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1 (* Copyright (c) 2006, 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 Arith List.
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12
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13 Require Import CpdtTactics Coinductive.
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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{General Recursion}% *)
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20
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21 (** Termination of all programs is a crucial property of Gallina. Non-terminating programs introduce logical inconsistency, where any theorem can be proved with an infinite loop. Coq uses a small set of conservative, syntactic criteria to check termination of all recursive definitions. These criteria are insufficient to support the natural encodings of a variety of important programming idioms. Further, since Coq makes it so convenient to encode mathematics computationally, with functional programs, we may find ourselves wanting to employ more complicated recursion in mathematical definitions.
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22
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23 What exactly are the conservative criteria that we run up against? For %\emph{%#<i>#recursive#</i>#%}% definitions, recursive calls are only allowed on %\emph{%#<i>#syntactic subterms#</i>#%}% of the original primary argument, a restriction known as %\index{primitive recursion}\emph{%#<i>#primitive recursion#</i>#%}%. In fact, Coq's handling of reflexive inductive types (those defined in terms of functions returning the same type) gives a bit more flexibility than in traditional primitive recursion, but the term is still applied commonly. In Chapter 5, we saw how %\emph{%#<i>#co-recursive#</i>#%}% definitions are checked against a syntactic guardness condition that guarantees productivity.
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24
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25 Many natural recursion patterns satisfy neither condition. For instance, there is our simple running example in this chapter, merge sort. We will study three different approaches to more flexible recursion, and the latter two of the approaches will even support definitions that may fail to terminate on certain inputs, without any up-front characterization of which inputs those may be.
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26
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27 Before proceeding, it is important to note that the problem here is not as fundamental as it may appear. The final example of Chapter 5 demonstrated what is called a %\index{deep embedding}\emph{%#<i>#deep embedding#</i>#%}% of the syntax and semantics of a programming language. That is, we gave a mathematical definition of a language of programs and their meanings. This language clearly admitted non-termination, and we could think of writing all our sophisticated recursive functions with such explicit syntax types. However, in doing so, we forfeit our chance to take advantage of Coq's very good built-in support for reasoning about Gallina programs. We would rather use a %\index{shallow embedding}\emph{%#<i>#shallow embedding#</i>#%}%, where we model informal constructs by encoding them as normal Gallina programs. Each of the three techniques of this chapter follows that style. *)
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28
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29
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30 (** * Well-Founded Recursion *)
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31
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32 (** The essence of terminating recursion is that there are no infinite chains of nested recursive calls. This intuition is commonly mapped to the mathematical idea of a %\index{well-founded relation}\emph{%#<i>#well-founded relation#</i>#%}%, and the associated standard technique in Coq is %\index{well-founded recursion}\emph{%#<i>#well-founded recursion#</i>#%}%. The syntactic-subterm relation that Coq applies by default is well-founded, but many cases demand alternate well-founded relations. To demonstrate, let us see where we get stuck on attempting a standard merge sort implementation. *)
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33
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34 Section mergeSort.
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35 Variable A : Type.
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36 Variable le : A -> A -> bool.
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37 (** We have a set equipped with some %``%#"#less-than-or-equal-to#"#%''% test. *)
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38
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39 (** A standard function inserts an element into a sorted list, preserving sortedness. *)
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40
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41 Fixpoint insert (x : A) (ls : list A) : list A :=
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42 match ls with
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43 | nil => x :: nil
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44 | h :: ls' =>
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45 if le x h
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46 then x :: ls
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47 else h :: insert x ls'
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48 end.
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49
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50 (** We will also need a function to merge two sorted lists. (We use a less efficient implementation than usual, because the more efficient implementation already forces us to think about well-founded recursion, while here we are only interested in setting up the example of merge sort.) *)
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51
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52 Fixpoint merge (ls1 ls2 : list A) : list A :=
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53 match ls1 with
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54 | nil => ls2
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55 | h :: ls' => insert h (merge ls' ls2)
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56 end.
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57
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58 (** The last helper function for classic merge sort is the one that follows, to partition a list arbitrarily into two pieces of approximately equal length. *)
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59
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60 Fixpoint partition (ls : list A) : list A * list A :=
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61 match ls with
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62 | nil => (nil, nil)
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63 | h :: nil => (h :: nil, nil)
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64 | h1 :: h2 :: ls' =>
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65 let (ls1, ls2) := partition ls' in
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66 (h1 :: ls1, h2 :: ls2)
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67 end.
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68
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69 (** Now, let us try to write the final sorting function, using a natural number %``%#"#[<=]#"#%''% test [leb] from the standard library.
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70 [[
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71 Fixpoint mergeSort (ls : list A) : list A :=
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72 if leb (length ls) 2
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73 then ls
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74 else let lss := partition ls in
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75 merge (mergeSort (fst lss)) (mergeSort (snd lss)).
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76 ]]
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77
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78 <<
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79 Recursive call to mergeSort has principal argument equal to
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80 "fst (partition ls)" instead of a subterm of "ls".
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81 >>
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82
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83 The definition is rejected for not following the simple primitive recursion criterion. In particular, it is not apparent that recursive calls to [mergeSort] are syntactic subterms of the original argument [ls]; indeed, they are not, yet we know this is a well-founded recursive definition.
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84
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85 To produce an acceptable definition, we need to choose a well-founded relation and prove that [mergeSort] respects it. A good starting point is an examination of how well-foundedness is formalized in the Coq standard library. *)
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86
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87 Print well_founded.
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88 (** %\vspace{-.15in}% [[
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89 well_founded =
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90 fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a
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91 ]]
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92
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93 The bulk of the definitional work devolves to the %\index{accessibility relation}\index{Gallina terms!Acc}\emph{%#<i>#accessibility#</i>#%}% relation [Acc], whose definition we may also examine. *)
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94
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95 Print Acc.
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96 (** %\vspace{-.15in}% [[
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97 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop :=
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98 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x
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99 ]]
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100
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101 In prose, an element [x] is accessible for a relation [R] if every element %``%#"#less than#"#%''% [x] according to [R] is also accessible. Since [Acc] is defined inductively, we know that any accessibility proof involves a finite chain of invocations, in a certain sense which we can make formal. Building on Chapter 5's examples, let us define a co-inductive relation that is closer to the usual informal notion of %``%#"#absence of infinite decreasing chains.#"#%''% *)
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102
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103 CoInductive isChain A (R : A -> A -> Prop) : stream A -> Prop :=
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104 | ChainCons : forall x y s, isChain R (Cons y s)
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105 -> R y x
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106 -> isChain R (Cons x (Cons y s)).
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107
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108 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *)
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109
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110 (* begin thide *)
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111 Lemma noChains' : forall A (R : A -> A -> Prop) x, Acc R x
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112 -> forall s, ~isChain R (Cons x s).
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113 induction 1; crush;
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114 match goal with
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115 | [ H : isChain _ _ |- _ ] => inversion H; eauto
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116 end.
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117 Qed.
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118
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119 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *)
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120
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121 Theorem noChains : forall A (R : A -> A -> Prop), well_founded R
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122 -> forall s, ~isChain R s.
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123 destruct s; apply noChains'; auto.
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124 Qed.
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125 (* end thide *)
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126
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127 (** Absence of infinite decreasing chains implies absence of infinitely nested recursive calls, for any recursive definition that respects the well-founded relation. The [Fix] combinator from the standard library formalizes that intuition: *)
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128
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129 Check Fix.
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130 (** %\vspace{-.15in}%[[
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131 Fix
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132 : forall (A : Type) (R : A -> A -> Prop),
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133 well_founded R ->
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134 forall P : A -> Type,
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135 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
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136 forall x : A, P x
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137 ]]
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138
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139 A call to %\index{Gallina terms!Fix}%[Fix] must present a relation [R] and a proof of its well-foundedness. The next argument, [P], is the possibly dependent range type of the function we build; the domain [A] of [R] is the function's domain. The following argument has this type:
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140
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141 [[
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142 forall x : A, (forall y : A, R y x -> P y) -> P x
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143 ]]
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144
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145 This is an encoding of the function body. The input [x] stands for the function argument, and the next input stands for the function we are defining. Recursive calls are encoded as calls to the second argument, whose type tells us it expects a value [y] and a proof that [y] is %``%#"#less than#"#%''% [x], according to [R]. In this way, we enforce the well-foundedness restriction on recursive calls.
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146
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147 The rest of [Fix]'s type tells us that it returns a function of exactly the type we expect, so we are now ready to use it to implement [mergeSort]. Careful readers may have noticed that [Fix] has a dependent type of the sort we met in the previous chapter.
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148
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149 Before writing [mergeSort], we need to settle on a well-founded relation. The right one for this example is based on lengths of lists. *)
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150
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151 Definition lengthOrder (ls1 ls2 : list A) :=
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152 length ls1 < length ls2.
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153
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154 (** We must prove that the relation is truly well-founded. To save some space in the rest of this chapter, we skip right to nice, automated proof scripts, though we postpone introducing the principles behind such scripts to Part III of the book. Curious readers may still replace semicolons with periods and newlines to step through these scripts interactively. *)
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155
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156 Hint Constructors Acc.
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157
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158 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
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159 unfold lengthOrder; induction len; crush.
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160 Defined.
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161
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162 Theorem lengthOrder_wf : well_founded lengthOrder.
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163 red; intro; eapply lengthOrder_wf'; eauto.
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164 Defined.
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165
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166 (** Notice that we end these proofs with %\index{Vernacular commands!Defined}%[Defined], not [Qed]. Recall that [Defined] marks the theorems as %\emph{transparent}%, so that the details of their proofs may be used during program execution. Why could such details possibly matter for computation? It turns out that [Fix] satisfies the primitive recursion restriction by declaring itself as %\emph{%#<i>#recursive in the structure of [Acc] proofs#</i>#%}%. This is possible because [Acc] proofs follow a predictable inductive structure. We must do work, as in the last theorem's proof, to establish that all elements of a type belong to [Acc], but the automatic unwinding of those proofs during recursion is straightforward. If we ended the proof with [Qed], the proof details would be hidden from computation, in which case the unwinding process would get stuck.
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167
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168 To justify our two recursive [mergeSort] calls, we will also need to prove that [partition] respects the [lengthOrder] relation. These proofs, too, must be kept transparent, to avoid stuckness of [Fix] evaluation. *)
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169
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170 Lemma partition_wf : forall len ls, 2 <= length ls <= len
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171 -> let (ls1, ls2) := partition ls in
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172 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
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173 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
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174 destruct (le_lt_dec 2 (length ls));
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175 repeat (match goal with
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176 | [ _ : length ?E < 2 |- _ ] => destruct E
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177 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
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178 | [ IH : _ |- context[partition ?L] ] =>
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179 specialize (IH L); destruct (partition L); destruct IH
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180 end; crush).
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181 Defined.
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182
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183 Ltac partition := intros ls ?; intros; generalize (@partition_wf (length ls) ls);
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184 destruct (partition ls); destruct 1; crush.
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185
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186 Lemma partition_wf1 : forall ls, 2 <= length ls
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187 -> lengthOrder (fst (partition ls)) ls.
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188 partition.
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189 Defined.
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190
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191 Lemma partition_wf2 : forall ls, 2 <= length ls
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192 -> lengthOrder (snd (partition ls)) ls.
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193 partition.
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194 Defined.
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195
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196 Hint Resolve partition_wf1 partition_wf2.
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197
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198 (** To write the function definition itself, we use the %\index{tactics!refine}%[refine] tactic as a convenient way to write a program that needs to manipulate proofs, without writing out those proofs manually. We also use a replacement [le_lt_dec] for [leb] that has a more interesting dependent type. *)
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199
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200 Definition mergeSort : list A -> list A.
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201 (* begin thide *)
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202 refine (Fix lengthOrder_wf (fun _ => list A)
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203 (fun (ls : list A)
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204 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
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205 if le_lt_dec 2 (length ls)
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206 then let lss := partition ls in
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207 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
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208 else ls)); subst lss; eauto.
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209 Defined.
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210 (* end thide *)
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211 End mergeSort.
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212
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213 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
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214
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215 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
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216 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
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217
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218 (** Since the subject of this chapter is merely how to define functions with unusual recursion structure, we will not prove any further correctness theorems about [mergeSort]. Instead, we stop at proving that [mergeSort] has the expected computational behavior, for all inputs, not merely the one we just tested. *)
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219
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220 (* begin thide *)
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221 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
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222 mergeSort le ls = if le_lt_dec 2 (length ls)
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223 then let lss := partition ls in
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224 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
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225 else ls.
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226 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
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227
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228 (** The library theorem [Fix_eq] imposes one more strange subgoal upon us. We must prove that the function body is unable to distinguish between %``%#"#self#"#%''% arguments that map equal inputs to equal outputs. One might think this should be true of any Gallina code, but in fact this general %\index{extensionality}\emph{%#<i>#function extensionality#</i>#%}% property is neither provable nor disprovable within Coq. The type of [Fix_eq] makes clear what we must show manually: *)
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229
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230 Check Fix_eq.
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231 (** %\vspace{-.15in}%[[
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232 Fix_eq
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233 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
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234 (P : A -> Type)
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235 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
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236 (forall (x : A) (f g : forall y : A, R y x -> P y),
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237 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
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238 forall x : A,
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239 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
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240 ]]
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241
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242 Most such obligations are dischargable with straightforward proof automation, and this example is no exception. *)
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243
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244 match goal with
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245 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
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246 end; simpl; f_equal; auto.
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247 Qed.
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248 (* end thide *)
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249
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250 (** As a final test of our definition's suitability, we can extract to OCaml. *)
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251
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252 Extraction mergeSort.
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253
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254 (** <<
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255 let rec mergeSort le x =
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256 match le_lt_dec (S (S O)) (length x) with
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257 | Left ->
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258 let lss = partition x in
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259 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
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260 | Right -> x
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adam@351
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261 >>
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262
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adam@353
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263 We see almost precisely the same definition we would have written manually in OCaml! It might be a good exercise for the reader to use the commands we saw in the previous chapter to clean up some remaining differences from idiomatic OCaml.
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264
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265 One more piece of the full picture is missing. To go on and prove correctness of [mergeSort], we would need more than a way of unfolding its definition. We also need an appropriate induction principle matched to the well-founded relation. Such a principle is available in the standard library, though we will say no more about its details here. *)
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266
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267 Check well_founded_induction.
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268 (** %\vspace{-.15in}%[[
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269 well_founded_induction
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270 : forall (A : Type) (R : A -> A -> Prop),
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271 well_founded R ->
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272 forall P : A -> Set,
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273 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
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274 forall a : A, P a
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275 ]]
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276
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277 Some more recent Coq features provide more convenient syntax for defining recursive functions. Interested readers can consult the Coq manual about the commands %\index{Function}%[Function] and %\index{Program Fixpoint}%[Program Fixpoint]. *)
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278
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279
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adam@354
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280 (** * A Non-Termination Monad Inspired by Domain Theory *)
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281
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282 (** The key insights of %\index{domain theory}%domain theory%~\cite{WinskelDomains}% inspire the next approach to modeling non-termination. Domain theory is based on %\emph{%#<i>#information orders#</i>#%}% that relate values representing computatiion results, according to how much information these values convey. For instance, a simple domain might include values %``%#"#the program does not terminate#"#%''% and %``%#"#the program terminates with the answer 5.#"#%''% The former is considered to be an %\emph{%#<i>#approximation#</i>#%}% of the latter, while the latter is %\emph{%#<i>#not#</i>#%}% an approximation of %``%#"#the program terminates with the answer 6.#"#%''% The details of domain theory will not be important in what follows; we merely borrow the notion of an approximation ordering on computation results.
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283
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284 Consider this definition of a type of computations. *)
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285
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286 Section computation.
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287 Variable A : Type.
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adam@355
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288 (** The type [A] describes the result a computation will yield, if it terminates.
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289
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290 We give a rich dependent type to computations themselves: *)
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291
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292 Definition computation :=
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293 {f : nat -> option A
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294 | forall (n : nat) (v : A),
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295 f n = Some v
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296 -> forall (n' : nat), n' >= n
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297 -> f n' = Some v}.
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298
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299 (** A computation is fundamentally a function [f] from an %\emph{%#<i>#approximation level#</i>#%}% [n] to an optional result. Intuitively, higher [n] values enable termination in more cases than lower values. A call to [f] may return [None] to indicate that [n] was not high enough to run the computation to completion; higher [n] values may yield [Some]. Further, the proof obligation within the sigma type asserts that [f] is %\emph{%#<i>#monotone#</i>#%}% in an appropriate sense: when some [n] is sufficient to produce termination, so are all higher [n] values, and they all yield the same program result [v].
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300
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301 It is easy to define a relation characterizing when a computation runs to a particular result at a particular approximation level. *)
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302
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303 Definition runTo (m : computation) (n : nat) (v : A) :=
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304 proj1_sig m n = Some v.
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305
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adam@355
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306 (** On top of [runTo], we also define [run], which is the most abstract notion of when a computation runs to a value. *)
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307
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308 Definition run (m : computation) (v : A) :=
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309 exists n, runTo m n v.
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310 End computation.
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311
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312 (** The book source code contains at this point some tactics, lemma proofs, and hint commands, to be used in proving facts about computations. Since their details are orthogonal to the message of this chapter, I have omitted them in the rendered version. *)
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313 (* begin hide *)
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314
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adam@352
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315 Hint Unfold runTo.
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316
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adam@352
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317 Ltac run' := unfold run, runTo in *; try red; crush;
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318 repeat (match goal with
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adam@352
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319 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
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320 match goal with
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adam@352
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321 | [ x : _ |- _ ] =>
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adam@352
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322 match x with
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323 | E => destruct E
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adam@352
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324 end
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325 end
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326 | [ |- context[match ?M with exist _ _ => _ end] ] => let Heq := fresh "Heq" in
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327 case_eq M; intros ? ? Heq; try rewrite Heq in *; try subst
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328 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
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329 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
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330 | [ H : forall n v, ?E n = Some v -> _,
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331 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
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332 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
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333 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
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334 end; simpl in *); eauto 7.
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335
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adam@352
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336 Ltac run := run'; repeat (match goal with
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adam@352
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337 | [ H : forall n v, ?E n = Some v -> _
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adam@352
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338 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
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adam@352
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339 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
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adam@352
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340 end; run').
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adam@352
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341
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adam@352
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342 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
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adam@352
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343 exists 0; auto.
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adam@352
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344 Qed.
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adam@352
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345
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adam@352
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346 Hint Resolve ex_irrelevant.
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adam@352
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347
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adam@352
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348 Require Import Max.
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adam@352
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349
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adam@352
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350 Ltac max := intros n m; generalize (max_spec_le n m); crush.
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adam@352
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351
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adam@352
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352 Lemma max_1 : forall n m, max n m >= n.
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adam@352
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353 max.
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adam@352
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354 Qed.
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adam@352
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355
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adam@352
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356 Lemma max_2 : forall n m, max n m >= m.
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adam@352
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357 max.
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adam@352
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358 Qed.
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adam@352
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359
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adam@352
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360 Hint Resolve max_1 max_2.
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adam@352
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361
|
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adam@352
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362 Lemma ge_refl : forall n, n >= n.
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adam@352
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363 crush.
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adam@352
|
364 Qed.
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adam@352
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365
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adam@352
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366 Hint Resolve ge_refl.
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adam@352
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367
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adam@352
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368 Hint Extern 1 => match goal with
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adam@352
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369 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
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adam@352
|
370 end.
|
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adam@355
|
371 (* end hide *)
|
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adam@355
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372 (** remove printing exists *)
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adam@355
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373
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adam@355
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374 (** Now, as a simple first example of a computation, we can define [Bottom], which corresponds to an infinite loop. For any approximation level, it fails to terminate (returns [None]). Note the use of [abstract] to create a new opaque lemma for the proof found by the [run] tactic. In contrast to the previous section, opauqe proofs are fine here, since the proof components of computations do not influence evaluation behavior. *)
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375
|
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adam@352
|
376 Section Bottom.
|
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adam@352
|
377 Variable A : Type.
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adam@352
|
378
|
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adam@352
|
379 Definition Bottom : computation A.
|
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adam@352
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380 exists (fun _ : nat => @None A); abstract run.
|
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adam@352
|
381 Defined.
|
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adam@352
|
382
|
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adam@352
|
383 Theorem run_Bottom : forall v, ~run Bottom v.
|
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adam@352
|
384 run.
|
|
adam@352
|
385 Qed.
|
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adam@352
|
386 End Bottom.
|
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adam@352
|
387
|
|
adam@355
|
388 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *)
|
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adam@355
|
389
|
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adam@352
|
390 Section Return.
|
|
adam@352
|
391 Variable A : Type.
|
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adam@352
|
392 Variable v : A.
|
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adam@352
|
393
|
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adam@352
|
394 Definition Return : computation A.
|
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adam@352
|
395 intros; exists (fun _ : nat => Some v); abstract run.
|
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adam@352
|
396 Defined.
|
|
adam@352
|
397
|
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adam@352
|
398 Theorem run_Return : run Return v.
|
|
adam@352
|
399 run.
|
|
adam@352
|
400 Qed.
|
|
adam@352
|
401 End Return.
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adam@352
|
402
|
|
adam@356
|
403 (** The name [Return] was meant to be suggestive of the standard operations of %\index{monad}%monads%~\cite{Monads}%. The other standard operation is [Bind], which lets us run one computation and, if it terminates, pass its result off to another computation. *)
|
|
adam@352
|
404
|
|
adam@352
|
405 Section Bind.
|
|
adam@352
|
406 Variables A B : Type.
|
|
adam@352
|
407 Variable m1 : computation A.
|
|
adam@352
|
408 Variable m2 : A -> computation B.
|
|
adam@352
|
409
|
|
adam@352
|
410 Definition Bind : computation B.
|
|
adam@352
|
411 exists (fun n =>
|
|
adam@352
|
412 let (f1, Hf1) := m1 in
|
|
adam@352
|
413 match f1 n with
|
|
adam@352
|
414 | None => None
|
|
adam@352
|
415 | Some v =>
|
|
adam@352
|
416 let (f2, Hf2) := m2 v in
|
|
adam@352
|
417 f2 n
|
|
adam@352
|
418 end); abstract run.
|
|
adam@352
|
419 Defined.
|
|
adam@352
|
420
|
|
adam@352
|
421 Theorem run_Bind : forall (v1 : A) (v2 : B),
|
|
adam@352
|
422 run m1 v1
|
|
adam@352
|
423 -> run (m2 v1) v2
|
|
adam@352
|
424 -> run Bind v2.
|
|
adam@352
|
425 run; match goal with
|
|
adam@352
|
426 | [ x : nat, y : nat |- _ ] => exists (max x y)
|
|
adam@352
|
427 end; run.
|
|
adam@352
|
428 Qed.
|
|
adam@352
|
429 End Bind.
|
|
adam@352
|
430
|
|
adam@355
|
431 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *)
|
|
adam@352
|
432
|
|
adam@352
|
433 Notation "x <- m1 ; m2" :=
|
|
adam@352
|
434 (Bind m1 (fun x => m2)) (right associativity, at level 70).
|
|
adam@352
|
435
|
|
adam@355
|
436 (** We can verify that we have indeed defined a monad, by proving the standard monad laws. Part of the exercise is choosing an appropriate notion of equality between computations. We use %``%#"#equality at all approximation levels.#"#%''% *)
|
|
adam@355
|
437
|
|
adam@352
|
438 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
|
|
adam@352
|
439
|
|
adam@352
|
440 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
|
|
adam@352
|
441 meq (Bind (Return a) f) (f a).
|
|
adam@352
|
442 run.
|
|
adam@352
|
443 Qed.
|
|
adam@352
|
444
|
|
adam@352
|
445 Theorem right_identity : forall A (m : computation A),
|
|
adam@352
|
446 meq (Bind m (@Return _)) m.
|
|
adam@352
|
447 run.
|
|
adam@352
|
448 Qed.
|
|
adam@352
|
449
|
|
adam@352
|
450 Theorem associativity : forall A B C (m : computation A) (f : A -> computation B) (g : B -> computation C),
|
|
adam@352
|
451 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
|
|
adam@352
|
452 run.
|
|
adam@352
|
453 Qed.
|
|
adam@352
|
454
|
|
adam@355
|
455 (** Now we come to the piece most directly inspired by domain theory. We want to support general recursive function definitions, but domain theory tells us that not all definitions are reasonable; some fail to be %\emph{%#<i>#continuous#</i>#%}% and thus represent unrealizable computations. To formalize an analogous notion of continuity for our non-termination monad, we write down the approximation relation on computation results that we have had in mind all along. *)
|
|
adam@352
|
456
|
|
adam@352
|
457 Section lattice.
|
|
adam@352
|
458 Variable A : Type.
|
|
adam@352
|
459
|
|
adam@352
|
460 Definition leq (x y : option A) :=
|
|
adam@352
|
461 forall v, x = Some v -> y = Some v.
|
|
adam@352
|
462 End lattice.
|
|
adam@352
|
463
|
|
adam@355
|
464 (** We now have the tools we need to define a new [Fix] combinator that, unlike the one we saw in the prior section, does not require a termination proof, and in fact admits recursive definition of functions that fail to terminate on some or all inputs. *)
|
|
adam@352
|
465
|
|
adam@352
|
466 Section Fix.
|
|
adam@355
|
467 (** First, we have the function domain and range types. *)
|
|
adam@355
|
468
|
|
adam@352
|
469 Variables A B : Type.
|
|
adam@355
|
470
|
|
adam@355
|
471 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *)
|
|
adam@355
|
472
|
|
adam@352
|
473 Variable f : (A -> computation B) -> (A -> computation B).
|
|
adam@352
|
474
|
|
adam@355
|
475 (** Finally, we impose an obligation to prove that the body [f] is continuous. That is, when [f] terminates according to one recursive version of itself, it also terminates with the same result at the same approximation level when passed a recursive version that refines the original, according to [leq]. *)
|
|
adam@355
|
476
|
|
adam@352
|
477 Hypothesis f_continuous : forall n v v1 x,
|
|
adam@352
|
478 runTo (f v1 x) n v
|
|
adam@352
|
479 -> forall (v2 : A -> computation B),
|
|
adam@352
|
480 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
|
|
adam@352
|
481 -> runTo (f v2 x) n v.
|
|
adam@352
|
482
|
|
adam@355
|
483 (** The computational part of the [Fix] combinator is easy to define. At approximation level 0, we diverge; at higher levels, we run the body with a functional argument drawn from the next lower level. *)
|
|
adam@355
|
484
|
|
adam@352
|
485 Fixpoint Fix' (n : nat) (x : A) : computation B :=
|
|
adam@352
|
486 match n with
|
|
adam@352
|
487 | O => Bottom _
|
|
adam@352
|
488 | S n' => f (Fix' n') x
|
|
adam@352
|
489 end.
|
|
adam@352
|
490
|
|
adam@355
|
491 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *)
|
|
adam@355
|
492
|
|
adam@352
|
493 Hint Extern 1 (_ >= _) => omega.
|
|
adam@352
|
494 Hint Unfold leq.
|
|
adam@352
|
495
|
|
adam@352
|
496 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
|
|
adam@352
|
497 -> forall n', n' >= n
|
|
adam@352
|
498 -> proj1_sig (Fix' n' x) steps = Some v.
|
|
adam@352
|
499 unfold runTo in *; induction n; crush;
|
|
adam@352
|
500 match goal with
|
|
adam@352
|
501 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
|
|
adam@352
|
502 end.
|
|
adam@352
|
503 Qed.
|
|
adam@352
|
504
|
|
adam@352
|
505 Hint Resolve Fix'_ok.
|
|
adam@352
|
506
|
|
adam@352
|
507 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
|
|
adam@352
|
508 match goal with
|
|
adam@352
|
509 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
|
|
adam@352
|
510 end.
|
|
adam@352
|
511
|
|
adam@352
|
512 Definition Fix : A -> computation B.
|
|
adam@352
|
513 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
|
|
adam@352
|
514 Defined.
|
|
adam@352
|
515
|
|
adam@355
|
516 (** Finally, we can prove that [Fix] obeys the expected computation rule. *)
|
|
adam@352
|
517
|
|
adam@352
|
518 Theorem run_Fix : forall x v,
|
|
adam@352
|
519 run (f Fix x) v
|
|
adam@352
|
520 -> run (Fix x) v.
|
|
adam@352
|
521 run; match goal with
|
|
adam@352
|
522 | [ n : nat |- _ ] => exists (S n); eauto
|
|
adam@352
|
523 end.
|
|
adam@352
|
524 Qed.
|
|
adam@352
|
525 End Fix.
|
|
adam@352
|
526
|
|
adam@355
|
527 (* begin hide *)
|
|
adam@352
|
528 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
|
|
adam@352
|
529 -> x = y.
|
|
adam@352
|
530 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
|
|
adam@352
|
531 Qed.
|
|
adam@352
|
532
|
|
adam@352
|
533 Lemma leq_None : forall A (x y : A), leq (Some x) None
|
|
adam@352
|
534 -> False.
|
|
adam@352
|
535 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
|
|
adam@352
|
536 Qed.
|
|
adam@352
|
537
|
|
adam@355
|
538 Ltac mergeSort' := run;
|
|
adam@355
|
539 repeat (match goal with
|
|
adam@355
|
540 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
|
|
adam@355
|
541 end; run);
|
|
adam@355
|
542 repeat match goal with
|
|
adam@355
|
543 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
|
|
adam@355
|
544 match goal with
|
|
adam@355
|
545 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
|
|
adam@355
|
546 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
|
|
adam@355
|
547 end
|
|
adam@355
|
548 end; run; repeat match goal with
|
|
adam@355
|
549 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
|
|
adam@355
|
550 end; auto.
|
|
adam@355
|
551 (* end hide *)
|
|
adam@355
|
552
|
|
adam@355
|
553 (** After all that work, it is now fairly painless to define a version of [mergeSort] that requires no proof of termination. We appeal to a program-specific tactic whose definition is hidden here but present in the book source. *)
|
|
adam@355
|
554
|
|
adam@352
|
555 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
|
|
adam@352
|
556 refine (fun A le => Fix
|
|
adam@352
|
557 (fun (mergeSort : list A -> computation (list A))
|
|
adam@352
|
558 (ls : list A) =>
|
|
adam@352
|
559 if le_lt_dec 2 (length ls)
|
|
adam@352
|
560 then let lss := partition ls in
|
|
adam@352
|
561 ls1 <- mergeSort (fst lss);
|
|
adam@352
|
562 ls2 <- mergeSort (snd lss);
|
|
adam@352
|
563 Return (merge le ls1 ls2)
|
|
adam@355
|
564 else Return ls) _); abstract mergeSort'.
|
|
adam@352
|
565 Defined.
|
|
adam@352
|
566
|
|
adam@355
|
567 (** Furthermore, %``%#"#running#"#%''% [mergeSort'] on concrete inputs is as easy as choosing a sufficiently high approximation level and letting Coq's computation rules do the rest. Contrast this with the proof work that goes into deriving an evaluation fact for a deeply embedded language, with one explicit proof rule application per execution step. *)
|
|
adam@352
|
568
|
|
adam@352
|
569 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
|
|
adam@352
|
570 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
|
|
adam@352
|
571 exists 4; reflexivity.
|
|
adam@352
|
572 Qed.
|
|
adam@352
|
573
|
|
adam@355
|
574 (** There is another benefit of our new [Fix] compared with one we used in the previous section: we can now write recursive functions that sometimes fail to terminate, without losing easy reasoning principles for the terminating cases. Consider this simple example, which appeals to another tactic whose definition we elide here. *)
|
|
adam@355
|
575
|
|
adam@355
|
576 (* begin hide *)
|
|
adam@355
|
577 Ltac looper := unfold leq in *; run;
|
|
adam@355
|
578 repeat match goal with
|
|
adam@355
|
579 | [ x : unit |- _ ] => destruct x
|
|
adam@355
|
580 | [ x : bool |- _ ] => destruct x
|
|
adam@355
|
581 end; auto.
|
|
adam@355
|
582 (* end hide *)
|
|
adam@355
|
583
|
|
adam@352
|
584 Definition looper : bool -> computation unit.
|
|
adam@352
|
585 refine (Fix (fun looper (b : bool) =>
|
|
adam@355
|
586 if b then Return tt else looper b) _); abstract looper.
|
|
adam@352
|
587 Defined.
|
|
adam@352
|
588
|
|
adam@352
|
589 Lemma test_looper : run (looper true) tt.
|
|
adam@352
|
590 exists 1; reflexivity.
|
|
adam@352
|
591 Qed.
|
|
adam@354
|
592
|
|
adam@355
|
593 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level.
|
|
adam@355
|
594
|
|
adam@355
|
595 There are other theorems that are important to prove about combinators like [Return], [Bind], and [Fix]. In general, for a computation [c], we sometimes have a hypothesis proving [run c v] for some [v], and we want to perform inversion to deduce what [v] must be. Each combinator should ideally have a theorem of that kind, for [c] built directly from that combinator. We have omitted such theorems here, but they are not hard to prove. In general, the domain theory-inspired approach avoids the type-theoretic %``%#"#gotchas#"#%''% that tend to show up in approaches that try to mix normal Coq computation with explicit syntax types. The next section of this chapter demonstrates two alternate approaches of that sort. *)
|
|
adam@355
|
596
|
|
adam@354
|
597
|
|
adam@354
|
598 (** * Co-Inductive Non-Termination Monads *)
|
|
adam@354
|
599
|
|
adam@356
|
600 (** There are two key downsides to both of the previous approaches: both require unusual syntax based on explicit calls to fixpoint combinators, and both generate immediate proof obligations about the bodies of recursive definitions. In Chapter 5, we have already seen how co-inductive types support recursive definitions that exhibit certain well-behaved varieties of non-termination. It turns out that we can leverage that co-induction support for encoding of general recursive definitions, by adding layers of co-inductive syntax. In effect, we mix elements of shallow and deep embeddings.
|
|
adam@356
|
601
|
|
adam@356
|
602 Our first example of this kind, proposed by Capretta%~\cite{Capretta}%, defines a silly-looking type of thunks; that is, computations that may be forced to yield results, if they terminate. *)
|
|
adam@356
|
603
|
|
adam@354
|
604 CoInductive thunk (A : Type) : Type :=
|
|
adam@354
|
605 | Answer : A -> thunk A
|
|
adam@354
|
606 | Think : thunk A -> thunk A.
|
|
adam@354
|
607
|
|
adam@356
|
608 (** A computation is either an immediate [Answer] or another computation wrapped inside [Think]. Since [thunk] is co-inductive, every [thunk] type is inhabited by an infinite nesting of [Think]s, standing for non-termination. Terminating results are [Answer] wrapped inside some finite number of [Think]s.
|
|
adam@356
|
609
|
|
adam@356
|
610 Why bother to write such a strange definition? The definition of [thunk] is motivated by the ability it gives us to define a %``%#"#bind#"#%''% operation, similar to the one we defined in the previous section. *)
|
|
adam@356
|
611
|
|
adam@356
|
612 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
|
|
adam@354
|
613 match m1 with
|
|
adam@354
|
614 | Answer x => m2 x
|
|
adam@354
|
615 | Think m1' => Think (TBind m1' m2)
|
|
adam@354
|
616 end.
|
|
adam@354
|
617
|
|
adam@356
|
618 (** Note that the definition would violate the co-recursion guardedness restriction if we left out the seemingly superfluous [Think] on the righthand side of the second [match] branch.
|
|
adam@356
|
619
|
|
adam@356
|
620 We can prove that [Answer] and [TBind] form a monad for [thunk]. The proof is omitted here but present in the book source. As usual for this sort of proof, a key element is choosing an appropriate notion of equality for [thunk]s. *)
|
|
adam@356
|
621
|
|
adam@356
|
622 (* begin hide *)
|
|
adam@354
|
623 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
|
|
adam@354
|
624 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
|
|
adam@354
|
625 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
|
|
adam@354
|
626 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
|
|
adam@354
|
627
|
|
adam@354
|
628 Section thunk_eq_coind.
|
|
adam@354
|
629 Variable A : Type.
|
|
adam@354
|
630 Variable P : thunk A -> thunk A -> Prop.
|
|
adam@354
|
631
|
|
adam@354
|
632 Hypothesis H : forall m1 m2, P m1 m2
|
|
adam@354
|
633 -> match m1, m2 with
|
|
adam@354
|
634 | Answer x1, Answer x2 => x1 = x2
|
|
adam@354
|
635 | Think m1', Think m2' => P m1' m2'
|
|
adam@354
|
636 | Think m1', _ => P m1' m2
|
|
adam@354
|
637 | _, Think m2' => P m1 m2'
|
|
adam@354
|
638 end.
|
|
adam@354
|
639
|
|
adam@354
|
640 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
|
|
adam@354
|
641 cofix; intros;
|
|
adam@354
|
642 match goal with
|
|
adam@354
|
643 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
|
|
adam@354
|
644 end; destruct m1; destruct m2; subst; repeat constructor; auto.
|
|
adam@354
|
645 Qed.
|
|
adam@354
|
646 End thunk_eq_coind.
|
|
adam@356
|
647 (* end hide *)
|
|
adam@356
|
648
|
|
adam@356
|
649 (** In the proofs to follow, we will need a function similar to one we saw in Chapter 5, to pull apart and reassemble a [thunk] in a way that provokes reduction of co-recursive calls. *)
|
|
adam@354
|
650
|
|
adam@354
|
651 Definition frob A (m : thunk A) : thunk A :=
|
|
adam@354
|
652 match m with
|
|
adam@354
|
653 | Answer x => Answer x
|
|
adam@354
|
654 | Think m' => Think m'
|
|
adam@354
|
655 end.
|
|
adam@354
|
656
|
|
adam@354
|
657 Theorem frob_eq : forall A (m : thunk A), frob m = m.
|
|
adam@354
|
658 destruct m; reflexivity.
|
|
adam@354
|
659 Qed.
|
|
adam@354
|
660
|
|
adam@356
|
661 (* begin hide *)
|
|
adam@354
|
662 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
|
|
adam@354
|
663 thunk_eq (frob m1) (frob m2)
|
|
adam@354
|
664 -> thunk_eq m1 m2.
|
|
adam@354
|
665 intros; repeat rewrite frob_eq in *; auto.
|
|
adam@354
|
666 Qed.
|
|
adam@354
|
667
|
|
adam@354
|
668 Ltac findDestr := match goal with
|
|
adam@354
|
669 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
|
|
adam@354
|
670 match E with
|
|
adam@354
|
671 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
|
|
adam@354
|
672 | _ => destruct E
|
|
adam@354
|
673 end
|
|
adam@354
|
674 end.
|
|
adam@354
|
675
|
|
adam@354
|
676 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
|
|
adam@354
|
677 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
|
|
adam@354
|
678 Qed.
|
|
adam@354
|
679
|
|
adam@354
|
680 Hint Resolve thunk_eq_refl.
|
|
adam@354
|
681
|
|
adam@354
|
682 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
|
|
adam@354
|
683 thunk_eq (TBind (Answer a) f) (f a).
|
|
adam@354
|
684 intros; apply thunk_eq_frob; crush.
|
|
adam@354
|
685 Qed.
|
|
adam@354
|
686
|
|
adam@354
|
687 Theorem tright_identity : forall A (m : thunk A),
|
|
adam@354
|
688 thunk_eq (TBind m (@Answer _)) m.
|
|
adam@354
|
689 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
|
|
adam@354
|
690 findDestr; reflexivity.
|
|
adam@354
|
691 Qed.
|
|
adam@354
|
692
|
|
adam@354
|
693 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
|
|
adam@354
|
694 TBind (Answer v) m2 = m2 v.
|
|
adam@354
|
695 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
|
|
adam@354
|
696 simpl; findDestr; reflexivity.
|
|
adam@354
|
697 Qed.
|
|
adam@354
|
698
|
|
adam@354
|
699 Hint Rewrite TBind_Answer : cpdt.
|
|
adam@354
|
700
|
|
adam@355
|
701 (** printing exists $\exists$ *)
|
|
adam@355
|
702
|
|
adam@354
|
703 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
|
|
adam@354
|
704 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
|
|
adam@354
|
705 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
|
|
adam@354
|
706 m1 = TBind (TBind m f) g
|
|
adam@354
|
707 /\ m2 = TBind m (fun x => TBind (f x) g))
|
|
adam@354
|
708 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
|
|
adam@354
|
709 Qed.
|
|
adam@356
|
710 (* end hide *)
|
|
adam@356
|
711
|
|
adam@356
|
712 (** As a simple example, here is how we might define a tail-recursive factorial function. *)
|
|
adam@354
|
713
|
|
adam@354
|
714 CoFixpoint fact (n acc : nat) : thunk nat :=
|
|
adam@354
|
715 match n with
|
|
adam@354
|
716 | O => Answer acc
|
|
adam@354
|
717 | S n' => Think (fact n' (S n' * acc))
|
|
adam@354
|
718 end.
|
|
adam@354
|
719
|
|
adam@356
|
720 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *)
|
|
adam@356
|
721
|
|
adam@354
|
722 Inductive eval A : thunk A -> A -> Prop :=
|
|
adam@354
|
723 | EvalAnswer : forall x, eval (Answer x) x
|
|
adam@354
|
724 | EvalThink : forall m x, eval m x -> eval (Think m) x.
|
|
adam@354
|
725
|
|
adam@354
|
726 Hint Rewrite frob_eq : cpdt.
|
|
adam@354
|
727
|
|
adam@354
|
728 Lemma eval_frob : forall A (c : thunk A) x,
|
|
adam@354
|
729 eval (frob c) x
|
|
adam@354
|
730 -> eval c x.
|
|
adam@354
|
731 crush.
|
|
adam@354
|
732 Qed.
|
|
adam@354
|
733
|
|
adam@354
|
734 Theorem eval_fact : eval (fact 5 1) 120.
|
|
adam@354
|
735 repeat (apply eval_frob; simpl; constructor).
|
|
adam@354
|
736 Qed.
|
|
adam@354
|
737
|
|
adam@356
|
738 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs.
|
|
adam@356
|
739
|
|
adam@356
|
740 Now consider another very similar definition, this time of a Fibonacci number funtion.
|
|
adam@356
|
741 [[
|
|
adam@354
|
742 CoFixpoint fib (n : nat) : thunk nat :=
|
|
adam@354
|
743 match n with
|
|
adam@354
|
744 | 0 => Answer 1
|
|
adam@354
|
745 | 1 => Answer 1
|
|
adam@354
|
746 | _ => TBind (fib (pred n)) (fun n1 =>
|
|
adam@354
|
747 TBind (fib (pred (pred n))) (fun n2 =>
|
|
adam@354
|
748 Answer (n1 + n2)))
|
|
adam@354
|
749 end.
|
|
adam@354
|
750 ]]
|
|
adam@354
|
751
|
|
adam@356
|
752 Coq complains that the guardedness condition is violated. The two recursive calls are immediate arguments to [TBind], but [TBind] is not a constructor of [thunk]. Rather, it is a defined function. This example shows a very serious limitation of [thunk] for traditional functional programming: it is not, in general, possible to make recursive calls and then make further recursive calls, depending on the first call's result. The [fact] example succeeded because it was already tail recursive, meaning no further computation is needed after a recursive call.
|
|
adam@356
|
753
|
|
adam@356
|
754 %\medskip%
|
|
adam@356
|
755
|
|
adam@356
|
756 I know no easy fix for this problem of [thunk], but we can define an alternate co-inductive monad that avoids the problem, based on a proposal by Megacz%~\cite{Megacz}%. We ran into trouble because [TBind] was not a constructor of [thunk], so let us define a new type family where %``%#"#bind#"#%''% is a constructor. *)
|
|
adam@354
|
757
|
|
adam@354
|
758 CoInductive comp (A : Type) : Type :=
|
|
adam@354
|
759 | Ret : A -> comp A
|
|
adam@354
|
760 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
|
|
adam@354
|
761
|
|
adam@356
|
762 (** This example shows off Coq's support for %\index{recursively non-uniform parameters}\emph{%#<i>#recursively non-uniform parameters#</i>#%}%, as in the case of the parameter [A] declared above, where each constructor's type ends in [comp A], but there is a recursive use of [comp] with a different parameter [B]. Beside that technical wrinkle, we see the simplest possible definition of a monad, via a type whose two constructors are precisely the monad operators.
|
|
adam@356
|
763
|
|
adam@356
|
764 It is easy to define the semantics of terminating [comp] computations. *)
|
|
adam@356
|
765
|
|
adam@354
|
766 Inductive exec A : comp A -> A -> Prop :=
|
|
adam@354
|
767 | ExecRet : forall x, exec (Ret x) x
|
|
adam@354
|
768 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
|
|
adam@354
|
769 -> exec (f x1) x2
|
|
adam@354
|
770 -> exec (Bnd c f) x2.
|
|
adam@354
|
771
|
|
adam@356
|
772 (** We can also prove that [Ret] and [Bnd] form a monad according to a notion of [comp] equality based on [exec], but we omit details here; they are in the book source at this point. *)
|
|
adam@356
|
773
|
|
adam@356
|
774 (* begin hide *)
|
|
adam@354
|
775 Hint Constructors exec.
|
|
adam@354
|
776
|
|
adam@354
|
777 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
|
|
adam@354
|
778
|
|
adam@354
|
779 Ltac inverter := repeat match goal with
|
|
adam@354
|
780 | [ H : exec _ _ |- _ ] => inversion H; []; crush
|
|
adam@354
|
781 end.
|
|
adam@354
|
782
|
|
adam@354
|
783 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
|
|
adam@354
|
784 comp_eq (Bnd (Ret a) f) (f a).
|
|
adam@354
|
785 red; crush; inverter; eauto.
|
|
adam@354
|
786 Qed.
|
|
adam@354
|
787
|
|
adam@354
|
788 Theorem cright_identity : forall A (m : comp A),
|
|
adam@354
|
789 comp_eq (Bnd m (@Ret _)) m.
|
|
adam@354
|
790 red; crush; inverter; eauto.
|
|
adam@354
|
791 Qed.
|
|
adam@354
|
792
|
|
adam@354
|
793 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
|
|
adam@354
|
794 exec c r
|
|
adam@354
|
795 -> forall m, c = Bnd (Bnd m f) g
|
|
adam@354
|
796 -> exec (Bnd m (fun x => Bnd (f x) g)) r.
|
|
adam@354
|
797 induction 1; crush.
|
|
adam@354
|
798 match goal with
|
|
adam@354
|
799 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
|
|
adam@354
|
800 end.
|
|
adam@354
|
801 move H3 after A.
|
|
adam@354
|
802 generalize dependent B0.
|
|
adam@354
|
803 do 2 intro.
|
|
adam@354
|
804 subst.
|
|
adam@354
|
805 crush.
|
|
adam@354
|
806 inversion H; clear H; crush.
|
|
adam@354
|
807 eauto.
|
|
adam@354
|
808 Qed.
|
|
adam@354
|
809
|
|
adam@354
|
810 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
|
|
adam@354
|
811 exec c r
|
|
adam@354
|
812 -> forall m, c = Bnd m (fun x => Bnd (f x) g)
|
|
adam@354
|
813 -> exec (Bnd (Bnd m f) g) r.
|
|
adam@354
|
814 induction 1; crush.
|
|
adam@354
|
815 match goal with
|
|
adam@354
|
816 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
|
|
adam@354
|
817 end.
|
|
adam@354
|
818 move H3 after B.
|
|
adam@354
|
819 generalize dependent B0.
|
|
adam@354
|
820 do 2 intro.
|
|
adam@354
|
821 subst.
|
|
adam@354
|
822 crush.
|
|
adam@354
|
823 inversion H0; clear H0; crush.
|
|
adam@354
|
824 eauto.
|
|
adam@354
|
825 Qed.
|
|
adam@354
|
826
|
|
adam@354
|
827 Hint Resolve cassociativity1 cassociativity2.
|
|
adam@354
|
828
|
|
adam@354
|
829 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
|
|
adam@354
|
830 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
|
|
adam@354
|
831 red; crush; eauto.
|
|
adam@354
|
832 Qed.
|
|
adam@356
|
833 (* end hide *)
|
|
adam@356
|
834
|
|
adam@356
|
835 (** Not only can we define the Fibonacci function with the new monad, but even our running example of merge sort becomes definable. By shadowing our previous notation for %``%#"#bind,#"#%''%, we can write almost exactly the same code as in our previous [mergeSort'] definition, but with less syntactic clutter. *)
|
|
adam@356
|
836
|
|
adam@356
|
837 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)).
|
|
adam@354
|
838
|
|
adam@354
|
839 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
|
|
adam@354
|
840 if le_lt_dec 2 (length ls)
|
|
adam@354
|
841 then let lss := partition ls in
|
|
adam@356
|
842 ls1 <- mergeSort'' le (fst lss);
|
|
adam@356
|
843 ls2 <- mergeSort'' le (snd lss);
|
|
adam@356
|
844 Ret (merge le ls1 ls2)
|
|
adam@354
|
845 else Ret ls.
|
|
adam@354
|
846
|
|
adam@356
|
847 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *)
|
|
adam@356
|
848
|
|
adam@354
|
849 Definition frob' A (c : comp A) :=
|
|
adam@354
|
850 match c with
|
|
adam@354
|
851 | Ret x => Ret x
|
|
adam@354
|
852 | Bnd _ c' f => Bnd c' f
|
|
adam@354
|
853 end.
|
|
adam@354
|
854
|
|
adam@354
|
855 Lemma exec_frob : forall A (c : comp A) x,
|
|
adam@354
|
856 exec (frob' c) x
|
|
adam@354
|
857 -> exec c x.
|
|
adam@356
|
858 destruct c; crush.
|
|
adam@354
|
859 Qed.
|
|
adam@354
|
860
|
|
adam@356
|
861 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *)
|
|
adam@356
|
862
|
|
adam@354
|
863 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
|
|
adam@354
|
864 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
|
|
adam@354
|
865 repeat (apply exec_frob; simpl; econstructor).
|
|
adam@354
|
866 Qed.
|
|
adam@354
|
867
|
|
adam@356
|
868 (** Have we finally reached the ideal solution for encoding general recursive definitions, with minimal hassle in syntax and proof obligations? Unfortunately, we have not, as [comp] has a serious expressivity weakness. Consider the following definition of a curried addition function: *)
|
|
adam@356
|
869
|
|
adam@354
|
870 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
|
|
adam@354
|
871
|
|
adam@356
|
872 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way.
|
|
adam@356
|
873 [[
|
|
adam@356
|
874 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3).
|
|
adam@354
|
875 ]]
|
|
adam@354
|
876
|
|
adam@354
|
877 <<
|
|
adam@354
|
878 Error: Universe inconsistency.
|
|
adam@354
|
879 >>
|
|
adam@356
|
880
|
|
adam@356
|
881 The problem has to do with rules for inductive definitions that we still study in more detail in Chapter 12. Briefly, recall that the type of the constructor [Bnd] quantifies over a type [B]. To make [testCurriedAdd] work, we would need to instantiate [B] as [nat -> comp nat]. However, Coq enforces a %\emph{predicativity restriction}% that (roughly) no quantifier in an inductive or co-inductive type's definition may ever be instantiated with a term that contains the type being defined. Chapter 12 presents the exact mechanism by which this restriction is enforced, but for now our conclusion is that [comp] is fatally flawed as a way of encoding interesting higher-order functional programs that use general recursion. *)
|
|
adam@354
|
882
|
|
adam@354
|
883
|
|
adam@354
|
884 (** * Comparing the Options *)
|
|
adam@354
|
885
|
|
adam@356
|
886 (** We have seen four different approaches to encoding general recursive definitions in Coq. Among them there is no clear champion that dominates the others in every important way. Instead, we close the chapter by comparing the techniques along a number of dimensions. Every technique allows recursive definitions with terminaton arguments that go beyond Coq's built-in termination checking, so we must turn to subtler points to highlight differences.
|
|
adam@356
|
887
|
|
adam@356
|
888 One useful property is automatic integration with normal Coq programming. That is, we would like the type of a function to be the same, whether or not that function is defined using an interesting recursion pattern. Only the first of the four techniques, well-founded recursion, meets this criterion. It is also the only one of the four to meet the related criterion that evaluation of function calls can take place entirely inside Coq's built-in computation machinery. The monad inspired by domain theory occupies some middle ground in this dimension, since generally standard computation is enough to evaluate a term once a high enough approximation level is provided.
|
|
adam@356
|
889
|
|
adam@356
|
890 Another useful property is that a function and its termination argument may be developed separately. We may even want to define functions that fail to terminate on some or all inputs. The well-founded recursion technique does not have this property, but the other three do.
|
|
adam@356
|
891
|
|
adam@356
|
892 One minor plus is the ability to write recursive definitions in natural syntax, rather than with calls to higher-order combinators. This downside of the first two techniques is actually rather easy to get around using Coq's notation mechanism, though we leave the details as an exercise for the reader.
|
|
adam@356
|
893
|
|
adam@356
|
894 The first two techniques impose proof obligations that are more basic than terminaton arguments, where well-founded recursion requires a proof of extensionality and domain-theoretic recursion requires a proof of continuity. A function may not be defined, and thus may not be computed with, until these obligations are proved. The co-inductive techniques avoid this problem, as recursive definitions may be made without any proof obligations.
|
|
adam@356
|
895
|
|
adam@356
|
896 We can also consider support for common idioms in functional programming. For instance, the [thunk] monad effectively only supports recursion that is tail recursion, while the others allow arbitrary recursion schemes.
|
|
adam@356
|
897
|
|
adam@356
|
898 On the other hand, the [comp] monad does not support the effective mixing of higher-order functions and general recursion, while all the other techniques do. For instance, we can finish the failed [curriedAdd] example in the domain-theoretic monad. *)
|
|
adam@356
|
899
|
|
adam@354
|
900 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
|
|
adam@354
|
901
|
|
adam@356
|
902 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3).
|
|
adam@356
|
903
|
|
adam@356
|
904 (** The same techniques also apply to more interesting higher-order functions like list map, and, as in all four techniques, we can mix regular and general recursion, preferring the former when possible to avoid proof obligations. *)
|
|
adam@354
|
905
|
|
adam@354
|
906 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
|
|
adam@354
|
907 match ls with
|
|
adam@354
|
908 | nil => Return nil
|
|
adam@354
|
909 | x :: ls' => Bind (f x) (fun x' =>
|
|
adam@354
|
910 Bind (map f ls') (fun ls'' =>
|
|
adam@354
|
911 Return (x' :: ls'')))
|
|
adam@354
|
912 end.
|
|
adam@354
|
913
|
|
adam@355
|
914 (** remove printing exists *)
|
|
adam@356
|
915 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil))
|
|
adam@356
|
916 (2 :: 3 :: 4 :: nil).
|
|
adam@354
|
917 exists 1; reflexivity.
|
|
adam@354
|
918 Qed.
|
|
adam@356
|
919
|
|
adam@356
|
920 (** One further disadvantage of [comp] is that we cannot prove an inversion lemma for executions of [Bind] without appealing to an %\emph{axiom}%, a logical complication that we discuss at more length in Chapter 12. The other three techniques allow proof of all the important theorems within the normal logic of Coq.
|
|
adam@356
|
921
|
|
adam@356
|
922 Perhaps one theme of our comparison is that one must trade off between on one hand, functional programming expressiveness and compatibility with normal Coq types and computation; and, on the other hand, the level of proof obligations one is willing to handle at function definition time. *)
|
