 (** [[
|
|
adamc@46
|
346 ls1 : list nat
|
|
adamc@46
|
347 ls2 : list nat
|
|
adamc@46
|
348 H0 : length ls1 + length ls2 = 6
|
|
adamc@46
|
349 ============================
|
|
adamc@46
|
350 length ls1 + length ls2 = 6 \/ length ls1 = length ls2
|
|
adamc@209
|
351
|
|
adamc@209
|
352 ]]
|
|
adamc@46
|
353
|
|
adamc@209
|
354 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. *)
|
|
adamc@46
|
355
|
|
adamc@46
|
356 tauto.
|
|
adamc@55
|
357 (* end thide *)
|
|
adamc@46
|
358 Qed.
|
|
adamc@46
|
359
|
|
adam@322
|
360 (** The [intuition] tactic 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. *)
|
|
adamc@46
|
361
|
|
adamc@55
|
362 (* begin thide *)
|
|
adamc@46
|
363 Theorem arith_comm' : forall ls1 ls2 : list nat,
|
|
adamc@46
|
364 length ls1 = length ls2 \/ length ls1 + length ls2 = 6
|
|
adamc@46
|
365 -> length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2.
|
|
adam@322
|
366 (* begin hide *)
|
|
adam@375
|
367 Hint Rewrite app_length.
|
|
adam@322
|
368 (* end hide *)
|
|
adam@375
|
369 (** %\hspace{.075in}%[Hint] %\coqdockw{%#<tt>#Rewrite#</tt>#%}% [app_length.] *)
|
|
adamc@46
|
370
|
|
adamc@46
|
371 crush.
|
|
adamc@46
|
372 Qed.
|
|
adamc@55
|
373 (* end thide *)
|
|
adamc@46
|
374
|
|
adamc@45
|
375 End Propositional.
|
|
adamc@45
|
376
|
|
adam@322
|
377 (** Ending the section here has the same effect as always. Each of our propositional theorems becomes universally quantified over the propositional variables that we used. *)
|
|
adam@322
|
378
|
|
adamc@46
|
379
|
|
adamc@47
|
380 (** * What Does It Mean to Be Constructive? *)
|
|
adamc@46
|
381
|
|
adamc@47
|
382 (** One potential point of confusion in the presentation so far is the distinction between [bool] and [Prop]. [bool] is a datatype whose two values are [true] and [false], while [Prop] is a more primitive type that includes among its members [True] and [False]. Why not collapse these two concepts into one, and why must there be more than two states of mathematical truth?
|
|
adamc@46
|
383
|
|
adam@322
|
384 The answer comes from the fact that Coq implements %\index{constructive logic}\textit{%#<i>#constructive#</i>#%}% or %\index{intuitionistic logic|see{constructive logic}}\textit{%#<i>#intuitionistic#</i>#%}% logic, in contrast to the %\index{classical logic}\textit{%#<i>#classical#</i>#%}% logic that you may be more familiar with. In constructive logic, classical tautologies like [~ ~ P -> P] and [P \/ ~ P] do not always hold. In general, we can only prove these tautologies when [P] is %\index{decidability}\textit{%#<i>#decidable#</i>#%}%, in the sense of %\index{computability|see{decidability}}%computability theory. The Curry-Howard encoding that Coq uses for [or] allows us to extract either a proof of [P] or a proof of [~ P] from any proof of [P \/ ~ P]. Since our proofs are just functional programs which we can run, a general %\index{law of the excluded middle}%law of the excluded middle would give us a decision procedure for the halting problem, where the instantiations of [P] would be formulas like %``%#"#this particular Turing machine halts.#"#%''%
|
|
adamc@47
|
385
|
|
adam@292
|
386 Hence the distinction between [bool] and [Prop]. Programs of type [bool] are computational by construction; we can always run them to determine their results. Many [Prop]s are undecidable, and so we can write more expressive formulas with [Prop]s than with [bool]s, but the inevitable consequence is that we cannot simply %``%#"#run a [Prop] to determine its truth.#"#%''%
|
|
adamc@47
|
387
|
|
adam@322
|
388 Constructive logic lets us define all of the logical connectives in an aesthetically-appealing way, with orthogonal inductive definitions. That is, each connective is defined independently using a simple, shared mechanism. Constructivity also enables a trick called %\index{program extraction}\textit{%#<i>#program extraction#</i>#%}%, where we write programs by phrasing them as theorems to be proved. Since our proofs are just functional programs, we can extract executable programs from our final proofs, which we could not do as naturally with classical proofs.
|
|
adamc@47
|
389
|
|
adamc@47
|
390 We will see more about Coq's program extraction facility in a later chapter. However, I think it is worth interjecting another warning at this point, following up on the prior warning about taking the Curry-Howard correspondence too literally. It is possible to write programs by theorem-proving methods in Coq, but hardly anyone does it. It is almost always most useful to maintain the distinction between programs and proofs. If you write a program by proving a theorem, you are likely to run into algorithmic inefficiencies that you introduced in your proof to make it easier to prove. It is a shame to have to worry about such situations while proving tricky theorems, and it is a happy state of affairs that you almost certainly will not need to, with the ideal of extracting programs from proofs being confined mostly to theoretical studies. *)
|
|
adamc@48
|
391
|
|
adamc@48
|
392
|
|
adamc@48
|
393 (** * First-Order Logic *)
|
|
adamc@48
|
394
|
|
adam@322
|
395 (** The %\index{Gallina terms!forall}%[forall] connective of first-order logic, which we have seen in many examples so far, is built into Coq. Getting ahead of ourselves a bit, we can see it as the dependent function type constructor. In fact, implication and universal quantification are just different syntactic shorthands for the same Coq mechanism. A formula [P -> Q] is equivalent to [forall x : P, Q], where [x] does not appear in [Q]. That is, the %``%#"#real#"#%''% type of the implication says %``%#"#for every proof of [P], there exists a proof of [Q].#"#%''%
|
|
adamc@48
|
396
|
|
adam@322
|
397 %\index{existential quantification}\index{Gallina terms!exists}\index{Gallina terms!ex}%Existential quantification is defined in the standard library. *)
|
|
adamc@48
|
398
|
|
adam@322
|
399 (* begin hide *)
|
|
adam@322
|
400 Print ex.
|
|
adam@322
|
401 (* end hide *)
|
|
adam@322
|
402 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#ex#</tt>#%}%[.]
|
|
adam@322
|
403 [[
|
|
adamc@209
|
404 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
|
|
adamc@209
|
405 ex_intro : forall x : A, P x -> ex P
|
|
adamc@209
|
406
|
|
adamc@209
|
407 ]]
|
|
adamc@48
|
408
|
|
adam@322
|
409 The family [ex] is parameterized by the type [A] that we quantify over, and by a predicate [P] over [A]s. We prove an existential by exhibiting some [x] of type [A], along with a proof of [P x]. As usual, there are tactics that save us from worrying about the low-level details most of the time. We use the equality operator [=], which, depending on the settings in which they learned logic, different people will say either is or is not part of first-order logic. For our purposes, it is. *)
|
|
adamc@48
|
410
|
|
adamc@48
|
411 Theorem exist1 : exists x : nat, x + 1 = 2.
|
|
adamc@55
|
412 (* begin thide *)
|
|
adamc@67
|
413 (** remove printing exists *)
|
|
adam@322
|
414 (** We can start this proof with a tactic %\index{tactics!exists}\coqdockw{%exists%}%, which should not be confused with the formula constructor shorthand of the same name. (In the PDF version of this document, the reverse %`%#'#E#'#%'% appears instead of the text %``%#"#exists#"#%''% in formulas.) *)
|
|
adamc@209
|
415
|
|
adam@322
|
416 (* begin hide *)
|
|
adamc@48
|
417 exists 1.
|
|
adam@322
|
418 (* end hide *)
|
|
adam@322
|
419 (** %\coqdockw{%#<tt>#exists#</tt>#%}% [1.] *)
|
|
adamc@48
|
420
|
|
adamc@209
|
421 (** The conclusion is replaced with a version using the existential witness that we announced.
|
|
adamc@48
|
422
|
|
adamc@209
|
423 [[
|
|
adamc@48
|
424 ============================
|
|
adamc@48
|
425 1 + 1 = 2
|
|
adam@302
|
426 ]]
|
|
adam@302
|
427 *)
|
|
adamc@48
|
428
|
|
adamc@48
|
429 reflexivity.
|
|
adamc@55
|
430 (* end thide *)
|
|
adamc@48
|
431 Qed.
|
|
adamc@48
|
432
|
|
adamc@48
|
433 (** printing exists $\exists$ *)
|
|
adamc@48
|
434
|
|
adamc@48
|
435 (** We can also use tactics to reason about existential hypotheses. *)
|
|
adamc@48
|
436
|
|
adamc@48
|
437 Theorem exist2 : forall n m : nat, (exists x : nat, n + x = m) -> n <= m.
|
|
adamc@55
|
438 (* begin thide *)
|
|
adamc@48
|
439 (** We start by case analysis on the proof of the existential fact. *)
|
|
adamc@209
|
440
|
|
adamc@48
|
441 destruct 1.
|
|
adamc@48
|
442 (** [[
|
|
adamc@48
|
443 n : nat
|
|
adamc@48
|
444 m : nat
|
|
adamc@48
|
445 x : nat
|
|
adamc@48
|
446 H : n + x = m
|
|
adamc@48
|
447 ============================
|
|
adamc@48
|
448 n <= m
|
|
adamc@209
|
449
|
|
adamc@209
|
450 ]]
|
|
adamc@48
|
451
|
|
adamc@209
|
452 The goal has been replaced by a form where there is a new free variable [x], and where we have a new hypothesis that the body of the existential holds with [x] substituted for the old bound variable. From here, the proof is just about arithmetic and is easy to automate. *)
|
|
adamc@48
|
453
|
|
adamc@48
|
454 crush.
|
|
adamc@55
|
455 (* end thide *)
|
|
adamc@48
|
456 Qed.
|
|
adamc@48
|
457
|
|
adamc@48
|
458
|
|
adamc@48
|
459 (* begin hide *)
|
|
adamc@48
|
460 (* In-class exercises *)
|
|
adamc@48
|
461
|
|
adamc@48
|
462 Theorem forall_exists_commute : forall (A B : Type) (P : A -> B -> Prop),
|
|
adamc@48
|
463 (exists x : A, forall y : B, P x y) -> (forall y : B, exists x : A, P x y).
|
|
adamc@52
|
464 (* begin thide *)
|
|
adamc@52
|
465 intros.
|
|
adamc@52
|
466 destruct H.
|
|
adamc@52
|
467 exists x.
|
|
adamc@52
|
468 apply H.
|
|
adamc@52
|
469 (* end thide *)
|
|
adamc@48
|
470 Admitted.
|
|
adamc@48
|
471
|
|
adamc@48
|
472 (* end hide *)
|
|
adamc@48
|
473
|
|
adamc@48
|
474
|
|
adam@322
|
475 (** The tactic [intuition] has a first-order cousin called %\index{tactics!firstorder}%[firstorder], which proves many formulas when only first-order reasoning is needed, and it tries to perform first-order simplifications in any case. First-order reasoning is much harder than propositional reasoning, so [firstorder] is much more likely than [intuition] to get stuck in a way that makes it run for long enough to be useless. *)
|
|
adamc@49
|
476
|
|
adamc@49
|
477
|
|
adamc@49
|
478 (** * Predicates with Implicit Equality *)
|
|
adamc@49
|
479
|
|
adamc@49
|
480 (** We start our exploration of a more complicated class of predicates with a simple example: an alternative way of characterizing when a natural number is zero. *)
|
|
adamc@49
|
481
|
|
adamc@49
|
482 Inductive isZero : nat -> Prop :=
|
|
adamc@49
|
483 | IsZero : isZero 0.
|
|
adamc@49
|
484
|
|
adamc@49
|
485 Theorem isZero_zero : isZero 0.
|
|
adamc@55
|
486 (* begin thide *)
|
|
adam@322
|
487 (* begin hide *)
|
|
adamc@49
|
488 constructor.
|
|
adam@322
|
489 (* end hide *)
|
|
adam@322
|
490 (** %\coqdockw{%#<tt>#constructor#</tt>#%}%[.]%\vspace{-.075in}% *)
|
|
adam@322
|
491
|
|
adamc@55
|
492 (* end thide *)
|
|
adamc@49
|
493 Qed.
|
|
adamc@49
|
494
|
|
adam@322
|
495 (** We can call [isZero] a %\index{judgment}\textit{%#<i>#judgment#</i>#%}%, in the sense often used in the semantics of programming languages. Judgments are typically defined in the style of %\index{natural deduction}\textit{%#<i>#natural deduction#</i>#%}%, where we write a number of %\index{inference rules}\textit{%#<i>#inference rules#</i>#%}% with premises appearing above a solid line and a conclusion appearing below the line. In this example, the sole constructor [IsZero] of [isZero] can be thought of as the single inference rule for deducing [isZero], with nothing above the line and [isZero 0] below it. The proof of [isZero_zero] demonstrates how we can apply an inference rule.
|
|
adamc@49
|
496
|
|
adamc@49
|
497 The definition of [isZero] differs in an important way from all of the other inductive definitions that we have seen in this and the previous chapter. Instead of writing just [Set] or [Prop] after the colon, here we write [nat -> Prop]. We saw examples of parameterized types like [list], but there the parameters appeared with names %\textit{%#<i>#before#</i>#%}% the colon. Every constructor of a parameterized inductive type must have a range type that uses the same parameter, whereas the form we use here enables us to use different arguments to the type for different constructors.
|
|
adamc@49
|
498
|
|
adam@322
|
499 For instance, our definition [isZero] makes the predicate provable only when the argument is [0]. We can see that the concept of equality is somehow implicit in the inductive definition mechanism. The way this is accomplished is similar to the way that logic variables are used in %\index{Prolog}%Prolog, and it is a very powerful mechanism that forms a foundation for formalizing all of mathematics. In fact, though it is natural to think of inductive types as folding in the functionality of equality, in Coq, the true situation is reversed, with equality defined as just another inductive type!%\index{Gallina terms!eq}\index{Gallina terms!refl\_equal}% *)
|
|
adamc@49
|
500
|
|
adam@322
|
501 (* begin hide *)
|
|
adamc@49
|
502 Print eq.
|
|
adam@322
|
503 (* end hide *)
|
|
adam@322
|
504 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#eq#</tt>#%}%[.]
|
|
adam@322
|
505 [[
|
|
adamc@209
|
506 Inductive eq (A : Type) (x : A) : A -> Prop := refl_equal : x = x
|
|
adamc@209
|
507
|
|
adamc@209
|
508 ]]
|
|
adamc@49
|
509
|
|
adam@322
|
510 Behind the scenes, uses of infix [=] are expanded to instances of [eq]. We see that [eq] has both a parameter [x] that is fixed and an extra unnamed argument of the same type. The type of [eq] allows us to state any equalities, even those that are provably false. However, examining the type of equality's sole constructor [refl_equal], we see that we can only %\textit{%#<i>#prove#</i>#%}% equality when its two arguments are syntactically equal. This definition turns out to capture all of the basic properties of equality, and the equality-manipulating tactics that we have seen so far, like [reflexivity] and [rewrite], are implemented treating [eq] as just another inductive type with a well-chosen definition. Another way of stating that definition is: equality is defined as the least reflexive relation.
|
|
adamc@49
|
511
|
|
adam@322
|
512 Returning to the example of [isZero], we can see how to work with hypotheses that use this predicate. *)
|
|
adamc@49
|
513
|
|
adamc@49
|
514 Theorem isZero_plus : forall n m : nat, isZero m -> n + m = n.
|
|
adamc@55
|
515 (* begin thide *)
|
|
adamc@49
|
516 (** We want to proceed by cases on the proof of the assumption about [isZero]. *)
|
|
adamc@209
|
517
|
|
adamc@49
|
518 destruct 1.
|
|
adamc@49
|
519 (** [[
|
|
adamc@49
|
520 n : nat
|
|
adamc@49
|
521 ============================
|
|
adamc@49
|
522 n + 0 = n
|
|
adamc@209
|
523
|
|
adamc@209
|
524 ]]
|
|
adamc@49
|
525
|
|
adamc@209
|
526 Since [isZero] has only one constructor, we are presented with only one subgoal. The argument [m] to [isZero] is replaced with that type's argument from the single constructor [IsZero]. From this point, the proof is trivial. *)
|
|
adamc@49
|
527
|
|
adamc@49
|
528 crush.
|
|
adamc@55
|
529 (* end thide *)
|
|
adamc@49
|
530 Qed.
|
|
adamc@49
|
531
|
|
adamc@49
|
532 (** Another example seems at first like it should admit an analogous proof, but in fact provides a demonstration of one of the most basic gotchas of Coq proving. *)
|
|
adamc@49
|
533
|
|
adamc@49
|
534 Theorem isZero_contra : isZero 1 -> False.
|
|
adamc@55
|
535 (* begin thide *)
|
|
adamc@49
|
536 (** Let us try a proof by cases on the assumption, as in the last proof. *)
|
|
adamc@209
|
537
|
|
adamc@49
|
538 destruct 1.
|
|
adamc@49
|
539 (** [[
|
|
adamc@49
|
540 ============================
|
|
adamc@49
|
541 False
|
|
adamc@209
|
542
|
|
adamc@209
|
543 ]]
|
|
adamc@49
|
544
|
|
adamc@209
|
545 It seems that case analysis has not helped us much at all! Our sole hypothesis disappears, leaving us, if anything, worse off than we were before. What went wrong? We have met an important restriction in tactics like [destruct] and [induction] when applied to types with arguments. If the arguments are not already free variables, they will be replaced by new free variables internally before doing the case analysis or induction. Since the argument [1] to [isZero] is replaced by a fresh variable, we lose the crucial fact that it is not equal to [0].
|
|
adamc@49
|
546
|
|
adam@322
|
547 Why does Coq use this restriction? We will discuss the issue in detail in a future chapter, when we see the dependently typed programming techniques that would allow us to write this proof term manually. For now, we just say that the algorithmic problem of %``%#"#logically complete case analysis#"#%''% is undecidable when phrased in Coq's logic. A few tactics and design patterns that we will present in this chapter suffice in almost all cases. For the current example, what we want is a tactic called %\index{tactics!inversion}%[inversion], which corresponds to the concept of inversion that is frequently used with natural deduction proof systems. *)
|
|
adamc@49
|
548
|
|
adamc@49
|
549 Undo.
|
|
adamc@49
|
550 inversion 1.
|
|
adamc@55
|
551 (* end thide *)
|
|
adamc@49
|
552 Qed.
|
|
adamc@49
|
553
|
|
adamc@49
|
554 (** What does [inversion] do? Think of it as a version of [destruct] that does its best to take advantage of the structure of arguments to inductive types. In this case, [inversion] completed the proof immediately, because it was able to detect that we were using [isZero] with an impossible argument.
|
|
adamc@49
|
555
|
|
adamc@49
|
556 Sometimes using [destruct] when you should have used [inversion] can lead to confusing results. To illustrate, consider an alternate proof attempt for the last theorem. *)
|
|
adamc@49
|
557
|
|
adamc@49
|
558 Theorem isZero_contra' : isZero 1 -> 2 + 2 = 5.
|
|
adamc@49
|
559 destruct 1.
|
|
adamc@49
|
560 (** [[
|
|
adamc@49
|
561 ============================
|
|
adamc@49
|
562 1 + 1 = 4
|
|
adamc@209
|
563
|
|
adamc@209
|
564 ]]
|
|
adamc@49
|
565
|
|
adam@280
|
566 What on earth happened here? Internally, [destruct] replaced [1] with a fresh variable, and, trying to be helpful, it also replaced the occurrence of [1] within the unary representation of each number in the goal. This has the net effect of decrementing each of these numbers. *)
|
|
adamc@209
|
567
|
|
adamc@49
|
568 Abort.
|
|
adamc@49
|
569
|
|
adam@280
|
570 (** To see more clearly what is happening, we can consider the type of [isZero]'s induction principle. *)
|
|
adam@280
|
571
|
|
adam@280
|
572 Check isZero_ind.
|
|
adam@280
|
573 (** %\vspace{-.15in}% [[
|
|
adam@280
|
574 isZero_ind
|
|
adam@280
|
575 : forall P : nat -> Prop, P 0 -> forall n : nat, isZero n -> P n
|
|
adam@280
|
576
|
|
adam@280
|
577 ]]
|
|
adam@280
|
578
|
|
adam@322
|
579 In our last proof script, [destruct] chose to instantiate [P] as [fun n => S n + S n = S (][S (][S (][S n)))]. You can verify for yourself that this specialization of the principle applies to the goal and that the hypothesis [P 0] then matches the subgoal we saw generated. If you are doing a proof and encounter a strange transmutation like this, there is a good chance that you should go back and replace a use of [destruct] with [inversion]. *)
|
|
adam@280
|
580
|
|
adamc@49
|
581
|
|
adamc@49
|
582 (* begin hide *)
|
|
adamc@49
|
583 (* In-class exercises *)
|
|
adamc@49
|
584
|
|
adamc@49
|
585 (* EX: Define an inductive type capturing when a list has exactly two elements. Prove that your predicate does not hold of the empty list, and prove that, whenever it holds of a list, the length of that list is two. *)
|
|
adamc@49
|
586
|
|
adamc@52
|
587 (* begin thide *)
|
|
adamc@52
|
588 Section twoEls.
|
|
adamc@52
|
589 Variable A : Type.
|
|
adamc@52
|
590
|
|
adamc@52
|
591 Inductive twoEls : list A -> Prop :=
|
|
adamc@52
|
592 | TwoEls : forall x y, twoEls (x :: y :: nil).
|
|
adamc@52
|
593
|
|
adamc@52
|
594 Theorem twoEls_nil : twoEls nil -> False.
|
|
adamc@52
|
595 inversion 1.
|
|
adamc@52
|
596 Qed.
|
|
adamc@52
|
597
|
|
adamc@52
|
598 Theorem twoEls_two : forall ls, twoEls ls -> length ls = 2.
|
|
adamc@52
|
599 inversion 1.
|
|
adamc@52
|
600 reflexivity.
|
|
adamc@52
|
601 Qed.
|
|
adamc@52
|
602 End twoEls.
|
|
adamc@52
|
603 (* end thide *)
|
|
adamc@52
|
604
|
|
adamc@49
|
605 (* end hide *)
|
|
adamc@49
|
606
|
|
adamc@50
|
607
|
|
adamc@50
|
608 (** * Recursive Predicates *)
|
|
adamc@50
|
609
|
|
adamc@50
|
610 (** We have already seen all of the ingredients we need to build interesting recursive predicates, like this predicate capturing even-ness. *)
|
|
adamc@50
|
611
|
|
adamc@50
|
612 Inductive even : nat -> Prop :=
|
|
adamc@50
|
613 | EvenO : even O
|
|
adamc@50
|
614 | EvenSS : forall n, even n -> even (S (S n)).
|
|
adamc@50
|
615
|
|
adam@322
|
616 (** Think of [even] as another judgment defined by natural deduction rules. [EvenO] is a rule with nothing above the line and [even O] below the line, and [EvenSS] is a rule with [even n] above the line and [even (][S (][S n))] below.
|
|
adamc@50
|
617
|
|
adamc@50
|
618 The proof techniques of the last section are easily adapted. *)
|
|
adamc@50
|
619
|
|
adamc@50
|
620 Theorem even_0 : even 0.
|
|
adamc@55
|
621 (* begin thide *)
|
|
adam@322
|
622 (* begin hide *)
|
|
adamc@50
|
623 constructor.
|
|
adam@322
|
624 (* end hide *)
|
|
adam@322
|
625 (** %\coqdockw{%#<tt>#constructor#</tt>#%}%[.]%\vspace{-.075in}% *)
|
|
adam@322
|
626
|
|
adamc@55
|
627 (* end thide *)
|
|
adamc@50
|
628 Qed.
|
|
adamc@50
|
629
|
|
adamc@50
|
630 Theorem even_4 : even 4.
|
|
adamc@55
|
631 (* begin thide *)
|
|
adam@322
|
632 (* begin hide *)
|
|
adamc@50
|
633 constructor; constructor; constructor.
|
|
adam@322
|
634 (* end hide *)
|
|
adam@322
|
635 (** %\coqdockw{%#<tt>#constructor#</tt>#%}%[; ]%\coqdockw{%#<tt>#constructor#</tt>#%}%[; ]%\coqdockw{%#<tt>#constructor#</tt>#%}%[.]%\vspace{-.075in}% *)
|
|
adam@322
|
636
|
|
adamc@55
|
637 (* end thide *)
|
|
adamc@50
|
638 Qed.
|
|
adamc@50
|
639
|
|
adam@375
|
640 (** It is not hard to see that sequences of constructor applications like the above can get tedious. We can avoid them using Coq's hint facility, with a new [Hint] variant that asks to consider all constructors of an inductive type during proof search. The tactic %\index{tactics!auto}%[auto] performs exhaustive proof search up to a fixed depth, considering only the proof steps we have registered as hints. *)
|
|
adamc@50
|
641
|
|
adamc@55
|
642 (* begin thide *)
|
|
adam@322
|
643 (* begin hide *)
|
|
adamc@50
|
644 Hint Constructors even.
|
|
adam@322
|
645 (* end hide *)
|
|
adam@322
|
646 (** %\noindent%[Hint] %\coqdockw{%#<tt>#Constructors#</tt>#%}% [even.] *)
|
|
adamc@50
|
647
|
|
adamc@50
|
648 Theorem even_4' : even 4.
|
|
adamc@50
|
649 auto.
|
|
adamc@50
|
650 Qed.
|
|
adamc@50
|
651
|
|
adamc@55
|
652 (* end thide *)
|
|
adamc@55
|
653
|
|
adam@322
|
654 (** We may also use [inversion] with [even]. *)
|
|
adam@322
|
655
|
|
adamc@50
|
656 Theorem even_1_contra : even 1 -> False.
|
|
adamc@55
|
657 (* begin thide *)
|
|
adamc@50
|
658 inversion 1.
|
|
adamc@55
|
659 (* end thide *)
|
|
adamc@50
|
660 Qed.
|
|
adamc@50
|
661
|
|
adamc@50
|
662 Theorem even_3_contra : even 3 -> False.
|
|
adamc@55
|
663 (* begin thide *)
|
|
adamc@50
|
664 inversion 1.
|
|
adamc@50
|
665 (** [[
|
|
adamc@50
|
666 H : even 3
|
|
adamc@50
|
667 n : nat
|
|
adamc@50
|
668 H1 : even 1
|
|
adamc@50
|
669 H0 : n = 1
|
|
adamc@50
|
670 ============================
|
|
adamc@50
|
671 False
|
|
adamc@209
|
672
|
|
adamc@209
|
673 ]]
|
|
adamc@50
|
674
|
|
adam@322
|
675 The [inversion] tactic can be a little overzealous at times, as we can see here with the introduction of the unused variable [n] and an equality hypothesis about it. For more complicated predicates, though, adding such assumptions is critical to dealing with the undecidability of general inversion. More complex inductive definitions and theorems can cause [inversion] to generate equalities where neither side is a variable. *)
|
|
adamc@50
|
676
|
|
adamc@50
|
677 inversion H1.
|
|
adamc@55
|
678 (* end thide *)
|
|
adamc@50
|
679 Qed.
|
|
adamc@50
|
680
|
|
adamc@50
|
681 (** We can also do inductive proofs about [even]. *)
|
|
adamc@50
|
682
|
|
adamc@50
|
683 Theorem even_plus : forall n m, even n -> even m -> even (n + m).
|
|
adamc@55
|
684 (* begin thide *)
|
|
adamc@50
|
685 (** It seems a reasonable first choice to proceed by induction on [n]. *)
|
|
adamc@209
|
686
|
|
adamc@50
|
687 induction n; crush.
|
|
adamc@50
|
688 (** [[
|
|
adamc@50
|
689 n : nat
|
|
adamc@50
|
690 IHn : forall m : nat, even n -> even m -> even (n + m)
|
|
adamc@50
|
691 m : nat
|
|
adamc@50
|
692 H : even (S n)
|
|
adamc@50
|
693 H0 : even m
|
|
adamc@50
|
694 ============================
|
|
adamc@50
|
695 even (S (n + m))
|
|
adamc@209
|
696
|
|
adamc@209
|
697 ]]
|
|
adamc@50
|
698
|
|
adamc@209
|
699 We will need to use the hypotheses [H] and [H0] somehow. The most natural choice is to invert [H]. *)
|
|
adamc@50
|
700
|
|
adamc@50
|
701 inversion H.
|
|
adamc@50
|
702 (** [[
|
|
adamc@50
|
703 n : nat
|
|
adamc@50
|
704 IHn : forall m : nat, even n -> even m -> even (n + m)
|
|
adamc@50
|
705 m : nat
|
|
adamc@50
|
706 H : even (S n)
|
|
adamc@50
|
707 H0 : even m
|
|
adamc@50
|
708 n0 : nat
|
|
adamc@50
|
709 H2 : even n0
|
|
adamc@50
|
710 H1 : S n0 = n
|
|
adamc@50
|
711 ============================
|
|
adamc@50
|
712 even (S (S n0 + m))
|
|
adamc@209
|
713
|
|
adamc@209
|
714 ]]
|
|
adamc@50
|
715
|
|
adamc@209
|
716 Simplifying the conclusion brings us to a point where we can apply a constructor. *)
|
|
adamc@209
|
717
|
|
adamc@50
|
718 simpl.
|
|
adamc@50
|
719 (** [[
|
|
adamc@50
|
720 ============================
|
|
adamc@50
|
721 even (S (S (n0 + m)))
|
|
adam@302
|
722 ]]
|
|
adam@302
|
723 *)
|
|
adamc@50
|
724
|
|
adam@322
|
725 (* begin hide *)
|
|
adamc@50
|
726 constructor.
|
|
adam@322
|
727 (* end hide *)
|
|
adam@322
|
728 (** %\coqdockw{%#<tt>#constructor#</tt>#%}%[.]
|
|
adam@322
|
729
|
|
adam@322
|
730 [[
|
|
adamc@50
|
731 ============================
|
|
adamc@50
|
732 even (n0 + m)
|
|
adamc@209
|
733
|
|
adamc@209
|
734 ]]
|
|
adamc@50
|
735
|
|
adamc@209
|
736 At this point, we would like to apply the inductive hypothesis, which is:
|
|
adamc@209
|
737
|
|
adamc@209
|
738 [[
|
|
adamc@50
|
739 IHn : forall m : nat, even n -> even m -> even (n + m)
|
|
adamc@209
|
740 ]]
|
|
adamc@50
|
741
|
|
adam@322
|
742 Unfortunately, the goal mentions [n0] where it would need to mention [n] to match [IHn]. We could keep looking for a way to finish this proof from here, but it turns out that we can make our lives much easier by changing our basic strategy. Instead of inducting on the structure of [n], we should induct %\textit{%#<i>#on the structure of one of the [even] proofs#</i>#%}%. This technique is commonly called %\index{rule induction}\textit{%#<i>#rule induction#</i>#%}% in programming language semantics. In the setting of Coq, we have already seen how predicates are defined using the same inductive type mechanism as datatypes, so the fundamental unity of rule induction with %``%#"#normal#"#%''% induction is apparent.
|
|
adamc@50
|
743
|
|
adam@322
|
744 Recall that tactics like [induction] and [destruct] may be passed numbers to refer to unnamed lefthand sides of implications in the conclusion, where the argument [n] refers to the [n]th such hypothesis. *)
|
|
adam@322
|
745
|
|
adam@322
|
746 (* begin hide *)
|
|
adamc@50
|
747 Restart.
|
|
adam@322
|
748 (* end hide *)
|
|
adam@322
|
749 (** %\noindent\coqdockw{%#<tt>#Restart#</tt>#%}%[.] *)
|
|
adamc@50
|
750
|
|
adamc@50
|
751 induction 1.
|
|
adamc@50
|
752 (** [[
|
|
adamc@50
|
753 m : nat
|
|
adamc@50
|
754 ============================
|
|
adamc@50
|
755 even m -> even (0 + m)
|
|
adam@322
|
756 ]]
|
|
adamc@50
|
757
|
|
adam@322
|
758 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
|
|
adam@322
|
759 [[
|
|
adamc@50
|
760 even m -> even (S (S n) + m)
|
|
adamc@209
|
761
|
|
adamc@209
|
762 ]]
|
|
adamc@50
|
763
|
|
adamc@209
|
764 The first case is easily discharged by [crush], based on the hint we added earlier to try the constructors of [even]. *)
|
|
adamc@50
|
765
|
|
adamc@50
|
766 crush.
|
|
adamc@50
|
767
|
|
adamc@50
|
768 (** Now we focus on the second case: *)
|
|
adamc@209
|
769
|
|
adamc@50
|
770 intro.
|
|
adamc@50
|
771 (** [[
|
|
adamc@50
|
772 m : nat
|
|
adamc@50
|
773 n : nat
|
|
adamc@50
|
774 H : even n
|
|
adamc@50
|
775 IHeven : even m -> even (n + m)
|
|
adamc@50
|
776 H0 : even m
|
|
adamc@50
|
777 ============================
|
|
adamc@50
|
778 even (S (S n) + m)
|
|
adamc@209
|
779
|
|
adamc@209
|
780 ]]
|
|
adamc@50
|
781
|
|
adamc@209
|
782 We simplify and apply a constructor, as in our last proof attempt. *)
|
|
adamc@50
|
783
|
|
adam@322
|
784 (* begin hide *)
|
|
adamc@50
|
785 simpl; constructor.
|
|
adam@322
|
786 (* end hide *)
|
|
adam@322
|
787 (** [simpl; ]%\coqdockw{%#<tt>#constructor#</tt>#%}%[.]
|
|
adam@322
|
788
|
|
adam@322
|
789 [[
|
|
adamc@50
|
790 ============================
|
|
adamc@50
|
791 even (n + m)
|
|
adamc@209
|
792
|
|
adamc@209
|
793 ]]
|
|
adamc@50
|
794
|
|
adamc@209
|
795 Now we have an exact match with our inductive hypothesis, and the remainder of the proof is trivial. *)
|
|
adamc@50
|
796
|
|
adamc@50
|
797 apply IHeven; assumption.
|
|
adamc@50
|
798
|
|
adamc@50
|
799 (** In fact, [crush] can handle all of the details of the proof once we declare the induction strategy. *)
|
|
adamc@50
|
800
|
|
adam@322
|
801 (* begin hide *)
|
|
adamc@50
|
802 Restart.
|
|
adam@322
|
803 (* end hide *)
|
|
adam@322
|
804 (** %\noindent\coqdockw{%#<tt>#Restart#</tt>#%}%[.] *)
|
|
adam@322
|
805
|
|
adamc@50
|
806 induction 1; crush.
|
|
adamc@55
|
807 (* end thide *)
|
|
adamc@50
|
808 Qed.
|
|
adamc@50
|
809
|
|
adamc@50
|
810 (** Induction on recursive predicates has similar pitfalls to those we encountered with inversion in the last section. *)
|
|
adamc@50
|
811
|
|
adamc@50
|
812 Theorem even_contra : forall n, even (S (n + n)) -> False.
|
|
adamc@55
|
813 (* begin thide *)
|
|
adamc@50
|
814 induction 1.
|
|
adamc@50
|
815 (** [[
|
|
adamc@50
|
816 n : nat
|
|
adamc@50
|
817 ============================
|
|
adamc@50
|
818 False
|
|
adam@322
|
819 ]]
|
|
adamc@50
|
820
|
|
adam@322
|
821 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
|
|
adam@322
|
822 [[
|
|
adamc@50
|
823 False
|
|
adamc@209
|
824
|
|
adamc@209
|
825 ]]
|
|
adamc@50
|
826
|
|
adam@280
|
827 We are already sunk trying to prove the first subgoal, since the argument to [even] was replaced by a fresh variable internally. This time, we find it easier to prove this theorem by way of a lemma. Instead of trusting [induction] to replace expressions with fresh variables, we do it ourselves, explicitly adding the appropriate equalities as new assumptions. *)
|
|
adamc@209
|
828
|
|
adamc@50
|
829 Abort.
|
|
adamc@50
|
830
|
|
adamc@50
|
831 Lemma even_contra' : forall n', even n' -> forall n, n' = S (n + n) -> False.
|
|
adamc@50
|
832 induction 1; crush.
|
|
adamc@50
|
833
|
|
adamc@54
|
834 (** At this point, it is useful to consider all cases of [n] and [n0] being zero or nonzero. Only one of these cases has any trickiness to it. *)
|
|
adamc@209
|
835
|
|
adamc@50
|
836 destruct n; destruct n0; crush.
|
|
adamc@50
|
837
|
|
adamc@50
|
838 (** [[
|
|
adamc@50
|
839 n : nat
|
|
adamc@50
|
840 H : even (S n)
|
|
adamc@50
|
841 IHeven : forall n0 : nat, S n = S (n0 + n0) -> False
|
|
adamc@50
|
842 n0 : nat
|
|
adamc@50
|
843 H0 : S n = n0 + S n0
|
|
adamc@50
|
844 ============================
|
|
adamc@50
|
845 False
|
|
adamc@209
|
846
|
|
adamc@209
|
847 ]]
|
|
adamc@50
|
848
|
|
adam@280
|
849 At this point it is useful to use a theorem from the standard library, which we also proved with a different name in the last chapter. We can search for a theorem that allows us to rewrite terms of the form [x + S y]. *)
|
|
adamc@209
|
850
|
|
adam@322
|
851 (* begin hide *)
|
|
adam@280
|
852 SearchRewrite (_ + S _).
|
|
adam@322
|
853 (* end hide *)
|
|
adam@322
|
854 (** %\coqdockw{%#<tt>#SearchRewrite#</tt>#%}% [(_ + S _).]
|
|
adam@322
|
855
|
|
adam@322
|
856 [[
|
|
adam@280
|
857 plus_n_Sm : forall n m : nat, S (n + m) = n + S m
|
|
adam@302
|
858 ]]
|
|
adam@302
|
859 *)
|
|
adamc@50
|
860
|
|
adamc@50
|
861 rewrite <- plus_n_Sm in H0.
|
|
adamc@50
|
862
|
|
adam@322
|
863 (** The induction hypothesis lets us complete the proof, if we use a variant of [apply] that has a %\index{tactics!with}%[with] clause to give instantiations of quantified variables. *)
|
|
adamc@209
|
864
|
|
adamc@50
|
865 apply IHeven with n0; assumption.
|
|
adamc@50
|
866
|
|
adam@322
|
867 (** As usual, we can rewrite the proof to avoid referencing any locally generated names, which makes our proof script more readable and more robust to changes in the theorem statement. We use the notation [<-] to request a hint that does right-to-left rewriting, just like we can with the [rewrite] tactic. *)
|
|
adamc@209
|
868
|
|
adam@322
|
869 (* begin hide *)
|
|
adamc@209
|
870 Restart.
|
|
adam@322
|
871 (* end hide *)
|
|
adam@322
|
872 (** %\hspace{-.075in}\coqdockw{%#<tt>#Restart#</tt>#%}%[.] *)
|
|
adam@322
|
873
|
|
adam@322
|
874 (* begin hide *)
|
|
adam@375
|
875 Hint Rewrite <- plus_n_Sm.
|
|
adam@322
|
876 (* end hide *)
|
|
adam@375
|
877 (** %\hspace{-.075in}%[Hint] %\noindent\coqdockw{%#<tt>#Rewrite#</tt>#%}% [<- plus_n_sm.] *)
|
|
adamc@50
|
878
|
|
adamc@50
|
879 induction 1; crush;
|
|
adamc@50
|
880 match goal with
|
|
adamc@50
|
881 | [ H : S ?N = ?N0 + ?N0 |- _ ] => destruct N; destruct N0
|
|
adamc@50
|
882 end; crush; eauto.
|
|
adamc@50
|
883 Qed.
|
|
adamc@50
|
884
|
|
adam@322
|
885 (** We write the proof in a way that avoids the use of local variable or hypothesis names, using the %\index{tactics!match}%[match] tactic form to do pattern-matching on the goal. We use unification variables prefixed by question marks in the pattern, and we take advantage of the possibility to mention a unification variable twice in one pattern, to enforce equality between occurrences. The hint to rewrite with [plus_n_Sm] in a particular direction saves us from having to figure out the right place to apply that theorem, and we also take critical advantage of a new tactic, %\index{tactics!eauto}%[eauto].
|
|
adamc@50
|
886
|
|
adam@322
|
887 The [crush] tactic uses the tactic [intuition], which, when it runs out of tricks to try using only propositional logic, by default tries the tactic [auto], which we saw in an earlier example. The [auto] tactic attempts %\index{Prolog}%Prolog-style logic programming, searching through all proof trees up to a certain depth that are built only out of hints that have been registered with [Hint] commands. Compared to Prolog, [auto] places an important restriction: it never introduces new unification variables during search. That is, every time a rule is applied during proof search, all of its arguments must be deducible by studying the form of the goal. This restriction is relaxed for [eauto], at the cost of possibly exponentially greater running time. In this particular case, we know that [eauto] has only a small space of proofs to search, so it makes sense to run it. It is common in effectively automated Coq proofs to see a bag of standard tactics applied to pick off the %``%#"#easy#"#%''% subgoals, finishing with [eauto] to handle the tricky parts that can benefit from ad-hoc exhaustive search.
|
|
adamc@50
|
888
|
|
adamc@50
|
889 The original theorem now follows trivially from our lemma. *)
|
|
adamc@50
|
890
|
|
adamc@50
|
891 Theorem even_contra : forall n, even (S (n + n)) -> False.
|
|
adamc@52
|
892 intros; eapply even_contra'; eauto.
|
|
adamc@50
|
893 Qed.
|
|
adamc@52
|
894
|
|
adam@322
|
895 (** We use a variant %\index{tactics!apply}%[eapply] of [apply] which has the same relationship to [apply] as [eauto] has to [auto]. An invocation of [apply] only succeeds if all arguments to the rule being used can be determined from the form of the goal, whereas [eapply] will introduce unification variables for undetermined arguments. In this case, [eauto] is able to determine the right values for those unification variables.
|
|
adamc@52
|
896
|
|
adamc@52
|
897 By considering an alternate attempt at proving the lemma, we can see another common pitfall of inductive proofs in Coq. Imagine that we had tried to prove [even_contra'] with all of the [forall] quantifiers moved to the front of the lemma statement. *)
|
|
adamc@52
|
898
|
|
adamc@52
|
899 Lemma even_contra'' : forall n' n, even n' -> n' = S (n + n) -> False.
|
|
adamc@52
|
900 induction 1; crush;
|
|
adamc@52
|
901 match goal with
|
|
adamc@52
|
902 | [ H : S ?N = ?N0 + ?N0 |- _ ] => destruct N; destruct N0
|
|
adamc@52
|
903 end; crush; eauto.
|
|
adamc@52
|
904
|
|
adamc@209
|
905 (** One subgoal remains:
|
|
adamc@52
|
906
|
|
adamc@209
|
907 [[
|
|
adamc@52
|
908 n : nat
|
|
adamc@52
|
909 H : even (S (n + n))
|
|
adamc@52
|
910 IHeven : S (n + n) = S (S (S (n + n))) -> False
|
|
adamc@52
|
911 ============================
|
|
adamc@52
|
912 False
|
|
adamc@209
|
913
|
|
adamc@209
|
914 ]]
|
|
adamc@52
|
915
|
|
adam@322
|
916 We are out of luck here. The inductive hypothesis is trivially true, since its assumption is false. In the version of this proof that succeeded, [IHeven] had an explicit quantification over [n]. This is because the quantification of [n] %\textit{%#<i>#appeared after the thing we are inducting on#</i>#%}% in the theorem statement. In general, quantified variables and hypotheses that appear before the induction object in the theorem statement stay fixed throughout the inductive proof. Variables and hypotheses that are quantified after the induction object may be varied explicitly in uses of inductive hypotheses. *)
|
|
adamc@52
|
917
|
|
adam@322
|
918 Abort.
|
|
adam@322
|
919
|
|
adam@322
|
920 (** Why should Coq implement [induction] this way? One answer is that it avoids burdening this basic tactic with additional heuristic smarts, but that is not the whole picture. Imagine that [induction] analyzed dependencies among variables and reordered quantifiers to preserve as much freedom as possible in later uses of inductive hypotheses. This could make the inductive hypotheses more complex, which could in turn cause particular automation machinery to fail when it would have succeeded before. In general, we want to avoid quantifiers in our proofs whenever we can, and that goal is furthered by the refactoring that the [induction] tactic forces us to do. *)
|
|
adamc@55
|
921 (* end thide *)
|
|
adamc@209
|
922
|
|
adam@322
|
923
|
|
adamc@51
|
924
|
|
adamc@52
|
925
|
|
adamc@52
|
926 (* begin hide *)
|
|
adamc@52
|
927 (* In-class exercises *)
|
|
adamc@52
|
928
|
|
adam@292
|
929 (* EX: Define a type [prop] of simple boolean formulas made up only of truth, falsehood, binary conjunction, and binary disjunction. Define an inductive predicate [holds] that captures when [prop]s are valid, and define a predicate [falseFree] that captures when a [prop] does not contain the %``%#"#false#"#%''% formula. Prove that every false-free [prop] is valid. *)
|
|
adamc@52
|
930
|
|
adamc@52
|
931 (* begin thide *)
|
|
adamc@52
|
932 Inductive prop : Set :=
|
|
adamc@52
|
933 | Tru : prop
|
|
adamc@52
|
934 | Fals : prop
|
|
adamc@52
|
935 | And : prop -> prop -> prop
|
|
adamc@52
|
936 | Or : prop -> prop -> prop.
|
|
adamc@52
|
937
|
|
adamc@52
|
938 Inductive holds : prop -> Prop :=
|
|
adamc@52
|
939 | HTru : holds Tru
|
|
adamc@52
|
940 | HAnd : forall p1 p2, holds p1 -> holds p2 -> holds (And p1 p2)
|
|
adamc@52
|
941 | HOr1 : forall p1 p2, holds p1 -> holds (Or p1 p2)
|
|
adamc@52
|
942 | HOr2 : forall p1 p2, holds p2 -> holds (Or p1 p2).
|
|
adamc@52
|
943
|
|
adamc@52
|
944 Inductive falseFree : prop -> Prop :=
|
|
adamc@52
|
945 | FFTru : falseFree Tru
|
|
adamc@52
|
946 | FFAnd : forall p1 p2, falseFree p1 -> falseFree p2 -> falseFree (And p1 p2)
|
|
adamc@52
|
947 | FFNot : forall p1 p2, falseFree p1 -> falseFree p2 -> falseFree (Or p1 p2).
|
|
adamc@52
|
948
|
|
adamc@52
|
949 Hint Constructors holds.
|
|
adamc@52
|
950
|
|
adamc@52
|
951 Theorem falseFree_holds : forall p, falseFree p -> holds p.
|
|
adamc@52
|
952 induction 1; crush.
|
|
adamc@52
|
953 Qed.
|
|
adamc@52
|
954 (* end thide *)
|
|
adamc@52
|
955
|
|
adamc@52
|
956
|
|
adamc@52
|
957 (* EX: Define an inductive type [prop'] that is the same as [prop] but omits the possibility for falsehood. Define a proposition [holds'] for [prop'] that is analogous to [holds]. Define a function [propify] for translating [prop']s to [prop]s. Prove that, for any [prop'] [p], if [propify p] is valid, then so is [p]. *)
|
|
adamc@52
|
958
|
|
adamc@52
|
959 (* begin thide *)
|
|
adamc@52
|
960 Inductive prop' : Set :=
|
|
adamc@52
|
961 | Tru' : prop'
|
|
adamc@52
|
962 | And' : prop' -> prop' -> prop'
|
|
adamc@52
|
963 | Or' : prop' -> prop' -> prop'.
|
|
adamc@52
|
964
|
|
adamc@52
|
965 Inductive holds' : prop' -> Prop :=
|
|
adamc@52
|
966 | HTru' : holds' Tru'
|
|
adamc@52
|
967 | HAnd' : forall p1 p2, holds' p1 -> holds' p2 -> holds' (And' p1 p2)
|
|
adamc@52
|
968 | HOr1' : forall p1 p2, holds' p1 -> holds' (Or' p1 p2)
|
|
adamc@52
|
969 | HOr2' : forall p1 p2, holds' p2 -> holds' (Or' p1 p2).
|
|
adamc@52
|
970
|
|
adamc@52
|
971 Fixpoint propify (p : prop') : prop :=
|
|
adamc@52
|
972 match p with
|
|
adamc@52
|
973 | Tru' => Tru
|
|
adamc@52
|
974 | And' p1 p2 => And (propify p1) (propify p2)
|
|
adamc@52
|
975 | Or' p1 p2 => Or (propify p1) (propify p2)
|
|
adamc@52
|
976 end.
|
|
adamc@52
|
977
|
|
adamc@52
|
978 Hint Constructors holds'.
|
|
adamc@52
|
979
|
|
adamc@52
|
980 Lemma propify_holds' : forall p', holds p' -> forall p, p' = propify p -> holds' p.
|
|
adamc@52
|
981 induction 1; crush; destruct p; crush.
|
|
adamc@52
|
982 Qed.
|
|
adamc@52
|
983
|
|
adamc@52
|
984 Theorem propify_holds : forall p, holds (propify p) -> holds' p.
|
|
adamc@52
|
985 intros; eapply propify_holds'; eauto.
|
|
adamc@52
|
986 Qed.
|
|
adamc@52
|
987 (* end thide *)
|
|
adamc@52
|
988
|
|
adamc@52
|
989 (* end hide *)
|
