adamc@70
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1 (* Copyright (c) 2008, Adam Chlipala
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2 *
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3 * This work is licensed under a
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4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
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5 * Unported License.
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6 * The license text is available at:
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7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
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8 *)
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9
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10 (* begin hide *)
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11 Require Import List.
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12
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13 Require Import Tactics.
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14
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15 Set Implicit Arguments.
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16 (* end hide *)
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17
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18
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19 (** %\part{Programming with Dependent Types}
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20
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21 \chapter{Subset Types and Variations}% *)
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22
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23 (** So far, we have seen many examples of what we might call "classical program verification." We write programs, write their specifications, and then prove that the programs satisfy their specifications. The programs that we have written in Coq have been normal functional programs that we could just as well have written in Haskell or ML. In this chapter, we start investigating uses of %\textit{%#<i>#dependent types#</i>#%}% to integrate programming, specification, and proving into a single phase. *)
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24
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25
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26 (** * Introducing Subset Types *)
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27
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28 (** Let us consider several ways of implementing the natural number predecessor function. We start by displaying the definition from the standard library: *)
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29
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30 Print pred.
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31 (** [[
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32
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33 pred = fun n : nat => match n with
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34 | 0 => 0
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35 | S u => u
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36 end
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37 : nat -> nat
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38 ]] *)
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39
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40 (** We can use a new command, [Extraction], to produce an OCaml version of this function. *)
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41
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42 Extraction pred.
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43
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44 (** %\begin{verbatim}
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45 (** val pred : nat -> nat **)
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46
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47 let pred = function
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48 | O -> O
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49 | S u -> u
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50 \end{verbatim}%
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51
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52 #<pre>
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53 (** val pred : nat -> nat **)
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54
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55 let pred = function
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56 | O -> O
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57 | S u -> u
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58 </pre># *)
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59
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60 (** Returning 0 as the predecessor of 0 can come across as somewhat of a hack. In some situations, we might like to be sure that we never try to take the predecessor of 0. We can enforce this by giving [pred] a stronger, dependent type. *)
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61
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62 Lemma zgtz : 0 > 0 -> False.
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63 crush.
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64 Qed.
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65
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66 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
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67 match n return (n > 0 -> nat) with
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68 | O => fun pf : 0 > 0 => match zgtz pf with end
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69 | S n' => fun _ => n'
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70 end.
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71
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72 (** We expand the type of [pred] to include a %\textit{%#<i>#proof#</i>#%}% that its argument [n] is greater than 0. When [n] is 0, we use the proof to derive a contradiction, which we can use to build a value of any type via a vacuous pattern match. When [n] is a successor, we have no need for the proof and just return the answer. The proof argument can be said to have a %\textit{%#<i>#dependent#</i>#%}% type, because its type depends on the %\textit{%#<i>#value#</i>#%}% of the argument [n].
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73
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74 There are two aspects of the definition of [pred_strong1] that may be surprising. First, we took advantage of [Definition]'s syntactic sugar for defining function arguments in the case of [n], but we bound the proofs later with explicit [fun] expressions. Second, there is the [return] clause for the [match], which we saw briefly in Chapter 2. Let us see what happens if we write this function in the way that at first seems most natural. *)
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75
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76 (** [[
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77 Definition pred_strong1' (n : nat) (pf : n > 0) : nat :=
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78 match n with
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79 | O => match zgtz pf with end
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80 | S n' => n'
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81 end.
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82
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83 [[
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84 Error: In environment
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85 n : nat
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86 pf : n > 0
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87 The term "pf" has type "n > 0" while it is expected to have type
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88 "0 > 0"
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89 ]]
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90
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91 The term [zgtz pf] fails to type-check. Somehow the type checker has failed to take into account information that follows from which [match] branch that term appears in. The problem is that, by default, [match] does not let us use such implied information. To get refined typing, we must always add special [match] annotations.
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92
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93 In this case, we must use a [return] annotation to declare the relationship between the %\textit{%#<i>#value#</i>#%}% of the [match] discriminee and the %\textit{%#<i>#type#</i>#%}% of the result. There is no annotation that lets us declare a relationship between the discriminee and the type of a variable that is already in scope; hence, we delay the binding of [pf], so that we can use the [return] annotation to express the needed relationship.
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94
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95 Why does Coq not infer this relationship for us? Certainly, it is not hard to imagine heuristics that would handle this particular case and many others. In general, however, the inference problem is undecidable. The known undecidable problem of %\textit{%#<i>#higher-order unification#</i>#%}% reduces to the [match] type inference problem. Over time, Coq is enhanced with more and more heuristics to get around this problem, but there must always exist [match]es whose types Coq cannot infer without annotations.
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96
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97 Let us now take a look at the OCaml code Coq generates for [pred_strong1]. *)
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98
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99 Extraction pred_strong1.
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100
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101 (** %\begin{verbatim}
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102 (** val pred_strong1 : nat -> nat **)
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103
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104 let pred_strong1 = function
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105 | O -> assert false (* absurd case *)
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106 | S n' -> n'
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107 \end{verbatim}%
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108
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109 #<pre>
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110 (** val pred_strong1 : nat -> nat **)
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111
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112 let pred_strong1 = function
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113 | O -> assert false (* absurd case *)
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114 | S n' -> n'
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115 </pre># *)
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116
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117 (** The proof argument has disappeared! We get exactly the OCaml code we would have written manually. This is our first demonstration of the main technically interesting feature of Coq program extraction: program components of type [Prop] are erased systematically.
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118
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119 We can reimplement our dependently-typed [pred] based on %\textit{%#<i>#subset types#</i>#%}%, defined in the standard library with the type family [sig]. *)
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120
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121 Print sig.
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122 (** [[
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123
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124 Inductive sig (A : Type) (P : A -> Prop) : Type :=
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125 exist : forall x : A, P x -> sig P
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126 For sig: Argument A is implicit
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127 For exist: Argument A is implicit
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128 ]]
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129
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130 [sig] is a Curry-Howard twin of [ex], except that [sig] is in [Type], while [ex] is in [Prop]. That means that [sig] values can survive extraction, while [ex] proofs will always be erased. The actual details of extraction of [sig]s are more subtle, as we will see shortly.
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131
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132 We rewrite [pred_strong1], using some syntactic sugar for subset types. *)
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133
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134 Locate "{ _ : _ | _ }".
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135 (** [[
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136
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137 Notation Scope
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138 "{ x : A | P }" := sig (fun x : A => P)
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139 : type_scope
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140 (default interpretation)
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141 ]] *)
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142
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143 Definition pred_strong2 (s : {n : nat | n > 0}) : nat :=
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144 match s with
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145 | exist O pf => match zgtz pf with end
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146 | exist (S n') _ => n'
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147 end.
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148
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149 Extraction pred_strong2.
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150
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151 (** %\begin{verbatim}
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152 (** val pred_strong2 : nat -> nat **)
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153
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154 let pred_strong2 = function
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155 | O -> assert false (* absurd case *)
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156 | S n' -> n'
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157 \end{verbatim}%
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158
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159 #<pre>
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160 (** val pred_strong2 : nat -> nat **)
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161
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162 let pred_strong2 = function
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163 | O -> assert false (* absurd case *)
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164 | S n' -> n'
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165 </pre>#
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166
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167 We arrive at the same OCaml code as was extracted from [pred_strong1], which may seem surprising at first. The reason is that a value of [sig] is a pair of two pieces, a value and a proof about it. Extraction erases the proof, which reduces the constructor [exist] of [sig] to taking just a single argument. An optimization eliminates uses of datatypes with single constructors taking single arguments, and we arrive back where we started.
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168
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169 We can continue on in the process of refining [pred]'s type. Let us change its result type to capture that the output is really the predecessor of the input. *)
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170
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171 Definition pred_strong3 (s : {n : nat | n > 0}) : {m : nat | proj1_sig s = S m} :=
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172 match s return {m : nat | proj1_sig s = S m} with
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173 | exist 0 pf => match zgtz pf with end
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174 | exist (S n') _ => exist _ n' (refl_equal _)
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175 end.
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176
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177 (** The function [proj1_sig] extracts the base value from a subset type. Besides the use of that function, the only other new thing is the use of the [exist] constructor to build a new [sig] value, and the details of how to do that follow from the output of our earlier [Print] command.
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178
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179 By now, the reader is probably ready to believe that the new [pred_strong] leads to the same OCaml code as we have seen several times so far, and Coq does not disappoint. *)
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180
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181 Extraction pred_strong3.
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182
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183 (** %\begin{verbatim}
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184 (** val pred_strong3 : nat -> nat **)
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185
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186 let pred_strong3 = function
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187 | O -> assert false (* absurd case *)
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188 | S n' -> n'
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189 \end{verbatim}%
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190
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191 #<pre>
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192 (** val pred_strong3 : nat -> nat **)
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193
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194 let pred_strong3 = function
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195 | O -> assert false (* absurd case *)
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196 | S n' -> n'
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197 </pre>#
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198
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199 We have managed to reach a type that is, in a formal sense, the most expressive possible for [pred]. Any other implementation of the same type must have the same input-output behavior. However, there is still room for improvement in making this kind of code easier to write. Here is a version that takes advantage of tactic-based theorem proving. We switch back to passing a separate proof argument instead of using a subset type for the function's input, because this leads to cleaner code. *)
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200
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201 Definition pred_strong4 (n : nat) : n > 0 -> {m : nat | n = S m}.
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202 refine (fun n =>
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203 match n return (n > 0 -> {m : nat | n = S m}) with
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204 | O => fun _ => False_rec _ _
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205 | S n' => fun _ => exist _ n' _
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206 end).
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207
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208 (** We build [pred_strong4] using tactic-based proving, beginning with a [Definition] command that ends in a period before a definition is given. Such a command enters the interactive proving mode, with the type given for the new identifier as our proof goal. We do most of the work with the [refine] tactic, to which we pass a partial "proof" of the type we are trying to prove. There may be some pieces left to fill in, indicated by underscores. Any underscore that Coq cannot reconstruct with type inference is added as a proof subgoal. In this case, we have two subgoals:
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209
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210 [[
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211
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212 2 subgoals
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213
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214 n : nat
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215 _ : 0 > 0
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216 ============================
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217 False
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218 ]]
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219
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220 [[
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221
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222 subgoal 2 is:
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223 S n' = S n'
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224 ]]
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225
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226 We can see that the first subgoal comes from the second underscore passed to [False_rec], and the second subgoal comes from the second underscore passed to [exist]. In the first case, we see that, though we bound the proof variable with an underscore, it is still available in our proof context. It is hard to refer to underscore-named variables in manual proofs, but automation makes short work of them. Both subgoals are easy to discharge that way, so let us back up and ask to prove all subgoals automatically. *)
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227
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228 Undo.
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229 refine (fun n =>
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230 match n return (n > 0 -> {m : nat | n = S m}) with
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231 | O => fun _ => False_rec _ _
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232 | S n' => fun _ => exist _ n' _
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233 end); crush.
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234 Defined.
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235
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236 (** We end the "proof" with [Defined] instead of [Qed], so that the definition we constructed remains visible. This contrasts to the case of ending a proof with [Qed], where the details of the proof are hidden afterward. Let us see what our prooof script constructed. *)
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237
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238 Print pred_strong4.
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239 (** [[
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240
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241 pred_strong4 =
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242 fun n : nat =>
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243 match n as n0 return (n0 > 0 -> {m : nat | n0 = S m}) with
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244 | 0 =>
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245 fun _ : 0 > 0 =>
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246 False_rec {m : nat | 0 = S m}
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247 (Bool.diff_false_true
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248 (Bool.absurd_eq_true false
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249 (Bool.diff_false_true
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250 (Bool.absurd_eq_true false (pred_strong4_subproof n _)))))
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251 | S n' =>
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252 fun _ : S n' > 0 =>
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253 exist (fun m : nat => S n' = S m) n' (refl_equal (S n'))
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254 end
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255 : forall n : nat, n > 0 -> {m : nat | n = S m}
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256 ]]
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257
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258 We see the code we entered, with some proofs filled in. The first proof obligation, the second argument to [False_rec], is filled in with a nasty-looking proof term that we can be glad we did not enter by hand. The second proof obligation is a simple reflexivity proof.
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259
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260 We are almost done with the ideal implementation of dependent predecessor. We can use Coq's syntax extension facility to arrive at code with almost no complexity beyond a Haskell or ML program with a complete specification in a comment. *)
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261
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262 Notation "!" := (False_rec _ _).
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263 Notation "[ e ]" := (exist _ e _).
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264
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265 Definition pred_strong5 (n : nat) : n > 0 -> {m : nat | n = S m}.
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266 refine (fun n =>
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267 match n return (n > 0 -> {m : nat | n = S m}) with
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268 | O => fun _ => !
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269 | S n' => fun _ => [n']
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270 end); crush.
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271 Defined.
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272
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273
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274 (** * Decidable Proposition Types *)
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275
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276 (** There is another type in the standard library which captures the idea of program values that indicate which of two propositions is true. *)
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277
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278 Print sumbool.
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279 (** [[
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280
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281 Inductive sumbool (A : Prop) (B : Prop) : Set :=
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282 left : A -> {A} + {B} | right : B -> {A} + {B}
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283 For left: Argument A is implicit
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284 For right: Argument B is implicit
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285 ]] *)
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286
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287 (** We can define some notations to make working with [sumbool] more convenient. *)
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288
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289 Notation "'Yes'" := (left _ _).
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290 Notation "'No'" := (right _ _).
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291 Notation "'Reduce' x" := (if x then Yes else No) (at level 50).
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292
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293 (** The [Reduce] notation is notable because it demonstrates how [if] is overloaded in Coq. The [if] form actually works when the test expression has any two-constructor inductive type. Moreover, in the [then] and [else] branches, the appropriate constructor arguments are bound. This is important when working with [sumbool]s, when we want to have the proof stored in the test expression available when proving the proof obligations generated in the appropriate branch.
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294
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295 Now we can write [eq_nat_dec], which compares two natural numbers, returning either a proof of their equality or a proof of their inequality. *)
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296
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297 Definition eq_nat_dec (n m : nat) : {n = m} + {n <> m}.
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298 refine (fix f (n m : nat) {struct n} : {n = m} + {n <> m} :=
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299 match n, m return {n = m} + {n <> m} with
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300 | O, O => Yes
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301 | S n', S m' => Reduce (f n' m')
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302 | _, _ => No
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303 end); congruence.
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304 Defined.
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305
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306 (** Our definition extracts to reasonable OCaml code. *)
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307
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308 Extraction eq_nat_dec.
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309
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310 (** %\begin{verbatim}
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311 (** val eq_nat_dec : nat -> nat -> sumbool **)
|
adamc@71
|
312
|
adamc@71
|
313 let rec eq_nat_dec n m =
|
adamc@71
|
314 match n with
|
adamc@71
|
315 | O -> (match m with
|
adamc@71
|
316 | O -> Left
|
adamc@71
|
317 | S n0 -> Right)
|
adamc@71
|
318 | S n' -> (match m with
|
adamc@71
|
319 | O -> Right
|
adamc@71
|
320 | S m' -> eq_nat_dec n' m')
|
adamc@71
|
321 \end{verbatim}%
|
adamc@71
|
322
|
adamc@71
|
323 #<pre>
|
adamc@71
|
324 (** val eq_nat_dec : nat -> nat -> sumbool **)
|
adamc@71
|
325
|
adamc@71
|
326 let rec eq_nat_dec n m =
|
adamc@71
|
327 match n with
|
adamc@71
|
328 | O -> (match m with
|
adamc@71
|
329 | O -> Left
|
adamc@71
|
330 | S n0 -> Right)
|
adamc@71
|
331 | S n' -> (match m with
|
adamc@71
|
332 | O -> Right
|
adamc@71
|
333 | S m' -> eq_nat_dec n' m')
|
adamc@71
|
334 </pre>#
|
adamc@71
|
335
|
adamc@71
|
336 Proving this kind of decidable equality result is so common that Coq comes with a tactic for automating it. *)
|
adamc@71
|
337
|
adamc@71
|
338 Definition eq_nat_dec' (n m : nat) : {n = m} + {n <> m}.
|
adamc@71
|
339 decide equality.
|
adamc@71
|
340 Defined.
|
adamc@71
|
341
|
adamc@71
|
342 (** Curious readers can verify that the [decide equality] version extracts to the same OCaml code as our more manual version does. That OCaml code had one undesirable property, which is that it uses %\texttt{%#<tt>#Left#</tt>#%}% and %\texttt{%#<tt>#Right#</tt>#%}% constructors instead of the boolean values built into OCaml. We can fix this, by using Coq's facility for mapping Coq inductive types to OCaml variant types. *)
|
adamc@71
|
343
|
adamc@71
|
344 Extract Inductive sumbool => "bool" ["true" "false"].
|
adamc@71
|
345 Extraction eq_nat_dec'.
|
adamc@71
|
346
|
adamc@71
|
347 (** %\begin{verbatim}
|
adamc@71
|
348 (** val eq_nat_dec' : nat -> nat -> bool **)
|
adamc@71
|
349
|
adamc@71
|
350 let rec eq_nat_dec' n m0 =
|
adamc@71
|
351 match n with
|
adamc@71
|
352 | O -> (match m0 with
|
adamc@71
|
353 | O -> true
|
adamc@71
|
354 | S n0 -> false)
|
adamc@71
|
355 | S n0 -> (match m0 with
|
adamc@71
|
356 | O -> false
|
adamc@71
|
357 | S n1 -> eq_nat_dec' n0 n1)
|
adamc@71
|
358 \end{verbatim}%
|
adamc@71
|
359
|
adamc@71
|
360 #<pre>
|
adamc@71
|
361 (** val eq_nat_dec' : nat -> nat -> bool **)
|
adamc@71
|
362
|
adamc@71
|
363 let rec eq_nat_dec' n m0 =
|
adamc@71
|
364 match n with
|
adamc@71
|
365 | O -> (match m0 with
|
adamc@71
|
366 | O -> true
|
adamc@71
|
367 | S n0 -> false)
|
adamc@71
|
368 | S n0 -> (match m0 with
|
adamc@71
|
369 | O -> false
|
adamc@71
|
370 | S n1 -> eq_nat_dec' n0 n1)
|
adamc@71
|
371 </pre># *)
|
adamc@72
|
372
|
adamc@72
|
373 (** %\smallskip%
|
adamc@72
|
374
|
adamc@72
|
375 We can build "smart" versions of the usual boolean operators and put them to good use in certified programming. For instance, here is a [sumbool] version of boolean "or." *)
|
adamc@72
|
376
|
adamc@72
|
377 Notation "x || y" := (if x then Yes else Reduce y) (at level 50).
|
adamc@72
|
378
|
adamc@72
|
379 (** Let us use it for building a function that decides list membership. We need to assume the existence of an equality decision procedure for the type of list elements. *)
|
adamc@72
|
380
|
adamc@72
|
381 Section In_dec.
|
adamc@72
|
382 Variable A : Set.
|
adamc@72
|
383 Variable A_eq_dec : forall x y : A, {x = y} + {x <> y}.
|
adamc@72
|
384
|
adamc@72
|
385 (** The final function is easy to write using the techniques we have developed so far. *)
|
adamc@72
|
386
|
adamc@72
|
387 Definition In_dec : forall (x : A) (ls : list A), {In x ls} + { ~In x ls}.
|
adamc@72
|
388 refine (fix f (x : A) (ls : list A) {struct ls}
|
adamc@72
|
389 : {In x ls} + { ~In x ls} :=
|
adamc@72
|
390 match ls return {In x ls} + { ~In x ls} with
|
adamc@72
|
391 | nil => No
|
adamc@72
|
392 | x' :: ls' => A_eq_dec x x' || f x ls'
|
adamc@72
|
393 end); crush.
|
adamc@72
|
394 Qed.
|
adamc@72
|
395 End In_dec.
|
adamc@72
|
396
|
adamc@72
|
397 (** [In_dec] has a reasonable extraction to OCaml. *)
|
adamc@72
|
398
|
adamc@72
|
399 Extraction In_dec.
|
adamc@72
|
400
|
adamc@72
|
401 (** %\begin{verbatim}
|
adamc@72
|
402 (** val in_dec : ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 list -> bool **)
|
adamc@72
|
403
|
adamc@72
|
404 let rec in_dec a_eq_dec x = function
|
adamc@72
|
405 | Nil -> false
|
adamc@72
|
406 | Cons (x', ls') ->
|
adamc@72
|
407 (match a_eq_dec x x' with
|
adamc@72
|
408 | true -> true
|
adamc@72
|
409 | false -> in_dec a_eq_dec x ls')
|
adamc@72
|
410 \end{verbatim}%
|
adamc@72
|
411
|
adamc@72
|
412 #<pre>
|
adamc@72
|
413 (** val in_dec : ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 list -> bool **)
|
adamc@72
|
414
|
adamc@72
|
415 let rec in_dec a_eq_dec x = function
|
adamc@72
|
416 | Nil -> false
|
adamc@72
|
417 | Cons (x', ls') ->
|
adamc@72
|
418 (match a_eq_dec x x' with
|
adamc@72
|
419 | true -> true
|
adamc@72
|
420 | false -> in_dec a_eq_dec x ls')
|
adamc@72
|
421 </pre># *)
|
adamc@72
|
422
|
adamc@72
|
423
|
adamc@72
|
424 (** * Partial Subset Types *)
|
adamc@72
|
425
|
adamc@73
|
426 (** Our final implementation of dependent predecessor used a very specific argument type to ensure that execution could always complete normally. Sometimes we want to allow execution to fail, and we want a more principled way of signaling that than returning a default value, as [pred] does for [0]. One approach is to define this type family [maybe], which is a version of [sig] that allows obligation-free failure. *)
|
adamc@73
|
427
|
adamc@72
|
428 Inductive maybe (A : Type) (P : A -> Prop) : Set :=
|
adamc@72
|
429 | Unknown : maybe P
|
adamc@72
|
430 | Found : forall x : A, P x -> maybe P.
|
adamc@72
|
431
|
adamc@73
|
432 (** We can define some new notations, analogous to those we defined for subset types. *)
|
adamc@73
|
433
|
adamc@72
|
434 Notation "{{ x | P }}" := (maybe (fun x => P)).
|
adamc@72
|
435 Notation "??" := (Unknown _).
|
adamc@72
|
436 Notation "[[ x ]]" := (Found _ x _).
|
adamc@72
|
437
|
adamc@73
|
438 (** Now our next version of [pred] is trivial to write. *)
|
adamc@73
|
439
|
adamc@73
|
440 Definition pred_strong6 (n : nat) : {{m | n = S m}}.
|
adamc@73
|
441 refine (fun n =>
|
adamc@73
|
442 match n return {{m | n = S m}} with
|
adamc@73
|
443 | O => ??
|
adamc@73
|
444 | S n' => [[n']]
|
adamc@73
|
445 end); trivial.
|
adamc@73
|
446 Defined.
|
adamc@73
|
447
|
adamc@73
|
448 (** Because we used [maybe], one valid implementation of the type we gave [pred_strong6] would return [??] in every case. We can strengthen the type to rule out such vacuous implementations, and the type family [sumor] from the standard library provides the easiest starting point. For type [A] and proposition [B], [A + {B}] desugars to [sumor A B], whose values are either values of [A] or proofs of [B]. *)
|
adamc@73
|
449
|
adamc@73
|
450 Print sumor.
|
adamc@73
|
451 (** [[
|
adamc@73
|
452
|
adamc@73
|
453 Inductive sumor (A : Type) (B : Prop) : Type :=
|
adamc@73
|
454 inleft : A -> A + {B} | inright : B -> A + {B}
|
adamc@73
|
455 For inleft: Argument A is implicit
|
adamc@73
|
456 For inright: Argument B is implicit
|
adamc@73
|
457 ]] *)
|
adamc@73
|
458
|
adamc@73
|
459 (** We add notations for easy use of the [sumor] constructors. The second notation is specialized to [sumor]s whose [A] parameters are instantiated with regular subset types, since this is how we will use [sumor] below. *)
|
adamc@73
|
460
|
adamc@73
|
461 Notation "!!" := (inright _ _).
|
adamc@73
|
462 Notation "[[[ x ]]]" := (inleft _ [x]).
|
adamc@73
|
463
|
adamc@73
|
464 (** Now we are ready to give the final version of possibly-failing predecessor. The [sumor]-based type that we use is maximally expressive; any implementation of the type has the same input-output behavior. *)
|
adamc@73
|
465
|
adamc@73
|
466 Definition pred_strong7 (n : nat) : {m : nat | n = S m} + {n = 0}.
|
adamc@73
|
467 refine (fun n =>
|
adamc@73
|
468 match n return {m : nat | n = S m} + {n = 0} with
|
adamc@73
|
469 | O => !!
|
adamc@73
|
470 | S n' => [[[n']]]
|
adamc@73
|
471 end); trivial.
|
adamc@73
|
472 Defined.
|
adamc@73
|
473
|
adamc@73
|
474
|
adamc@73
|
475 (** * Monadic Notations *)
|
adamc@73
|
476
|
adamc@73
|
477 (** We can treat [maybe] like a monad, in the same way that the Haskell [Maybe] type is interpreted as a failure monad. Our [maybe] has the wrong type to be a literal monad, but a "bind"-like notation will still be helpful. *)
|
adamc@73
|
478
|
adamc@72
|
479 Notation "x <- e1 ; e2" := (match e1 with
|
adamc@72
|
480 | Unknown => ??
|
adamc@72
|
481 | Found x _ => e2
|
adamc@72
|
482 end)
|
adamc@72
|
483 (right associativity, at level 60).
|
adamc@72
|
484
|
adamc@73
|
485 (** The meaning of [x <- e1; e2] is: First run [e1]. If it fails to find an answer, then announce failure for our derived computation, too. If [e1] %\textit{%#<i>#does#</i>#%}% find an answer, pass that answer on to [e2] to find the final result. The variable [x] can be considered bound in [e2].
|
adamc@73
|
486
|
adamc@73
|
487 This notation is very helpful for composing richly-typed procedures. For instance, here is a very simple implementation of a function to take the predecessors of two naturals at once. *)
|
adamc@73
|
488
|
adamc@73
|
489 Definition doublePred (n1 n2 : nat) : {{p | n1 = S (fst p) /\ n2 = S (snd p)}}.
|
adamc@73
|
490 refine (fun n1 n2 =>
|
adamc@73
|
491 m1 <- pred_strong6 n1;
|
adamc@73
|
492 m2 <- pred_strong6 n2;
|
adamc@73
|
493 [[(m1, m2)]]); tauto.
|
adamc@73
|
494 Defined.
|
adamc@73
|
495
|
adamc@73
|
496 (** We can build a [sumor] version of the "bind" notation and use it to write a similarly straightforward version of this function. *)
|
adamc@73
|
497
|
adamc@73
|
498 (** printing <-- $\longleftarrow$ *)
|
adamc@73
|
499
|
adamc@73
|
500 Notation "x <-- e1 ; e2" := (match e1 with
|
adamc@73
|
501 | inright _ => !!
|
adamc@73
|
502 | inleft (exist x _) => e2
|
adamc@73
|
503 end)
|
adamc@73
|
504 (right associativity, at level 60).
|
adamc@73
|
505
|
adamc@73
|
506 (** printing * $\times$ *)
|
adamc@73
|
507
|
adamc@73
|
508 Definition doublePred' (n1 n2 : nat) : {p : nat * nat | n1 = S (fst p) /\ n2 = S (snd p)}
|
adamc@73
|
509 + {n1 = 0 \/ n2 = 0}.
|
adamc@73
|
510 refine (fun n1 n2 =>
|
adamc@73
|
511 m1 <-- pred_strong7 n1;
|
adamc@73
|
512 m2 <-- pred_strong7 n2;
|
adamc@73
|
513 [[[(m1, m2)]]]); tauto.
|
adamc@73
|
514 Defined.
|
adamc@72
|
515
|
adamc@72
|
516
|
adamc@72
|
517 (** * A Type-Checking Example *)
|
adamc@72
|
518
|
adamc@75
|
519 (** We can apply these specification types to build a certified type-checker for a simple expression language. *)
|
adamc@75
|
520
|
adamc@72
|
521 Inductive exp : Set :=
|
adamc@72
|
522 | Nat : nat -> exp
|
adamc@72
|
523 | Plus : exp -> exp -> exp
|
adamc@72
|
524 | Bool : bool -> exp
|
adamc@72
|
525 | And : exp -> exp -> exp.
|
adamc@72
|
526
|
adamc@75
|
527 (** We define a simple language of types and its typing rules, in the style introduced in Chapter 4. *)
|
adamc@75
|
528
|
adamc@72
|
529 Inductive type : Set := TNat | TBool.
|
adamc@72
|
530
|
adamc@72
|
531 Inductive hasType : exp -> type -> Prop :=
|
adamc@72
|
532 | HtNat : forall n,
|
adamc@72
|
533 hasType (Nat n) TNat
|
adamc@72
|
534 | HtPlus : forall e1 e2,
|
adamc@72
|
535 hasType e1 TNat
|
adamc@72
|
536 -> hasType e2 TNat
|
adamc@72
|
537 -> hasType (Plus e1 e2) TNat
|
adamc@72
|
538 | HtBool : forall b,
|
adamc@72
|
539 hasType (Bool b) TBool
|
adamc@72
|
540 | HtAnd : forall e1 e2,
|
adamc@72
|
541 hasType e1 TBool
|
adamc@72
|
542 -> hasType e2 TBool
|
adamc@72
|
543 -> hasType (And e1 e2) TBool.
|
adamc@72
|
544
|
adamc@75
|
545 (** It will be helpful to have a function for comparing two types. We build one using [decide equality]. *)
|
adamc@75
|
546
|
adamc@75
|
547 Definition eq_type_dec : forall t1 t2 : type, {t1 = t2} + {t1 <> t2}.
|
adamc@72
|
548 decide equality.
|
adamc@72
|
549 Defined.
|
adamc@72
|
550
|
adamc@75
|
551 (** Another notation complements the monadic notation for [maybe] that we defined earlier. Sometimes we want to be to include "assertions" in our procedures. That is, we want to run a decision procedure and fail if it fails; otherwise, we want to continue, with the proof that it produced made available to us. This infix notation captures that, for a procedure that returns an arbitrary two-constructor type. *)
|
adamc@75
|
552
|
adamc@73
|
553 Notation "e1 ;; e2" := (if e1 then e2 else ??)
|
adamc@73
|
554 (right associativity, at level 60).
|
adamc@73
|
555
|
adamc@75
|
556 (** With that notation defined, we can implement a [typeCheck] function, whose code is only more complex than what we would write in ML because it needs to include some extra type annotations. Every [[[e]]] expression adds a [hasType] proof obligation, and [crush] makes short work of them when we add [hasType]'s constructors as hints. *)
|
adamc@75
|
557
|
adamc@72
|
558 Definition typeCheck (e : exp) : {{t | hasType e t}}.
|
adamc@72
|
559 Hint Constructors hasType.
|
adamc@72
|
560
|
adamc@72
|
561 refine (fix F (e : exp) : {{t | hasType e t}} :=
|
adamc@72
|
562 match e return {{t | hasType e t}} with
|
adamc@72
|
563 | Nat _ => [[TNat]]
|
adamc@72
|
564 | Plus e1 e2 =>
|
adamc@72
|
565 t1 <- F e1;
|
adamc@72
|
566 t2 <- F e2;
|
adamc@72
|
567 eq_type_dec t1 TNat;;
|
adamc@72
|
568 eq_type_dec t2 TNat;;
|
adamc@72
|
569 [[TNat]]
|
adamc@72
|
570 | Bool _ => [[TBool]]
|
adamc@72
|
571 | And e1 e2 =>
|
adamc@72
|
572 t1 <- F e1;
|
adamc@72
|
573 t2 <- F e2;
|
adamc@72
|
574 eq_type_dec t1 TBool;;
|
adamc@72
|
575 eq_type_dec t2 TBool;;
|
adamc@72
|
576 [[TBool]]
|
adamc@72
|
577 end); crush.
|
adamc@72
|
578 Defined.
|
adamc@72
|
579
|
adamc@75
|
580 (** Despite manipulating proofs, our type checker is easy to run. *)
|
adamc@75
|
581
|
adamc@72
|
582 Eval simpl in typeCheck (Nat 0).
|
adamc@75
|
583 (** [[
|
adamc@75
|
584
|
adamc@75
|
585 = [[TNat]]
|
adamc@75
|
586 : {{t | hasType (Nat 0) t}}
|
adamc@75
|
587 ]] *)
|
adamc@75
|
588
|
adamc@72
|
589 Eval simpl in typeCheck (Plus (Nat 1) (Nat 2)).
|
adamc@75
|
590 (** [[
|
adamc@75
|
591
|
adamc@75
|
592 = [[TNat]]
|
adamc@75
|
593 : {{t | hasType (Plus (Nat 1) (Nat 2)) t}}
|
adamc@75
|
594 ]] *)
|
adamc@75
|
595
|
adamc@72
|
596 Eval simpl in typeCheck (Plus (Nat 1) (Bool false)).
|
adamc@75
|
597 (** [[
|
adamc@75
|
598
|
adamc@75
|
599 = ??
|
adamc@75
|
600 : {{t | hasType (Plus (Nat 1) (Bool false)) t}}
|
adamc@75
|
601 ]] *)
|
adamc@75
|
602
|
adamc@75
|
603 (** The type-checker also extracts to some reasonable OCaml code. *)
|
adamc@75
|
604
|
adamc@75
|
605 Extraction typeCheck.
|
adamc@75
|
606
|
adamc@75
|
607 (** %\begin{verbatim}
|
adamc@75
|
608 (** val typeCheck : exp -> type0 maybe **)
|
adamc@75
|
609
|
adamc@75
|
610 let rec typeCheck = function
|
adamc@75
|
611 | Nat n -> Found TNat
|
adamc@75
|
612 | Plus (e1, e2) ->
|
adamc@75
|
613 (match typeCheck e1 with
|
adamc@75
|
614 | Unknown -> Unknown
|
adamc@75
|
615 | Found t1 ->
|
adamc@75
|
616 (match typeCheck e2 with
|
adamc@75
|
617 | Unknown -> Unknown
|
adamc@75
|
618 | Found t2 ->
|
adamc@75
|
619 (match eq_type_dec t1 TNat with
|
adamc@75
|
620 | true ->
|
adamc@75
|
621 (match eq_type_dec t2 TNat with
|
adamc@75
|
622 | true -> Found TNat
|
adamc@75
|
623 | false -> Unknown)
|
adamc@75
|
624 | false -> Unknown)))
|
adamc@75
|
625 | Bool b -> Found TBool
|
adamc@75
|
626 | And (e1, e2) ->
|
adamc@75
|
627 (match typeCheck e1 with
|
adamc@75
|
628 | Unknown -> Unknown
|
adamc@75
|
629 | Found t1 ->
|
adamc@75
|
630 (match typeCheck e2 with
|
adamc@75
|
631 | Unknown -> Unknown
|
adamc@75
|
632 | Found t2 ->
|
adamc@75
|
633 (match eq_type_dec t1 TBool with
|
adamc@75
|
634 | true ->
|
adamc@75
|
635 (match eq_type_dec t2 TBool with
|
adamc@75
|
636 | true -> Found TBool
|
adamc@75
|
637 | false -> Unknown)
|
adamc@75
|
638 | false -> Unknown)))
|
adamc@75
|
639 \end{verbatim}%
|
adamc@75
|
640
|
adamc@75
|
641 #<pre>
|
adamc@75
|
642 (** val typeCheck : exp -> type0 maybe **)
|
adamc@75
|
643
|
adamc@75
|
644 let rec typeCheck = function
|
adamc@75
|
645 | Nat n -> Found TNat
|
adamc@75
|
646 | Plus (e1, e2) ->
|
adamc@75
|
647 (match typeCheck e1 with
|
adamc@75
|
648 | Unknown -> Unknown
|
adamc@75
|
649 | Found t1 ->
|
adamc@75
|
650 (match typeCheck e2 with
|
adamc@75
|
651 | Unknown -> Unknown
|
adamc@75
|
652 | Found t2 ->
|
adamc@75
|
653 (match eq_type_dec t1 TNat with
|
adamc@75
|
654 | true ->
|
adamc@75
|
655 (match eq_type_dec t2 TNat with
|
adamc@75
|
656 | true -> Found TNat
|
adamc@75
|
657 | false -> Unknown)
|
adamc@75
|
658 | false -> Unknown)))
|
adamc@75
|
659 | Bool b -> Found TBool
|
adamc@75
|
660 | And (e1, e2) ->
|
adamc@75
|
661 (match typeCheck e1 with
|
adamc@75
|
662 | Unknown -> Unknown
|
adamc@75
|
663 | Found t1 ->
|
adamc@75
|
664 (match typeCheck e2 with
|
adamc@75
|
665 | Unknown -> Unknown
|
adamc@75
|
666 | Found t2 ->
|
adamc@75
|
667 (match eq_type_dec t1 TBool with
|
adamc@75
|
668 | true ->
|
adamc@75
|
669 (match eq_type_dec t2 TBool with
|
adamc@75
|
670 | true -> Found TBool
|
adamc@75
|
671 | false -> Unknown)
|
adamc@75
|
672 | false -> Unknown)))
|
adamc@75
|
673 </pre># *)
|
adamc@75
|
674
|
adamc@75
|
675 (** %\smallskip%
|
adamc@75
|
676
|
adamc@75
|
677 We can adapt this implementation to use [sumor], so that we know our type-checker only fails on ill-typed inputs. First, we define an analogue to the "assertion" notation. *)
|
adamc@73
|
678
|
adamc@73
|
679 Notation "e1 ;;; e2" := (if e1 then e2 else !!)
|
adamc@73
|
680 (right associativity, at level 60).
|
adamc@73
|
681
|
adamc@75
|
682 (** Next, we prove a helpful lemma, which states that a given expression can have at most one type. *)
|
adamc@75
|
683
|
adamc@75
|
684 Lemma hasType_det : forall e t1,
|
adamc@73
|
685 hasType e t1
|
adamc@73
|
686 -> forall t2,
|
adamc@73
|
687 hasType e t2
|
adamc@73
|
688 -> t1 = t2.
|
adamc@73
|
689 induction 1; inversion 1; crush.
|
adamc@73
|
690 Qed.
|
adamc@73
|
691
|
adamc@75
|
692 (** Now we can define the type-checker. Its type expresses that it only fails on untypable expressions. *)
|
adamc@75
|
693
|
adamc@73
|
694 Definition typeCheck' (e : exp) : {t : type | hasType e t} + {forall t, ~hasType e t}.
|
adamc@73
|
695 Hint Constructors hasType.
|
adamc@75
|
696 (** We register all of the typing rules as hints. *)
|
adamc@75
|
697
|
adamc@73
|
698 Hint Resolve hasType_det.
|
adamc@75
|
699 (** [hasType_det] will also be useful for proving proof obligations with contradictory contexts. Since its statement includes [forall]-bound variables that do not appear in its conclusion, only [eauto] will apply this hint. *)
|
adamc@73
|
700
|
adamc@75
|
701 (** Finally, the implementation of [typeCheck] can be transcribed literally, simply switching notations as needed. *)
|
adamc@73
|
702 refine (fix F (e : exp) : {t : type | hasType e t} + {forall t, ~hasType e t} :=
|
adamc@73
|
703 match e return {t : type | hasType e t} + {forall t, ~hasType e t} with
|
adamc@73
|
704 | Nat _ => [[[TNat]]]
|
adamc@73
|
705 | Plus e1 e2 =>
|
adamc@73
|
706 t1 <-- F e1;
|
adamc@73
|
707 t2 <-- F e2;
|
adamc@73
|
708 eq_type_dec t1 TNat;;;
|
adamc@73
|
709 eq_type_dec t2 TNat;;;
|
adamc@73
|
710 [[[TNat]]]
|
adamc@73
|
711 | Bool _ => [[[TBool]]]
|
adamc@73
|
712 | And e1 e2 =>
|
adamc@73
|
713 t1 <-- F e1;
|
adamc@73
|
714 t2 <-- F e2;
|
adamc@73
|
715 eq_type_dec t1 TBool;;;
|
adamc@73
|
716 eq_type_dec t2 TBool;;;
|
adamc@73
|
717 [[[TBool]]]
|
adamc@73
|
718 end); clear F; crush' tt hasType; eauto.
|
adamc@75
|
719
|
adamc@75
|
720 (** We clear [F], the local name for the recursive function, to avoid strange proofs that refer to recursive calls that we never make. The [crush] variant [crush'] helps us by performing automatic inversion on instances of the predicates specified in its second argument. Once we throw in [eauto] to apply [hasType_det] for us, we have discharged all the subgoals. *)
|
adamc@73
|
721 Defined.
|
adamc@73
|
722
|
adamc@75
|
723 (** The short implementation here hides just how time-saving automation is. Every use of one of the notations adds a proof obligation, giving us 12 in total. Most of these obligations require multiple inversions and either uses of [hasType_det] or applications of [hasType] rules.
|
adamc@75
|
724
|
adamc@75
|
725 The results of simplifying calls to [typeCheck'] look deceptively similar to the results for [typeCheck], but now the types of the results provide more information. *)
|
adamc@75
|
726
|
adamc@73
|
727 Eval simpl in typeCheck' (Nat 0).
|
adamc@75
|
728 (** [[
|
adamc@75
|
729
|
adamc@75
|
730 = [[[TNat]]]
|
adamc@75
|
731 : {t : type | hasType (Nat 0) t} +
|
adamc@75
|
732 {(forall t : type, ~ hasType (Nat 0) t)}
|
adamc@75
|
733 ]] *)
|
adamc@75
|
734
|
adamc@73
|
735 Eval simpl in typeCheck' (Plus (Nat 1) (Nat 2)).
|
adamc@75
|
736 (** [[
|
adamc@75
|
737
|
adamc@75
|
738 = [[[TNat]]]
|
adamc@75
|
739 : {t : type | hasType (Plus (Nat 1) (Nat 2)) t} +
|
adamc@75
|
740 {(forall t : type, ~ hasType (Plus (Nat 1) (Nat 2)) t)}
|
adamc@75
|
741 ]] *)
|
adamc@75
|
742
|
adamc@73
|
743 Eval simpl in typeCheck' (Plus (Nat 1) (Bool false)).
|
adamc@75
|
744 (** [[
|
adamc@75
|
745
|
adamc@75
|
746 = !!
|
adamc@75
|
747 : {t : type | hasType (Plus (Nat 1) (Bool false)) t} +
|
adamc@75
|
748 {(forall t : type, ~ hasType (Plus (Nat 1) (Bool false)) t)}
|
adamc@75
|
749 ]] *)
|