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adam@346
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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 (* end hide *)
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17
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18
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19 (** %\chapter{Infinite Data and Proofs}% *)
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20
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21 (** In lazy functional programming languages like %\index{Haskell}%Haskell, infinite data structures are everywhere. Infinite lists and more exotic datatypes provide convenient abstractions for communication between parts of a program. Achieving similar convenience without infinite lazy structures would, in many cases, require acrobatic inversions of control flow.
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22
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23 %\index{laziness}%Laziness is easy to implement in Haskell, where all the definitions in a program may be thought of as mutually recursive. In such an unconstrained setting, it is easy to implement an infinite loop when you really meant to build an infinite list, where any finite prefix of the list should be forceable in finite time. Haskell programmers learn how to avoid such slip-ups. In Coq, such a laissez-faire policy is not good enough.
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24
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25 We spent some time in the last chapter discussing the %\index{Curry-Howard correspondence}%Curry-Howard isomorphism, where proofs are identified with functional programs. In such a setting, infinite loops, intended or otherwise, are disastrous. If Coq allowed the full breadth of definitions that Haskell did, we could code up an infinite loop and use it to prove any proposition vacuously. That is, the addition of general recursion would make CIC %\textit{%#<i>#inconsistent#</i>#%}%. For an arbitrary proposition [P], we could write:
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26 [[
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27 Fixpoint bad (u : unit) : P := bad u.
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28 ]]
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29
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30 This would leave us with [bad tt] as a proof of [P].
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31
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32 There are also algorithmic considerations that make universal termination very desirable. We have seen how tactics like [reflexivity] compare terms up to equivalence under computational rules. Calls to recursive, pattern-matching functions are simplified automatically, with no need for explicit proof steps. It would be very hard to hold onto that kind of benefit if it became possible to write non-terminating programs; we would be running smack into the halting problem.
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33
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34 One solution is to use types to contain the possibility of non-termination. For instance, we can create a %``%#"#non-termination monad,#"#%''% inside which we must write all of our general-recursive programs. This is a heavyweight solution, and so we would like to avoid it whenever possible.
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35
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36 Luckily, Coq has special support for a class of lazy data structures that happens to contain most examples found in Haskell. That mechanism, %\index{co-inductive types}\textit{%#<i>#co-inductive types#</i>#%}%, is the subject of this chapter. *)
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37
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38
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39 (** * Computing with Infinite Data *)
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40
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41 (** Let us begin with the most basic type of infinite data, %\textit{%#<i>#streams#</i>#%}%, or lazy lists.%\index{Vernacular commands!CoInductive}% *)
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42
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43 Section stream.
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44 Variable A : Set.
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45
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46 CoInductive stream : Set :=
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47 | Cons : A -> stream -> stream.
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48 End stream.
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49
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50 (** The definition is surprisingly simple. Starting from the definition of [list], we just need to change the keyword [Inductive] to [CoInductive]. We could have left a [Nil] constructor in our definition, but we will leave it out to force all of our streams to be infinite.
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51
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52 How do we write down a stream constant? Obviously simple application of constructors is not good enough, since we could only denote finite objects that way. Rather, whereas recursive definitions were necessary to %\textit{%#<i>#use#</i>#%}% values of recursive inductive types effectively, here we find that we need %\index{co-recursive definitions}\textit{%#<i>#co-recursive definitions#</i>#%}% to %\textit{%#<i>#build#</i>#%}% values of co-inductive types effectively.
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53
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54 We can define a stream consisting only of zeroes.%\index{Vernacular commands!CoFixpoint}% *)
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55
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56 CoFixpoint zeroes : stream nat := Cons 0 zeroes.
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57
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58 (* EX: Define a stream that alternates between [true] and [false]. *)
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59 (* begin thide *)
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60
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61 (** We can also define a stream that alternates between [true] and [false]. *)
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62
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63 CoFixpoint trues_falses : stream bool := Cons true falses_trues
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64 with falses_trues : stream bool := Cons false trues_falses.
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65 (* end thide *)
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66
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67 (** Co-inductive values are fair game as arguments to recursive functions, and we can use that fact to write a function to take a finite approximation of a stream. *)
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68
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69 (* EX: Define a function to calculate a finite approximation of a stream, to a particular length. *)
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70 (* begin thide *)
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71
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72 Fixpoint approx A (s : stream A) (n : nat) : list A :=
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73 match n with
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74 | O => nil
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75 | S n' =>
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76 match s with
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77 | Cons h t => h :: approx t n'
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78 end
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79 end.
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80
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81 Eval simpl in approx zeroes 10.
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82 (** %\vspace{-.15in}% [[
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83 = 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: nil
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84 : list nat
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85 ]]
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86 *)
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87
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88 Eval simpl in approx trues_falses 10.
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89 (** %\vspace{-.15in}% [[
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90 = true
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91 :: false
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92 :: true
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93 :: false
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94 :: true :: false :: true :: false :: true :: false :: nil
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95 : list bool
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96 ]]
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97 *)
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98
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99 (* end thide *)
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100
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101 (** So far, it looks like co-inductive types might be a magic bullet, allowing us to import all of the Haskeller's usual tricks. However, there are important restrictions that are dual to the restrictions on the use of inductive types. Fixpoints %\textit{%#<i>#consume#</i>#%}% values of inductive types, with restrictions on which %\textit{%#<i>#arguments#</i>#%}% may be passed in recursive calls. Dually, co-fixpoints %\textit{%#<i>#produce#</i>#%}% values of co-inductive types, with restrictions on what may be done with the %\textit{%#<i>#results#</i>#%}% of co-recursive calls.
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102
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103 The restriction for co-inductive types shows up as the %\index{guardedness condition}\textit{%#<i>#guardedness condition#</i>#%}%, and it can be broken into two parts. First, consider this stream definition, which would be legal in Haskell.
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104 [[
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105 CoFixpoint looper : stream nat := looper.
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106 ]]
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107
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108 <<
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109 Error:
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110 Recursive definition of looper is ill-formed.
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111 In environment
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112 looper : stream nat
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113
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114 unguarded recursive call in "looper"
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115 >>
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116
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117 The rule we have run afoul of here is that %\textit{%#<i>#every co-recursive call must be guarded by a constructor#</i>#%}%; that is, every co-recursive call must be a direct argument to a constructor of the co-inductive type we are generating. It is a good thing that this rule is enforced. If the definition of [looper] were accepted, our [approx] function would run forever when passed [looper], and we would have fallen into inconsistency.
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118
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119 Some familiar functions are easy to write in co-recursive fashion. *)
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120
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121 Section map.
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122 Variables A B : Set.
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123 Variable f : A -> B.
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124
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125 CoFixpoint map (s : stream A) : stream B :=
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126 match s with
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127 | Cons h t => Cons (f h) (map t)
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128 end.
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129 End map.
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130
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131 (** This code is a literal copy of that for the list [map] function, with the [Nil] case removed and [Fixpoint] changed to [CoFixpoint]. Many other standard functions on lazy data structures can be implemented just as easily. Some, like [filter], cannot be implemented. Since the predicate passed to [filter] may reject every element of the stream, we cannot satisfy the guardedness condition.
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132
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133 The implications of the condition can be subtle. To illustrate how, we start off with another co-recursive function definition that %\textit{%#<i>#is#</i>#%}% legal. The function [interleave] takes two streams and produces a new stream that alternates between their elements. *)
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134
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135 Section interleave.
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136 Variable A : Set.
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137
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138 CoFixpoint interleave (s1 s2 : stream A) : stream A :=
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139 match s1, s2 with
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140 | Cons h1 t1, Cons h2 t2 => Cons h1 (Cons h2 (interleave t1 t2))
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141 end.
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142 End interleave.
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143
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144 (** Now say we want to write a weird stuttering version of [map] that repeats elements in a particular way, based on interleaving. *)
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145
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146 Section map'.
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147 Variables A B : Set.
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148 Variable f : A -> B.
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149 (* begin thide *)
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150 (** [[
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151 CoFixpoint map' (s : stream A) : stream B :=
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152 match s with
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153 | Cons h t => interleave (Cons (f h) (map' t)) (Cons (f h) (map' t))
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154 end.
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155 ]]
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156
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157 We get another error message about an unguarded recursive call. *)
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158
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159 End map'.
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160
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161 (** What is going wrong here? Imagine that, instead of [interleave], we had called some other, less well-behaved function on streams. Here is one simpler example demonstrating the essential pitfall. We start defining a standard function for taking the tail of a stream. Since streams are infinite, this operation is total. *)
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162
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163 Definition tl A (s : stream A) : stream A :=
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164 match s with
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165 | Cons _ s' => s'
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166 end.
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167
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168 (** Coq rejects the following definition that uses [tl].
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169 [[
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170 CoFixpoint bad : stream nat := tl (Cons 0 bad).
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171 ]]
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172
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173 Imagine that Coq had accepted our definition, and consider how we might evaluate [approx bad 1]. We would be trying to calculate the first element in the stream [bad]. However, it is not hard to see that the definition of [bad] %``%#"#begs the question#"#%''%: unfolding the definition of [tl], we see that we essentially say %``%#"#define [bad] to equal itself#"#%''%! Of course such an equation admits no single well-defined solution, which does not fit well with the determinism of Gallina reduction.
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174
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175 Since Coq can be considered to check definitions after inlining and simplification of previously defined identifiers, the basic guardedness condition rules out our definition of [bad]. Such an inlining reduces [bad] to:
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176 [[
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177 CoFixpoint bad : stream nat := bad.
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178 ]]
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179 This is the same looping definition we rejected earlier. A similar inlining process reveals the way that Coq saw our failed definition of [map']:
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180 [[
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181 CoFixpoint map' (s : stream A) : stream B :=
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182 match s with
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183 | Cons h t => Cons (f h) (Cons (f h) (interleave (map' t) (map' t)))
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184 end.
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185 ]]
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186 Clearly in this case the [map'] calls are not immediate arguments to constructors, so we violate the guardedness condition. *)
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187 (* end thide *)
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188
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189 (** A more interesting question is why that condition is the right one. We can make an intuitive argument that the original [map'] definition is perfectly reasonable and denotes a well-understood transformation on streams, such that every output would behave properly with [approx]. The guardedness condition is an example of a syntactic check for %\index{productivity}\emph{%#<i>#productivity#</i>#%}% of co-recursive definitions. A productive definition can be thought of as one whose outputs can be forced in finite time to any finite approximation level, as with [approx]. If we replaced the guardedness condition with more involved checks, we might be able to detect and allow a broader range of productive definitions. However, mistakes in these checks could cause inconsistency, and programmers would need to understand the new, more complex checks. Coq's design strikes a balance between consistency and simplicity with its choice of guard condition, though we can imagine other worthwhile balances being struck, too. *)
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190
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191
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192 (** * Infinite Proofs *)
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193
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194 (** Let us say we want to give two different definitions of a stream of all ones, and then we want to prove that they are equivalent. *)
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195
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196 CoFixpoint ones : stream nat := Cons 1 ones.
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197 Definition ones' := map S zeroes.
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198
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199 (** The obvious statement of the equality is this: *)
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200
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201 Theorem ones_eq : ones = ones'.
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202
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203 (** However, faced with the initial subgoal, it is not at all clear how this theorem can be proved. In fact, it is unprovable. The [eq] predicate that we use is fundamentally limited to equalities that can be demonstrated by finite, syntactic arguments. To prove this equivalence, we will need to introduce a new relation. *)
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204 (* begin thide *)
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205
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206 Abort.
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207
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208 (** Co-inductive datatypes make sense by analogy from Haskell. What we need now is a %\textit{%#<i>#co-inductive proposition#</i>#%}%. That is, we want to define a proposition whose proofs may be infinite, subject to the guardedness condition. The idea of infinite proofs does not show up in usual mathematics, but it can be very useful (unsurprisingly) for reasoning about infinite data structures. Besides examples from Haskell, infinite data and proofs will also turn out to be useful for modelling inherently infinite mathematical objects, like program executions.
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209
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210 We are ready for our first %\index{co-inductive predicates}%co-inductive predicate. *)
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211
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212 Section stream_eq.
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213 Variable A : Set.
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214
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215 CoInductive stream_eq : stream A -> stream A -> Prop :=
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216 | Stream_eq : forall h t1 t2,
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217 stream_eq t1 t2
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218 -> stream_eq (Cons h t1) (Cons h t2).
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219 End stream_eq.
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220
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221 (** We say that two streams are equal if and only if they have the same heads and their tails are equal. We use the normal finite-syntactic equality for the heads, and we refer to our new equality recursively for the tails.
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222
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223 We can try restating the theorem with [stream_eq]. *)
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224
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225 Theorem ones_eq : stream_eq ones ones'.
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226 (** Coq does not support tactical co-inductive proofs as well as it supports tactical inductive proofs. The usual starting point is the [cofix] tactic, which asks to structure this proof as a co-fixpoint. *)
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227
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228 cofix.
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229 (** [[
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230 ones_eq : stream_eq ones ones'
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231 ============================
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232 stream_eq ones ones'
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233
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234 ]]
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235
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236 It looks like this proof might be easier than we expected! *)
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237
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238 assumption.
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239 (** [[
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240 Proof completed.
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241 ]]
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242
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243 Unfortunately, we are due for some disappointment in our victory lap.
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244 [[
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245 Qed.
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246 ]]
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247
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248 <<
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249 Error:
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250 Recursive definition of ones_eq is ill-formed.
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251
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252 In environment
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253 ones_eq : stream_eq ones ones'
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254
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255 unguarded recursive call in "ones_eq"
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256 >>
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257
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258 Via the Curry-Howard correspondence, the same guardedness condition applies to our co-inductive proofs as to our co-inductive data structures. We should be grateful that this proof is rejected, because, if it were not, the same proof structure could be used to prove any co-inductive theorem vacuously, by direct appeal to itself!
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259
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260 Thinking about how Coq would generate a proof term from the proof script above, we see that the problem is that we are violating the guardedness condition. During our proofs, Coq can help us check whether we have yet gone wrong in this way. We can run the command [Guarded] in any context to see if it is possible to finish the proof in a way that will yield a properly guarded proof term.%\index{Vernacular commands!Guarded}%
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261 [[
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262 Guarded.
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263 ]]
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264
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265 Running [Guarded] here gives us the same error message that we got when we tried to run [Qed]. In larger proofs, [Guarded] can be helpful in detecting problems %\textit{%#<i>#before#</i>#%}% we think we are ready to run [Qed].
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adamc@63
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266
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adam@281
|
267 We need to start the co-induction by applying [stream_eq]'s constructor. To do that, we need to know that both arguments to the predicate are [Cons]es. Informally, this is trivial, but [simpl] is not able to help us. *)
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adamc@63
|
268
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adamc@63
|
269 Undo.
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adamc@63
|
270 simpl.
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adamc@63
|
271 (** [[
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adamc@63
|
272 ones_eq : stream_eq ones ones'
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adamc@63
|
273 ============================
|
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adamc@63
|
274 stream_eq ones ones'
|
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adamc@211
|
275
|
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adamc@211
|
276 ]]
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adamc@63
|
277
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adamc@211
|
278 It turns out that we are best served by proving an auxiliary lemma. *)
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adamc@211
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279
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adamc@63
|
280 Abort.
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adamc@63
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281
|
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adamc@63
|
282 (** First, we need to define a function that seems pointless on first glance. *)
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adamc@63
|
283
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adamc@63
|
284 Definition frob A (s : stream A) : stream A :=
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adamc@63
|
285 match s with
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adamc@63
|
286 | Cons h t => Cons h t
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adamc@63
|
287 end.
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adamc@63
|
288
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adamc@63
|
289 (** Next, we need to prove a theorem that seems equally pointless. *)
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adamc@63
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290
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adamc@63
|
291 Theorem frob_eq : forall A (s : stream A), s = frob s.
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adamc@63
|
292 destruct s; reflexivity.
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adamc@63
|
293 Qed.
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adamc@63
|
294
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adamc@63
|
295 (** But, miraculously, this theorem turns out to be just what we needed. *)
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adamc@63
|
296
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adamc@63
|
297 Theorem ones_eq : stream_eq ones ones'.
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adamc@63
|
298 cofix.
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adamc@63
|
299
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adamc@63
|
300 (** We can use the theorem to rewrite the two streams. *)
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adamc@211
|
301
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adamc@63
|
302 rewrite (frob_eq ones).
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adamc@63
|
303 rewrite (frob_eq ones').
|
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adamc@63
|
304 (** [[
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adamc@63
|
305 ones_eq : stream_eq ones ones'
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adamc@63
|
306 ============================
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adamc@63
|
307 stream_eq (frob ones) (frob ones')
|
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adamc@211
|
308
|
|
adamc@211
|
309 ]]
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adamc@63
|
310
|
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adamc@211
|
311 Now [simpl] is able to reduce the streams. *)
|
|
adamc@63
|
312
|
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adamc@63
|
313 simpl.
|
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adamc@63
|
314 (** [[
|
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adamc@63
|
315 ones_eq : stream_eq ones ones'
|
|
adamc@63
|
316 ============================
|
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adamc@63
|
317 stream_eq (Cons 1 ones)
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adamc@63
|
318 (Cons 1
|
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adamc@63
|
319 ((cofix map (s : stream nat) : stream nat :=
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adamc@63
|
320 match s with
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adamc@63
|
321 | Cons h t => Cons (S h) (map t)
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adamc@63
|
322 end) zeroes))
|
|
adamc@211
|
323
|
|
adamc@211
|
324 ]]
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adamc@63
|
325
|
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adam@346
|
326 Note that [cofix] notation for anonymous co-recursion, which is analogous to the [fix] notation we have already seen for recursion. Since we have exposed the [Cons] structure of each stream, we can apply the constructor of [stream_eq]. *)
|
|
adamc@63
|
327
|
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adamc@63
|
328 constructor.
|
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adamc@63
|
329 (** [[
|
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adamc@63
|
330 ones_eq : stream_eq ones ones'
|
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adamc@63
|
331 ============================
|
|
adamc@63
|
332 stream_eq ones
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adamc@63
|
333 ((cofix map (s : stream nat) : stream nat :=
|
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adamc@63
|
334 match s with
|
|
adamc@63
|
335 | Cons h t => Cons (S h) (map t)
|
|
adamc@63
|
336 end) zeroes)
|
|
adamc@211
|
337
|
|
adamc@211
|
338 ]]
|
|
adamc@63
|
339
|
|
adamc@211
|
340 Now, modulo unfolding of the definition of [map], we have matched our assumption. *)
|
|
adamc@211
|
341
|
|
adamc@63
|
342 assumption.
|
|
adamc@63
|
343 Qed.
|
|
adamc@63
|
344
|
|
adamc@63
|
345 (** Why did this silly-looking trick help? The answer has to do with the constraints placed on Coq's evaluation rules by the need for termination. The [cofix]-related restriction that foiled our first attempt at using [simpl] is dual to a restriction for [fix]. In particular, an application of an anonymous [fix] only reduces when the top-level structure of the recursive argument is known. Otherwise, we would be unfolding the recursive definition ad infinitum.
|
|
adamc@63
|
346
|
|
adamc@63
|
347 Fixpoints only reduce when enough is known about the %\textit{%#<i>#definitions#</i>#%}% of their arguments. Dually, co-fixpoints only reduce when enough is known about %\textit{%#<i>#how their results will be used#</i>#%}%. In particular, a [cofix] is only expanded when it is the discriminee of a [match]. Rewriting with our superficially silly lemma wrapped new [match]es around the two [cofix]es, triggering reduction.
|
|
adamc@63
|
348
|
|
adamc@63
|
349 If [cofix]es reduced haphazardly, it would be easy to run into infinite loops in evaluation, since we are, after all, building infinite objects.
|
|
adamc@63
|
350
|
|
adamc@63
|
351 One common source of difficulty with co-inductive proofs is bad interaction with standard Coq automation machinery. If we try to prove [ones_eq'] with automation, like we have in previous inductive proofs, we get an invalid proof. *)
|
|
adamc@63
|
352
|
|
adamc@63
|
353 Theorem ones_eq' : stream_eq ones ones'.
|
|
adamc@63
|
354 cofix; crush.
|
|
adamc@63
|
355 (** [[
|
|
adamc@205
|
356 Guarded.
|
|
adam@302
|
357 ]]
|
|
adam@302
|
358 *)
|
|
adamc@63
|
359 Abort.
|
|
adam@346
|
360
|
|
adam@346
|
361 (** The standard [auto] machinery sees that our goal matches an assumption and so applies that assumption, even though this violates guardedness. One usually starts a proof like this by [destruct]ing some parameter and running a custom tactic to figure out the first proof rule to apply for each case. Alternatively, there are tricks that can be played with %``%#"#hiding#"#%''% the co-inductive hypothesis.
|
|
adam@346
|
362
|
|
adam@346
|
363 %\medskip%
|
|
adam@346
|
364
|
|
adam@346
|
365 Must we always be cautious with automation in proofs by co-induction? Induction seems to have dual versions the same pitfalls inherent in it, and yet we avoid those pitfalls by encapsulating safe Curry-Howard recursion schemes inside named induction principles. It turns out that we can usually do the same with %\index{co-induction principles}\emph{%#<i>#co-induction principles#</i>#%}%. Let us take that tack here, so that we can arrive at an [induction x; crush]-style proof for [ones_eq'].
|
|
adam@346
|
366
|
|
adam@346
|
367 An induction principle is parameterized over a predicate characterizing what we mean to prove, %\emph{%#<i>#as a function of the inductive fact that we already know#</i>#%}%. Dually, a co-induction principle ought to be parameterized over a predicate characterizing what we mean to prove, %\emph{%#<i>#as a function of the arguments to the co-inductive predicate that we are trying to prove#</i>#%}%.
|
|
adam@346
|
368
|
|
adam@346
|
369 To state a useful principle for [stream_eq], it will be useful first to define the stream head function. *)
|
|
adam@346
|
370
|
|
adam@346
|
371 Definition hd A (s : stream A) : A :=
|
|
adam@346
|
372 match s with
|
|
adam@346
|
373 | Cons x _ => x
|
|
adam@346
|
374 end.
|
|
adam@346
|
375
|
|
adam@346
|
376 (** Now we enter a section for the co-induction principle, based on %\index{Park's principle}%Park's principle as introduced in a tutorial by Gim%\'%enez%~\cite{IT}%. *)
|
|
adam@346
|
377
|
|
adam@346
|
378 Section stream_eq_coind.
|
|
adam@346
|
379 Variable A : Set.
|
|
adam@346
|
380 Variable R : stream A -> stream A -> Prop.
|
|
adam@346
|
381 (** This relation generalizes the theorem we want to prove, characterizinge exactly which two arguments to [stream_eq] we want to consider. *)
|
|
adam@346
|
382
|
|
adam@346
|
383 Hypothesis Cons_case_hd : forall s1 s2, R s1 s2 -> hd s1 = hd s2.
|
|
adam@346
|
384 Hypothesis Cons_case_tl : forall s1 s2, R s1 s2 -> R (tl s1) (tl s2).
|
|
adam@346
|
385 (** Two hypotheses characterize what makes a good choice of [R]: it enforces equality of stream heads, and it is %``%#<i>#hereditary#</i>#%''% in the sense that a [R] stream pair passes on %``%#"#[R]-ness#"#%''% to its tails. An established technical term for such a relation is %\index{bisimulation}\emph{%#<i>#bisimulation#</i>#%}%. *)
|
|
adam@346
|
386
|
|
adam@346
|
387 (** Now it is straightforward to prove the principle, which says that any stream pair in [R] is equal. The reader may wish to step through the proof script to see what is going on. *)
|
|
adam@346
|
388 Theorem stream_eq_coind : forall s1 s2, R s1 s2 -> stream_eq s1 s2.
|
|
adam@346
|
389 cofix; destruct s1; destruct s2; intro.
|
|
adam@346
|
390 generalize (Cons_case_hd H); intro Heq; simpl in Heq; rewrite Heq.
|
|
adam@346
|
391 constructor.
|
|
adam@346
|
392 apply stream_eq_coind.
|
|
adam@346
|
393 apply (Cons_case_tl H).
|
|
adam@346
|
394 Qed.
|
|
adam@346
|
395 End stream_eq_coind.
|
|
adam@346
|
396
|
|
adam@346
|
397 (** To see why this proof is guarded, we can print it and verify that the one co-recursive call is an immediate argument to a constructor. *)
|
|
adam@346
|
398 Print stream_eq_coind.
|
|
adam@346
|
399
|
|
adam@346
|
400 (** We omit the output and proceed to proving [ones_eq''] again. The only bit of ingenuity is in choosing [R], and in this case the most restrictive predicate works. *)
|
|
adam@346
|
401
|
|
adam@346
|
402 Theorem ones_eq'' : stream_eq ones ones'.
|
|
adam@346
|
403 apply (stream_eq_coind (fun s1 s2 => s1 = ones /\ s2 = ones')); crush.
|
|
adam@346
|
404 Qed.
|
|
adam@346
|
405
|
|
adam@346
|
406 (** Note that this proof achieves the proper reduction behavior via [hd] and [tl], rather than [frob] as in the last proof. All three functions pattern match on their arguments, catalyzing computation steps.
|
|
adam@346
|
407
|
|
adam@346
|
408 Compared to the inductive proofs that we are used to, it still seems unsatisfactory that we had to write out a choice of [R] in the last proof. An alternate is to capture a common pattern of co-recursion in a more specialized co-induction principle. For the current example, that pattern is: prove [stream_eq s1 s2] where [s1] and [s2] are defined as their own tails. *)
|
|
adam@346
|
409
|
|
adam@346
|
410 Section stream_eq_loop.
|
|
adam@346
|
411 Variable A : Set.
|
|
adam@346
|
412 Variables s1 s2 : stream A.
|
|
adam@346
|
413
|
|
adam@346
|
414 Hypothesis Cons_case_hd : hd s1 = hd s2.
|
|
adam@346
|
415 Hypothesis loop1 : tl s1 = s1.
|
|
adam@346
|
416 Hypothesis loop2 : tl s2 = s2.
|
|
adam@346
|
417
|
|
adam@346
|
418 (** The proof of the principle includes a choice of [R], so that we no longer need to make such choices thereafter. *)
|
|
adam@346
|
419
|
|
adam@346
|
420 Theorem stream_eq_loop : stream_eq s1 s2.
|
|
adam@346
|
421 apply (stream_eq_coind (fun s1' s2' => s1' = s1 /\ s2' = s2)); crush.
|
|
adam@346
|
422 Qed.
|
|
adam@346
|
423 End stream_eq_loop.
|
|
adam@346
|
424
|
|
adam@346
|
425 Theorem ones_eq''' : stream_eq ones ones'.
|
|
adam@346
|
426 apply stream_eq_loop; crush.
|
|
adam@346
|
427 Qed.
|
|
adamc@68
|
428 (* end thide *)
|
|
adamc@63
|
429
|
|
adam@346
|
430 (** Let us put [stream_eq_ind] through its paces a bit more, considering two different ways to compute infinite streams of all factorial values. First, we import the [fact] factorial function from the standard library. *)
|
|
adam@346
|
431
|
|
adam@346
|
432 Require Import Arith.
|
|
adam@346
|
433 Print fact.
|
|
adam@346
|
434 (** %\vspace{-.15in}%[[
|
|
adam@346
|
435 fact =
|
|
adam@346
|
436 fix fact (n : nat) : nat :=
|
|
adam@346
|
437 match n with
|
|
adam@346
|
438 | 0 => 1
|
|
adam@346
|
439 | S n0 => S n0 * fact n0
|
|
adam@346
|
440 end
|
|
adam@346
|
441 : nat -> nat
|
|
adam@346
|
442 ]]
|
|
adam@346
|
443 *)
|
|
adam@346
|
444
|
|
adam@346
|
445 (** The simplest way to compute the factorial stream involves calling [fact] afresh at each position. *)
|
|
adam@346
|
446
|
|
adam@346
|
447 CoFixpoint fact_slow' (n : nat) := Cons (fact n) (fact_slow' (S n)).
|
|
adam@346
|
448 Definition fact_slow := fact_slow' 1.
|
|
adam@346
|
449
|
|
adam@346
|
450 (** A more clever, optimized method maintains an accumulator of the previous factorial, so that each new entry can be computed with a single multiplication. *)
|
|
adam@346
|
451
|
|
adam@346
|
452 CoFixpoint fact_iter' (cur acc : nat) := Cons acc (fact_iter' (S cur) (acc * cur)).
|
|
adam@346
|
453 Definition fact_iter := fact_iter' 2 1.
|
|
adam@346
|
454
|
|
adam@346
|
455 (** We can verify that the streams are equal up to particular finite bounds. *)
|
|
adam@346
|
456
|
|
adam@346
|
457 Eval simpl in approx fact_iter 5.
|
|
adam@346
|
458 (** %\vspace{-.15in}%[[
|
|
adam@346
|
459 = 1 :: 2 :: 6 :: 24 :: 120 :: nil
|
|
adam@346
|
460 : list nat
|
|
adam@346
|
461 ]]
|
|
adam@346
|
462 *)
|
|
adam@346
|
463 Eval simpl in approx fact_slow 5.
|
|
adam@346
|
464 (** %\vspace{-.15in}%[[
|
|
adam@346
|
465 = 1 :: 2 :: 6 :: 24 :: 120 :: nil
|
|
adam@346
|
466 : list nat
|
|
adam@346
|
467 ]]
|
|
adam@346
|
468
|
|
adam@346
|
469 Now, to prove that the two versions are equivalent, it is helpful to prove (and add as a proof hint) a quick lemma about the computational behavior of [fact]. *)
|
|
adam@346
|
470
|
|
adam@346
|
471 (* begin thide *)
|
|
adam@346
|
472 Lemma fact_def : forall x n,
|
|
adam@346
|
473 fact_iter' x (fact n * S n) = fact_iter' x (fact (S n)).
|
|
adam@346
|
474 simpl; intros; f_equal; ring.
|
|
adam@346
|
475 Qed.
|
|
adam@346
|
476
|
|
adam@346
|
477 Hint Resolve fact_def.
|
|
adam@346
|
478
|
|
adam@346
|
479 (** With the hint added, it is easy to prove an auxiliary lemma relating [fact_iter'] and [fact_slow']. The key bit of ingenuity is introduction of an existential quantifier for the shared parameter [n]. *)
|
|
adam@346
|
480
|
|
adam@346
|
481 Lemma fact_eq' : forall n, stream_eq (fact_iter' (S n) (fact n)) (fact_slow' n).
|
|
adam@346
|
482 intro; apply (stream_eq_coind (fun s1 s2 => exists n, s1 = fact_iter' (S n) (fact n)
|
|
adam@346
|
483 /\ s2 = fact_slow' n)); crush; eauto.
|
|
adam@346
|
484 Qed.
|
|
adam@346
|
485
|
|
adam@346
|
486 (** The final theorem is a direct corollary of [fact_eq']. *)
|
|
adam@346
|
487
|
|
adam@346
|
488 Theorem fact_eq : stream_eq fact_iter fact_slow.
|
|
adam@346
|
489 apply fact_eq'.
|
|
adam@346
|
490 Qed.
|
|
adam@346
|
491
|
|
adam@346
|
492 (** As in the case of [ones_eq'], we may be unsatisfied that we needed to write down a choice of [R] that seems to duplicate information already present in a lemma statement. We can facilitate a simpler proof by defining a co-induction principle specialized to goals that begin with single universal quantifiers, and the strategy can be extended in a straightforward way to principles for other counts of quantifiers. (Our [stream_eq_loop] principle is effectively the instantiation of this technique to zero quantifiers.) *)
|
|
adam@346
|
493
|
|
adam@346
|
494 Section stream_eq_onequant.
|
|
adam@346
|
495 Variables A B : Set.
|
|
adam@346
|
496 (** We have the types [A], the domain of the one quantifier; and [B], the type of data found in the streams. *)
|
|
adam@346
|
497
|
|
adam@346
|
498 Variables f g : A -> stream B.
|
|
adam@346
|
499 (** The two streams we compare must be of the forms [f x] and [g x], for some shared [x]. Note that this falls out naturally when [x] is a shared universally quantified variable in a lemma statement. *)
|
|
adam@346
|
500
|
|
adam@346
|
501 Hypothesis Cons_case_hd : forall x, hd (f x) = hd (g x).
|
|
adam@346
|
502 Hypothesis Cons_case_tl : forall x, exists y, tl (f x) = f y /\ tl (g x) = g y.
|
|
adam@346
|
503 (** These conditions are inspired by the bisimulation requirements, with a more general version of the [R] choice we made for [fact_eq'] inlined into the hypotheses of [stream_eq_coind]. *)
|
|
adam@346
|
504
|
|
adam@346
|
505 Theorem stream_eq_onequant : forall x, stream_eq (f x) (g x).
|
|
adam@346
|
506 intro; apply (stream_eq_coind (fun s1 s2 => exists x, s1 = f x /\ s2 = g x)); crush; eauto.
|
|
adam@346
|
507 Qed.
|
|
adam@346
|
508 End stream_eq_onequant.
|
|
adam@346
|
509
|
|
adam@346
|
510 Lemma fact_eq'' : forall n, stream_eq (fact_iter' (S n) (fact n)) (fact_slow' n).
|
|
adam@346
|
511 apply stream_eq_onequant; crush; eauto.
|
|
adam@346
|
512 Qed.
|
|
adam@346
|
513
|
|
adam@346
|
514 (** We have arrived at one of our customary automated proofs, thanks to the new principle. *)
|
|
adam@346
|
515 (* end thide *)
|
|
adamc@64
|
516
|
|
adamc@64
|
517
|
|
adamc@64
|
518 (** * Simple Modeling of Non-Terminating Programs *)
|
|
adamc@64
|
519
|
|
adam@347
|
520 (** We close the chapter with a quick motivating example for more complex uses of co-inductive types. We will define a co-inductive semantics for a simple imperative programming language and use that semantics to prove the correctness of a trivial optimization that removes spurious additions by 0. We follow the technique of %\index{co-inductive big-step operational semantics}\emph{%#<i>#co-inductive big-step operational semantics#</i>#%}~\cite{BigStep}%.
|
|
adamc@64
|
521
|
|
adam@347
|
522 We define a suggestive synonym for [nat], as we will consider programs with infinitely many variables, represented as [nat]s. *)
|
|
adamc@211
|
523
|
|
adam@347
|
524 Definition var := nat.
|
|
adamc@64
|
525
|
|
adam@347
|
526 (** We define a type [vars] of maps from variables to values. To define a function [set] for setting a variable's value in a map, we import the [Arith] module from Coq's standard library, and we use its function [beq_nat] for comparing natural numbers. *)
|
|
adamc@64
|
527
|
|
adam@347
|
528 Definition vars := var -> nat.
|
|
adam@347
|
529 Require Import Arith.
|
|
adam@347
|
530 Definition set (vs : vars) (v : var) (n : nat) : vars :=
|
|
adam@347
|
531 fun v' => if beq_nat v v' then n else vs v'.
|
|
adamc@67
|
532
|
|
adam@347
|
533 (** We define a simple arithmetic expression language with variables, and we give it a semantics via an interpreter. *)
|
|
adamc@67
|
534
|
|
adam@347
|
535 Inductive exp : Set :=
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adam@347
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536 | Const : nat -> exp
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adam@347
|
537 | Var : var -> exp
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adam@347
|
538 | Plus : exp -> exp -> exp.
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adamc@64
|
539
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adam@347
|
540 Fixpoint evalExp (vs : vars) (e : exp) : nat :=
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|
adam@347
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541 match e with
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adam@347
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542 | Const n => n
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adam@347
|
543 | Var v => vs v
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adam@347
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544 | Plus e1 e2 => evalExp vs e1 + evalExp vs e2
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|
adam@347
|
545 end.
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adamc@64
|
546
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adam@347
|
547 (** Finally, we define a language of commands. It includes variable assignment, sequencing, and a %\texttt{%#<tt>#while#</tt>#%}% form that repeats as long as its test expression evaluates to a nonzero value. *)
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adamc@64
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548
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|
adam@347
|
549 Inductive cmd : Set :=
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adam@347
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550 | Assign : var -> exp -> cmd
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adam@347
|
551 | Seq : cmd -> cmd -> cmd
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adam@347
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552 | While : exp -> cmd -> cmd.
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|
adamc@64
|
553
|
|
adam@347
|
554 (** We could define an inductive relation to characterize the results of command evaluation. However, such a relation would not capture %\emph{%#<i>#nonterminating#</i>#%}% executions. With a co-inductive relation, we can capture both cases. The parameters of the relation are an initial state, a command, and a final state. A program that does not terminate in a particular initial state is related to %\emph{%#<i>#any#</i>#%}% final state. *)
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|
adamc@67
|
555
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|
adam@347
|
556 CoInductive evalCmd : vars -> cmd -> vars -> Prop :=
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adam@347
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557 | EvalAssign : forall vs v e, evalCmd vs (Assign v e) (set vs v (evalExp vs e))
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adam@347
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558 | EvalSeq : forall vs1 vs2 vs3 c1 c2, evalCmd vs1 c1 vs2
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adam@347
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559 -> evalCmd vs2 c2 vs3
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adam@347
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560 -> evalCmd vs1 (Seq c1 c2) vs3
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adam@347
|
561 | EvalWhileFalse : forall vs e c, evalExp vs e = 0
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|
adam@347
|
562 -> evalCmd vs (While e c) vs
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adam@347
|
563 | EvalWhileTrue : forall vs1 vs2 vs3 e c, evalExp vs1 e <> 0
|
|
adam@347
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564 -> evalCmd vs1 c vs2
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adam@347
|
565 -> evalCmd vs2 (While e c) vs3
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|
adam@347
|
566 -> evalCmd vs1 (While e c) vs3.
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|
adam@347
|
567
|
|
adam@347
|
568 (** Having learned our lesson in the last section, before proceeding, we build a co-induction principle for [evalCmd]. *)
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adam@347
|
569
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adam@347
|
570 Section evalCmd_coind.
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adam@347
|
571 Variable R : vars -> cmd -> vars -> Prop.
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adam@347
|
572
|
|
adam@347
|
573 Hypothesis AssignCase : forall vs1 vs2 v e, R vs1 (Assign v e) vs2
|
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adam@347
|
574 -> vs2 = set vs1 v (evalExp vs1 e).
|
|
adam@347
|
575
|
|
adam@347
|
576 Hypothesis SeqCase : forall vs1 vs3 c1 c2, R vs1 (Seq c1 c2) vs3
|
|
adam@347
|
577 -> exists vs2, R vs1 c1 vs2 /\ R vs2 c2 vs3.
|
|
adam@347
|
578
|
|
adam@347
|
579 Hypothesis WhileCase : forall vs1 vs3 e c, R vs1 (While e c) vs3
|
|
adam@347
|
580 -> (evalExp vs1 e = 0 /\ vs3 = vs1)
|
|
adam@347
|
581 \/ exists vs2, evalExp vs1 e <> 0 /\ R vs1 c vs2 /\ R vs2 (While e c) vs3.
|
|
adam@347
|
582
|
|
adam@347
|
583 (** The proof is routine. We make use of a form of %\index{tactics!destruct}%[destruct] that takes an %\index{intro pattern}\emph{%#<i>#intro pattern#</i>#%}% in an [as] clause. These patterns control how deeply we break apart the components of an inductive value, and we refer the reader to the Coq manual for more details. *)
|
|
adam@347
|
584
|
|
adam@347
|
585 Theorem evalCmd_coind : forall vs1 c vs2, R vs1 c vs2 -> evalCmd vs1 c vs2.
|
|
adam@347
|
586 cofix; intros; destruct c.
|
|
adam@347
|
587 rewrite (AssignCase H); constructor.
|
|
adam@347
|
588 destruct (SeqCase H) as [? [? ?]]; econstructor; eauto.
|
|
adam@347
|
589 destruct (WhileCase H) as [[? ?] | [? [? [? ?]]]]; subst;
|
|
adam@347
|
590 [ econstructor | econstructor 4 ]; eauto.
|
|
adam@347
|
591 Qed.
|
|
adam@347
|
592 End evalCmd_coind.
|
|
adam@347
|
593
|
|
adam@347
|
594 (** Now that we have a co-induction principle, we should use it to prove something! Our example is a trivial program optimizer that finds places to replace [0 + e] with [e]. *)
|
|
adam@347
|
595
|
|
adam@347
|
596 Fixpoint optExp (e : exp) : exp :=
|
|
adam@347
|
597 match e with
|
|
adam@347
|
598 | Plus (Const 0) e => optExp e
|
|
adam@347
|
599 | Plus e1 e2 => Plus (optExp e1) (optExp e2)
|
|
adam@347
|
600 | _ => e
|
|
adam@347
|
601 end.
|
|
adam@347
|
602
|
|
adam@347
|
603 Fixpoint optCmd (c : cmd) : cmd :=
|
|
adam@347
|
604 match c with
|
|
adam@347
|
605 | Assign v e => Assign v (optExp e)
|
|
adam@347
|
606 | Seq c1 c2 => Seq (optCmd c1) (optCmd c2)
|
|
adam@347
|
607 | While e c => While (optExp e) (optCmd c)
|
|
adam@347
|
608 end.
|
|
adam@347
|
609
|
|
adam@347
|
610 (** Before proving correctness of [optCmd], we prove a lemma about [optExp]. This is where we have to do the most work, choosing pattern match opportunities automatically. *)
|
|
adam@347
|
611
|
|
adam@347
|
612 (* begin thide *)
|
|
adam@347
|
613 Lemma optExp_correct : forall vs e, evalExp vs (optExp e) = evalExp vs e.
|
|
adam@347
|
614 induction e; crush;
|
|
adam@347
|
615 repeat (match goal with
|
|
adam@347
|
616 | [ |- context[match ?E with Const _ => _ | Var _ => _
|
|
adam@347
|
617 | Plus _ _ => _ end] ] => destruct E
|
|
adam@347
|
618 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
|
|
adam@347
|
619 end; crush).
|
|
adamc@64
|
620 Qed.
|
|
adamc@64
|
621
|
|
adam@347
|
622 Hint Rewrite optExp_correct : cpdt.
|
|
adamc@64
|
623
|
|
adam@347
|
624 (** The final theorem is easy to establish, using our co-induction principle and a bit of Ltac smarts that we leave unexplained for now. Curious readers can consult the Coq manual, or wait for the later chapters of this book about proof automation. *)
|
|
adamc@64
|
625
|
|
adam@347
|
626 Theorem optCmd_correct : forall vs1 c vs2, evalCmd vs1 c vs2
|
|
adam@347
|
627 -> evalCmd vs1 (optCmd c) vs2.
|
|
adam@347
|
628 intros; apply (evalCmd_coind (fun vs1 c' vs2 => exists c, evalCmd vs1 c vs2
|
|
adam@347
|
629 /\ c' = optCmd c)); eauto; crush;
|
|
adam@347
|
630 match goal with
|
|
adam@347
|
631 | [ H : _ = optCmd ?E |- _ ] => destruct E; simpl in *; discriminate
|
|
adam@347
|
632 || injection H; intros; subst
|
|
adam@347
|
633 end; match goal with
|
|
adam@347
|
634 | [ H : evalCmd _ _ _ |- _ ] => ((inversion H; [])
|
|
adam@347
|
635 || (inversion H; [|])); subst
|
|
adam@347
|
636 end; crush; eauto 10.
|
|
adam@347
|
637 Qed.
|
|
adam@347
|
638 (* end thide *)
|
|
adamc@64
|
639
|
|
adam@347
|
640 (** In this form, the theorem tells us that the optimizer preserves observable behavior of both terminating and nonterminating programs, but we did not have to do more work than for the case of terminating programs alone. We merely took the natural inductive definition for terminating executions, made it co-inductive, and applied the appropriate co-induction principle. Curious readers might experiment with adding command constructs like %\texttt{%#<tt>#if#</tt>#%}%; the same proof should continue working, after the co-induction principle is extended to the new evaluation rules. *)
|
|
adamc@81
|
641
|
|
adamc@81
|
642
|
|
adamc@81
|
643 (** * Exercises *)
|
|
adamc@81
|
644
|
|
adamc@81
|
645 (** %\begin{enumerate}%#<ol>#
|
|
adamc@81
|
646
|
|
adamc@81
|
647 %\item%#<li># %\begin{enumerate}%#<ol>#
|
|
adamc@81
|
648 %\item%#<li># Define a co-inductive type of infinite trees carrying data of a fixed parameter type. Each node should contain a data value and two child trees.#</li>#
|
|
adamc@81
|
649 %\item%#<li># Define a function [everywhere] for building a tree with the same data value at every node.#</li>#
|
|
adamc@81
|
650 %\item%#<li># Define a function [map] for building an output tree out of two input trees by traversing them in parallel and applying a two-argument function to their corresponding data values.#</li>#
|
|
adamc@104
|
651 %\item%#<li># Define a tree [falses] where every node has the value [false].#</li>#
|
|
adamc@104
|
652 %\item%#<li># Define a tree [true_false] where the root node has value [true], its children have value [false], all nodes at the next have the value [true], and so on, alternating boolean values from level to level.#</li>#
|
|
adam@292
|
653 %\item%#<li># Prove that [true_false] is equal to the result of mapping the boolean %``%#"#or#"#%''% function [orb] over [true_false] and [falses]. You can make [orb] available with [Require Import Bool.]. You may find the lemma [orb_false_r] from the same module helpful. Your proof here should not be about the standard equality [=], but rather about some new equality relation that you define.#</li>#
|
|
adamc@81
|
654 #</ol>#%\end{enumerate}% #</li>#
|
|
adamc@81
|
655
|
|
adamc@81
|
656 #</ol>#%\end{enumerate}% *)
|
