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1 (* Copyright (c) 2006, 2011-2012, 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 _recursive_ definitions, recursive calls are only allowed on _syntactic subterms_ of the original primary argument, a restriction known as%\index{primitive recursion}% _primitive recursion_. 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 _co-recursive_ 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}% _deep embedding_ 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}% _shallow embedding_, 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}% _well-founded relation_, and the associated standard technique in Coq is%\index{well-founded recursion}% _well-founded recursion_. 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}% _accessibility_ 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 that 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 forall x : A, (forall y : A, R y x -> P y) -> P x
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142 ]]
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143
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144 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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145
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146 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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147
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148 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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149
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150 Definition lengthOrder (ls1 ls2 : list A) :=
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151 length ls1 < length ls2.
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152
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153 (** 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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154
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155 Hint Constructors Acc.
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156
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157 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
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158 unfold lengthOrder; induction len; crush.
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159 Defined.
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160
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161 Theorem lengthOrder_wf : well_founded lengthOrder.
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162 red; intro; eapply lengthOrder_wf'; eauto.
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163 Defined.
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164
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165 (** 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 _recursive in the structure of [Acc] proofs_. 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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166
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167 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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168
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169 Lemma partition_wf : forall len ls, 2 <= length ls <= len
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170 -> let (ls1, ls2) := partition ls in
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171 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
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172 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
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173 destruct (le_lt_dec 2 (length ls));
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174 repeat (match goal with
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175 | [ _ : length ?E < 2 |- _ ] => destruct E
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176 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
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177 | [ IH : _ |- context[partition ?L] ] =>
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178 specialize (IH L); destruct (partition L); destruct IH
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179 end; crush).
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180 Defined.
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181
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182 Ltac partition := intros ls ?; intros; generalize (@partition_wf (length ls) ls);
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183 destruct (partition ls); destruct 1; crush.
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184
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185 Lemma partition_wf1 : forall ls, 2 <= length ls
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186 -> lengthOrder (fst (partition ls)) ls.
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187 partition.
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188 Defined.
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189
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190 Lemma partition_wf2 : forall ls, 2 <= length ls
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191 -> lengthOrder (snd (partition ls)) ls.
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192 partition.
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193 Defined.
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194
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195 Hint Resolve partition_wf1 partition_wf2.
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196
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197 (** 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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198
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199 Definition mergeSort : list A -> list A.
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200 (* begin thide *)
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201 refine (Fix lengthOrder_wf (fun _ => list A)
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202 (fun (ls : list A)
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203 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
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204 if le_lt_dec 2 (length ls)
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205 then let lss := partition ls in
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206 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
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207 else ls)); subst lss; eauto.
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208 Defined.
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209 (* end thide *)
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210 End mergeSort.
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211
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212 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
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213
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214 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
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215 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
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216
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217 (** 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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218
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219 (* begin thide *)
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220 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
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221 mergeSort le ls = if le_lt_dec 2 (length ls)
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222 then let lss := partition ls in
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223 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
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224 else ls.
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225 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
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226
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227 (** 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}% _function extensionality_ property is neither provable nor disprovable within Coq. The type of [Fix_eq] makes clear what we must show manually: *)
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228
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229 Check Fix_eq.
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230 (** %\vspace{-.15in}%[[
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231 Fix_eq
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232 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
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233 (P : A -> Type)
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234 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
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235 (forall (x : A) (f g : forall y : A, R y x -> P y),
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236 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
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237 forall x : A,
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238 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
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239 ]]
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240
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241 Most such obligations are dischargable with straightforward proof automation, and this example is no exception. *)
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242
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243 match goal with
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244 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
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245 end; simpl; f_equal; auto.
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246 Qed.
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247 (* end thide *)
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248
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249 (** As a final test of our definition's suitability, we can extract to OCaml. *)
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250
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251 Extraction mergeSort.
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252
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253 (** <<
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254 let rec mergeSort le x =
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255 match le_lt_dec (S (S O)) (length x) with
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256 | Left ->
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257 let lss = partition x in
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258 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
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259 | Right -> x
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260 >>
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261
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262 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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263
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264 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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265
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adam@351
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266 Check well_founded_induction.
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adam@351
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267 (** %\vspace{-.15in}%[[
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|
268 well_founded_induction
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adam@351
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269 : forall (A : Type) (R : A -> A -> Prop),
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270 well_founded R ->
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adam@351
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271 forall P : A -> Set,
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272 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
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adam@351
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273 forall a : A, P a
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adam@351
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274 ]]
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275
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adam@351
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276 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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277
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278
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adam@354
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279 (** * A Non-Termination Monad Inspired by Domain Theory *)
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280
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adam@398
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281 (** The key insights of %\index{domain theory}%domain theory%~\cite{WinskelDomains}% inspire the next approach to modeling non-termination. Domain theory is based on _information orders_ that relate values representing computation 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 _approximation_ of the latter, while the latter is _not_ 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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282
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adam@355
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283 Consider this definition of a type of computations. *)
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284
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adam@352
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285 Section computation.
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adam@352
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286 Variable A : Type.
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adam@355
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287 (** The type [A] describes the result a computation will yield, if it terminates.
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288
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289 We give a rich dependent type to computations themselves: *)
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290
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adam@352
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291 Definition computation :=
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292 {f : nat -> option A
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293 | forall (n : nat) (v : A),
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294 f n = Some v
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adam@352
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295 -> forall (n' : nat), n' >= n
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adam@352
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296 -> f n' = Some v}.
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297
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adam@398
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298 (** A computation is fundamentally a function [f] from an _approximation level_ [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 _monotone_ 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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299
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300 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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301
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302 Definition runTo (m : computation) (n : nat) (v : A) :=
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303 proj1_sig m n = Some v.
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304
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adam@355
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305 (** 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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306
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adam@352
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307 Definition run (m : computation) (v : A) :=
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308 exists n, runTo m n v.
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adam@352
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309 End computation.
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310
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adam@355
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311 (** 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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adam@355
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312 (* begin hide *)
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adam@355
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313
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adam@352
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314 Hint Unfold runTo.
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315
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adam@352
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316 Ltac run' := unfold run, runTo in *; try red; crush;
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adam@352
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317 repeat (match goal with
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adam@352
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318 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
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adam@352
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319 match goal with
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adam@352
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320 | [ x : _ |- _ ] =>
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adam@352
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321 match x with
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adam@352
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322 | E => destruct E
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adam@352
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323 end
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adam@352
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324 end
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adam@352
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325 | [ |- context[match ?M with exist _ _ => _ end] ] => let Heq := fresh "Heq" in
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326 case_eq M; intros ? ? Heq; try rewrite Heq in *; try subst
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327 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
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328 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
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329 | [ H : forall n v, ?E n = Some v -> _,
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330 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
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331 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
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332 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
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adam@352
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333 end; simpl in *); eauto 7.
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adam@352
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334
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adam@352
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335 Ltac run := run'; repeat (match goal with
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adam@352
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336 | [ H : forall n v, ?E n = Some v -> _
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adam@352
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337 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
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adam@352
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338 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
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adam@352
|
339 end; run').
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adam@352
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340
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adam@352
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341 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
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adam@352
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342 exists 0; auto.
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adam@352
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343 Qed.
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adam@352
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344
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adam@352
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345 Hint Resolve ex_irrelevant.
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adam@352
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346
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adam@352
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347 Require Import Max.
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adam@352
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348
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adam@380
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349 Theorem max_spec_le : forall n m, n <= m /\ max n m = m \/ m <= n /\ max n m = n.
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adam@380
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350 induction n; destruct m; simpl; intuition;
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adam@380
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351 specialize (IHn m); intuition.
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adam@380
|
352 Qed.
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adam@380
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353
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adam@352
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354 Ltac max := intros n m; generalize (max_spec_le n m); crush.
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adam@352
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355
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adam@352
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356 Lemma max_1 : forall n m, max n m >= n.
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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 Lemma max_2 : forall n m, max n m >= m.
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adam@352
|
361 max.
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adam@352
|
362 Qed.
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adam@352
|
363
|
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adam@352
|
364 Hint Resolve max_1 max_2.
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adam@352
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365
|
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adam@352
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366 Lemma ge_refl : forall n, n >= n.
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adam@352
|
367 crush.
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adam@352
|
368 Qed.
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adam@352
|
369
|
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adam@352
|
370 Hint Resolve ge_refl.
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adam@352
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371
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adam@352
|
372 Hint Extern 1 => match goal with
|
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adam@352
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373 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
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adam@352
|
374 end.
|
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adam@355
|
375 (* end hide *)
|
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adam@355
|
376 (** remove printing exists *)
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adam@355
|
377
|
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adam@357
|
378 (** 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, opaque proofs are fine here, since the proof components of computations do not influence evaluation behavior. *)
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adam@352
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379
|
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adam@352
|
380 Section Bottom.
|
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adam@352
|
381 Variable A : Type.
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adam@352
|
382
|
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adam@352
|
383 Definition Bottom : computation A.
|
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adam@352
|
384 exists (fun _ : nat => @None A); abstract run.
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adam@352
|
385 Defined.
|
|
adam@352
|
386
|
|
adam@352
|
387 Theorem run_Bottom : forall v, ~run Bottom v.
|
|
adam@352
|
388 run.
|
|
adam@352
|
389 Qed.
|
|
adam@352
|
390 End Bottom.
|
|
adam@352
|
391
|
|
adam@355
|
392 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *)
|
|
adam@355
|
393
|
|
adam@352
|
394 Section Return.
|
|
adam@352
|
395 Variable A : Type.
|
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adam@352
|
396 Variable v : A.
|
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adam@352
|
397
|
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adam@352
|
398 Definition Return : computation A.
|
|
adam@352
|
399 intros; exists (fun _ : nat => Some v); abstract run.
|
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adam@352
|
400 Defined.
|
|
adam@352
|
401
|
|
adam@352
|
402 Theorem run_Return : run Return v.
|
|
adam@352
|
403 run.
|
|
adam@352
|
404 Qed.
|
|
adam@352
|
405 End Return.
|
|
adam@352
|
406
|
|
adam@356
|
407 (** 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
|
408
|
|
adam@352
|
409 Section Bind.
|
|
adam@352
|
410 Variables A B : Type.
|
|
adam@352
|
411 Variable m1 : computation A.
|
|
adam@352
|
412 Variable m2 : A -> computation B.
|
|
adam@352
|
413
|
|
adam@352
|
414 Definition Bind : computation B.
|
|
adam@352
|
415 exists (fun n =>
|
|
adam@357
|
416 let (f1, _) := m1 in
|
|
adam@352
|
417 match f1 n with
|
|
adam@352
|
418 | None => None
|
|
adam@352
|
419 | Some v =>
|
|
adam@357
|
420 let (f2, _) := m2 v in
|
|
adam@352
|
421 f2 n
|
|
adam@352
|
422 end); abstract run.
|
|
adam@352
|
423 Defined.
|
|
adam@352
|
424
|
|
adam@352
|
425 Theorem run_Bind : forall (v1 : A) (v2 : B),
|
|
adam@352
|
426 run m1 v1
|
|
adam@352
|
427 -> run (m2 v1) v2
|
|
adam@352
|
428 -> run Bind v2.
|
|
adam@352
|
429 run; match goal with
|
|
adam@352
|
430 | [ x : nat, y : nat |- _ ] => exists (max x y)
|
|
adam@352
|
431 end; run.
|
|
adam@352
|
432 Qed.
|
|
adam@352
|
433 End Bind.
|
|
adam@352
|
434
|
|
adam@355
|
435 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *)
|
|
adam@352
|
436
|
|
adam@352
|
437 Notation "x <- m1 ; m2" :=
|
|
adam@352
|
438 (Bind m1 (fun x => m2)) (right associativity, at level 70).
|
|
adam@352
|
439
|
|
adam@355
|
440 (** 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
|
441
|
|
adam@352
|
442 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
|
|
adam@352
|
443
|
|
adam@352
|
444 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
|
|
adam@352
|
445 meq (Bind (Return a) f) (f a).
|
|
adam@352
|
446 run.
|
|
adam@352
|
447 Qed.
|
|
adam@352
|
448
|
|
adam@352
|
449 Theorem right_identity : forall A (m : computation A),
|
|
adam@352
|
450 meq (Bind m (@Return _)) m.
|
|
adam@352
|
451 run.
|
|
adam@352
|
452 Qed.
|
|
adam@352
|
453
|
|
adam@357
|
454 Theorem associativity : forall A B C (m : computation A)
|
|
adam@357
|
455 (f : A -> computation B) (g : B -> computation C),
|
|
adam@352
|
456 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
|
|
adam@352
|
457 run.
|
|
adam@352
|
458 Qed.
|
|
adam@352
|
459
|
|
adam@398
|
460 (** 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 _continuous_ 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
|
461
|
|
adam@352
|
462 Section lattice.
|
|
adam@352
|
463 Variable A : Type.
|
|
adam@352
|
464
|
|
adam@352
|
465 Definition leq (x y : option A) :=
|
|
adam@352
|
466 forall v, x = Some v -> y = Some v.
|
|
adam@352
|
467 End lattice.
|
|
adam@352
|
468
|
|
adam@355
|
469 (** 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
|
470
|
|
adam@352
|
471 Section Fix.
|
|
adam@355
|
472 (** First, we have the function domain and range types. *)
|
|
adam@355
|
473
|
|
adam@352
|
474 Variables A B : Type.
|
|
adam@355
|
475
|
|
adam@355
|
476 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *)
|
|
adam@355
|
477
|
|
adam@352
|
478 Variable f : (A -> computation B) -> (A -> computation B).
|
|
adam@352
|
479
|
|
adam@355
|
480 (** 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
|
481
|
|
adam@352
|
482 Hypothesis f_continuous : forall n v v1 x,
|
|
adam@352
|
483 runTo (f v1 x) n v
|
|
adam@352
|
484 -> forall (v2 : A -> computation B),
|
|
adam@352
|
485 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
|
|
adam@352
|
486 -> runTo (f v2 x) n v.
|
|
adam@352
|
487
|
|
adam@355
|
488 (** 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
|
489
|
|
adam@352
|
490 Fixpoint Fix' (n : nat) (x : A) : computation B :=
|
|
adam@352
|
491 match n with
|
|
adam@352
|
492 | O => Bottom _
|
|
adam@352
|
493 | S n' => f (Fix' n') x
|
|
adam@352
|
494 end.
|
|
adam@352
|
495
|
|
adam@355
|
496 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *)
|
|
adam@355
|
497
|
|
adam@352
|
498 Hint Extern 1 (_ >= _) => omega.
|
|
adam@352
|
499 Hint Unfold leq.
|
|
adam@352
|
500
|
|
adam@352
|
501 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
|
|
adam@352
|
502 -> forall n', n' >= n
|
|
adam@352
|
503 -> proj1_sig (Fix' n' x) steps = Some v.
|
|
adam@352
|
504 unfold runTo in *; induction n; crush;
|
|
adam@352
|
505 match goal with
|
|
adam@352
|
506 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
|
|
adam@352
|
507 end.
|
|
adam@352
|
508 Qed.
|
|
adam@352
|
509
|
|
adam@352
|
510 Hint Resolve Fix'_ok.
|
|
adam@352
|
511
|
|
adam@352
|
512 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
|
|
adam@352
|
513 match goal with
|
|
adam@352
|
514 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
|
|
adam@352
|
515 end.
|
|
adam@352
|
516
|
|
adam@352
|
517 Definition Fix : A -> computation B.
|
|
adam@352
|
518 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
|
|
adam@352
|
519 Defined.
|
|
adam@352
|
520
|
|
adam@355
|
521 (** Finally, we can prove that [Fix] obeys the expected computation rule. *)
|
|
adam@352
|
522
|
|
adam@352
|
523 Theorem run_Fix : forall x v,
|
|
adam@352
|
524 run (f Fix x) v
|
|
adam@352
|
525 -> run (Fix x) v.
|
|
adam@352
|
526 run; match goal with
|
|
adam@352
|
527 | [ n : nat |- _ ] => exists (S n); eauto
|
|
adam@352
|
528 end.
|
|
adam@352
|
529 Qed.
|
|
adam@352
|
530 End Fix.
|
|
adam@352
|
531
|
|
adam@355
|
532 (* begin hide *)
|
|
adam@352
|
533 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
|
|
adam@352
|
534 -> x = y.
|
|
adam@352
|
535 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
|
|
adam@352
|
536 Qed.
|
|
adam@352
|
537
|
|
adam@352
|
538 Lemma leq_None : forall A (x y : A), leq (Some x) None
|
|
adam@352
|
539 -> False.
|
|
adam@352
|
540 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
|
|
adam@352
|
541 Qed.
|
|
adam@352
|
542
|
|
adam@355
|
543 Ltac mergeSort' := run;
|
|
adam@355
|
544 repeat (match goal with
|
|
adam@355
|
545 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
|
|
adam@355
|
546 end; run);
|
|
adam@355
|
547 repeat match goal with
|
|
adam@355
|
548 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
|
|
adam@355
|
549 match goal with
|
|
adam@355
|
550 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
|
|
adam@355
|
551 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
|
|
adam@355
|
552 end
|
|
adam@355
|
553 end; run; repeat match goal with
|
|
adam@355
|
554 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
|
|
adam@355
|
555 end; auto.
|
|
adam@355
|
556 (* end hide *)
|
|
adam@355
|
557
|
|
adam@355
|
558 (** 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
|
559
|
|
adam@352
|
560 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
|
|
adam@352
|
561 refine (fun A le => Fix
|
|
adam@352
|
562 (fun (mergeSort : list A -> computation (list A))
|
|
adam@352
|
563 (ls : list A) =>
|
|
adam@352
|
564 if le_lt_dec 2 (length ls)
|
|
adam@352
|
565 then let lss := partition ls in
|
|
adam@352
|
566 ls1 <- mergeSort (fst lss);
|
|
adam@352
|
567 ls2 <- mergeSort (snd lss);
|
|
adam@352
|
568 Return (merge le ls1 ls2)
|
|
adam@355
|
569 else Return ls) _); abstract mergeSort'.
|
|
adam@352
|
570 Defined.
|
|
adam@352
|
571
|
|
adam@355
|
572 (** 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
|
573
|
|
adam@352
|
574 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
|
|
adam@352
|
575 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
|
|
adam@352
|
576 exists 4; reflexivity.
|
|
adam@352
|
577 Qed.
|
|
adam@352
|
578
|
|
adam@355
|
579 (** 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
|
580
|
|
adam@355
|
581 (* begin hide *)
|
|
adam@355
|
582 Ltac looper := unfold leq in *; run;
|
|
adam@355
|
583 repeat match goal with
|
|
adam@355
|
584 | [ x : unit |- _ ] => destruct x
|
|
adam@355
|
585 | [ x : bool |- _ ] => destruct x
|
|
adam@355
|
586 end; auto.
|
|
adam@355
|
587 (* end hide *)
|
|
adam@355
|
588
|
|
adam@352
|
589 Definition looper : bool -> computation unit.
|
|
adam@352
|
590 refine (Fix (fun looper (b : bool) =>
|
|
adam@355
|
591 if b then Return tt else looper b) _); abstract looper.
|
|
adam@352
|
592 Defined.
|
|
adam@352
|
593
|
|
adam@352
|
594 Lemma test_looper : run (looper true) tt.
|
|
adam@352
|
595 exists 1; reflexivity.
|
|
adam@352
|
596 Qed.
|
|
adam@354
|
597
|
|
adam@355
|
598 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level.
|
|
adam@355
|
599
|
|
adam@355
|
600 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
|
601
|
|
adam@354
|
602
|
|
adam@354
|
603 (** * Co-Inductive Non-Termination Monads *)
|
|
adam@354
|
604
|
|
adam@356
|
605 (** 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
|
606
|
|
adam@356
|
607 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
|
608
|
|
adam@354
|
609 CoInductive thunk (A : Type) : Type :=
|
|
adam@354
|
610 | Answer : A -> thunk A
|
|
adam@354
|
611 | Think : thunk A -> thunk A.
|
|
adam@354
|
612
|
|
adam@356
|
613 (** 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
|
614
|
|
adam@356
|
615 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
|
616
|
|
adam@356
|
617 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
|
|
adam@354
|
618 match m1 with
|
|
adam@354
|
619 | Answer x => m2 x
|
|
adam@354
|
620 | Think m1' => Think (TBind m1' m2)
|
|
adam@354
|
621 end.
|
|
adam@354
|
622
|
|
adam@356
|
623 (** 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
|
624
|
|
adam@356
|
625 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
|
626
|
|
adam@356
|
627 (* begin hide *)
|
|
adam@354
|
628 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
|
|
adam@354
|
629 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
|
|
adam@354
|
630 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
|
|
adam@354
|
631 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
|
|
adam@354
|
632
|
|
adam@354
|
633 Section thunk_eq_coind.
|
|
adam@354
|
634 Variable A : Type.
|
|
adam@354
|
635 Variable P : thunk A -> thunk A -> Prop.
|
|
adam@354
|
636
|
|
adam@354
|
637 Hypothesis H : forall m1 m2, P m1 m2
|
|
adam@354
|
638 -> match m1, m2 with
|
|
adam@354
|
639 | Answer x1, Answer x2 => x1 = x2
|
|
adam@354
|
640 | Think m1', Think m2' => P m1' m2'
|
|
adam@354
|
641 | Think m1', _ => P m1' m2
|
|
adam@354
|
642 | _, Think m2' => P m1 m2'
|
|
adam@354
|
643 end.
|
|
adam@354
|
644
|
|
adam@354
|
645 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
|
|
adam@354
|
646 cofix; intros;
|
|
adam@354
|
647 match goal with
|
|
adam@354
|
648 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
|
|
adam@354
|
649 end; destruct m1; destruct m2; subst; repeat constructor; auto.
|
|
adam@354
|
650 Qed.
|
|
adam@354
|
651 End thunk_eq_coind.
|
|
adam@356
|
652 (* end hide *)
|
|
adam@356
|
653
|
|
adam@356
|
654 (** 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
|
655
|
|
adam@354
|
656 Definition frob A (m : thunk A) : thunk A :=
|
|
adam@354
|
657 match m with
|
|
adam@354
|
658 | Answer x => Answer x
|
|
adam@354
|
659 | Think m' => Think m'
|
|
adam@354
|
660 end.
|
|
adam@354
|
661
|
|
adam@354
|
662 Theorem frob_eq : forall A (m : thunk A), frob m = m.
|
|
adam@354
|
663 destruct m; reflexivity.
|
|
adam@354
|
664 Qed.
|
|
adam@354
|
665
|
|
adam@356
|
666 (* begin hide *)
|
|
adam@354
|
667 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
|
|
adam@354
|
668 thunk_eq (frob m1) (frob m2)
|
|
adam@354
|
669 -> thunk_eq m1 m2.
|
|
adam@354
|
670 intros; repeat rewrite frob_eq in *; auto.
|
|
adam@354
|
671 Qed.
|
|
adam@354
|
672
|
|
adam@354
|
673 Ltac findDestr := match goal with
|
|
adam@354
|
674 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
|
|
adam@354
|
675 match E with
|
|
adam@354
|
676 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
|
|
adam@354
|
677 | _ => destruct E
|
|
adam@354
|
678 end
|
|
adam@354
|
679 end.
|
|
adam@354
|
680
|
|
adam@354
|
681 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
|
|
adam@354
|
682 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
|
|
adam@354
|
683 Qed.
|
|
adam@354
|
684
|
|
adam@354
|
685 Hint Resolve thunk_eq_refl.
|
|
adam@354
|
686
|
|
adam@354
|
687 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
|
|
adam@354
|
688 thunk_eq (TBind (Answer a) f) (f a).
|
|
adam@354
|
689 intros; apply thunk_eq_frob; crush.
|
|
adam@354
|
690 Qed.
|
|
adam@354
|
691
|
|
adam@354
|
692 Theorem tright_identity : forall A (m : thunk A),
|
|
adam@354
|
693 thunk_eq (TBind m (@Answer _)) m.
|
|
adam@354
|
694 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
|
|
adam@354
|
695 findDestr; reflexivity.
|
|
adam@354
|
696 Qed.
|
|
adam@354
|
697
|
|
adam@354
|
698 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
|
|
adam@354
|
699 TBind (Answer v) m2 = m2 v.
|
|
adam@354
|
700 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
|
|
adam@354
|
701 simpl; findDestr; reflexivity.
|
|
adam@354
|
702 Qed.
|
|
adam@354
|
703
|
|
adam@375
|
704 Hint Rewrite TBind_Answer.
|
|
adam@354
|
705
|
|
adam@355
|
706 (** printing exists $\exists$ *)
|
|
adam@355
|
707
|
|
adam@354
|
708 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
|
|
adam@354
|
709 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
|
|
adam@354
|
710 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
|
|
adam@354
|
711 m1 = TBind (TBind m f) g
|
|
adam@354
|
712 /\ m2 = TBind m (fun x => TBind (f x) g))
|
|
adam@354
|
713 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
|
|
adam@354
|
714 Qed.
|
|
adam@356
|
715 (* end hide *)
|
|
adam@356
|
716
|
|
adam@356
|
717 (** As a simple example, here is how we might define a tail-recursive factorial function. *)
|
|
adam@354
|
718
|
|
adam@354
|
719 CoFixpoint fact (n acc : nat) : thunk nat :=
|
|
adam@354
|
720 match n with
|
|
adam@354
|
721 | O => Answer acc
|
|
adam@354
|
722 | S n' => Think (fact n' (S n' * acc))
|
|
adam@354
|
723 end.
|
|
adam@354
|
724
|
|
adam@356
|
725 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *)
|
|
adam@356
|
726
|
|
adam@354
|
727 Inductive eval A : thunk A -> A -> Prop :=
|
|
adam@354
|
728 | EvalAnswer : forall x, eval (Answer x) x
|
|
adam@354
|
729 | EvalThink : forall m x, eval m x -> eval (Think m) x.
|
|
adam@354
|
730
|
|
adam@375
|
731 Hint Rewrite frob_eq.
|
|
adam@354
|
732
|
|
adam@354
|
733 Lemma eval_frob : forall A (c : thunk A) x,
|
|
adam@354
|
734 eval (frob c) x
|
|
adam@354
|
735 -> eval c x.
|
|
adam@354
|
736 crush.
|
|
adam@354
|
737 Qed.
|
|
adam@354
|
738
|
|
adam@354
|
739 Theorem eval_fact : eval (fact 5 1) 120.
|
|
adam@354
|
740 repeat (apply eval_frob; simpl; constructor).
|
|
adam@354
|
741 Qed.
|
|
adam@354
|
742
|
|
adam@356
|
743 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs.
|
|
adam@356
|
744
|
|
adam@357
|
745 Now consider another very similar definition, this time of a Fibonacci number funtion. *)
|
|
adam@357
|
746
|
|
adam@357
|
747 Notation "x <- m1 ; m2" :=
|
|
adam@357
|
748 (TBind m1 (fun x => m2)) (right associativity, at level 70).
|
|
adam@357
|
749
|
|
adam@404
|
750 (* begin hide *)
|
|
adam@404
|
751 Definition fib := 0.
|
|
adam@404
|
752 (* end hide *)
|
|
adam@404
|
753
|
|
adam@357
|
754 (** %\vspace{-.15in}%[[
|
|
adam@354
|
755 CoFixpoint fib (n : nat) : thunk nat :=
|
|
adam@354
|
756 match n with
|
|
adam@354
|
757 | 0 => Answer 1
|
|
adam@354
|
758 | 1 => Answer 1
|
|
adam@357
|
759 | _ => n1 <- fib (pred n);
|
|
adam@357
|
760 n2 <- fib (pred (pred n));
|
|
adam@357
|
761 Answer (n1 + n2)
|
|
adam@354
|
762 end.
|
|
adam@354
|
763 ]]
|
|
adam@354
|
764
|
|
adam@356
|
765 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
|
766
|
|
adam@356
|
767 %\medskip%
|
|
adam@356
|
768
|
|
adam@356
|
769 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
|
770
|
|
adam@354
|
771 CoInductive comp (A : Type) : Type :=
|
|
adam@354
|
772 | Ret : A -> comp A
|
|
adam@354
|
773 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
|
|
adam@354
|
774
|
|
adam@404
|
775 (** This example shows off Coq's support for%\index{recursively non-uniform parameters}% _recursively non-uniform parameters_, 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
|
776
|
|
adam@356
|
777 It is easy to define the semantics of terminating [comp] computations. *)
|
|
adam@356
|
778
|
|
adam@354
|
779 Inductive exec A : comp A -> A -> Prop :=
|
|
adam@354
|
780 | ExecRet : forall x, exec (Ret x) x
|
|
adam@354
|
781 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
|
|
adam@354
|
782 -> exec (f x1) x2
|
|
adam@354
|
783 -> exec (Bnd c f) x2.
|
|
adam@354
|
784
|
|
adam@356
|
785 (** 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
|
786
|
|
adam@356
|
787 (* begin hide *)
|
|
adam@354
|
788 Hint Constructors exec.
|
|
adam@354
|
789
|
|
adam@354
|
790 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
|
|
adam@354
|
791
|
|
adam@354
|
792 Ltac inverter := repeat match goal with
|
|
adam@354
|
793 | [ H : exec _ _ |- _ ] => inversion H; []; crush
|
|
adam@354
|
794 end.
|
|
adam@354
|
795
|
|
adam@354
|
796 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
|
|
adam@354
|
797 comp_eq (Bnd (Ret a) f) (f a).
|
|
adam@354
|
798 red; crush; inverter; eauto.
|
|
adam@354
|
799 Qed.
|
|
adam@354
|
800
|
|
adam@354
|
801 Theorem cright_identity : forall A (m : comp A),
|
|
adam@354
|
802 comp_eq (Bnd m (@Ret _)) m.
|
|
adam@354
|
803 red; crush; inverter; eauto.
|
|
adam@354
|
804 Qed.
|
|
adam@354
|
805
|
|
adam@354
|
806 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
|
|
adam@354
|
807 exec c r
|
|
adam@354
|
808 -> forall m, c = Bnd (Bnd m f) g
|
|
adam@354
|
809 -> exec (Bnd m (fun x => Bnd (f x) g)) r.
|
|
adam@354
|
810 induction 1; crush.
|
|
adam@354
|
811 match goal with
|
|
adam@354
|
812 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
|
|
adam@354
|
813 end.
|
|
adam@354
|
814 move H3 after A.
|
|
adam@354
|
815 generalize dependent B0.
|
|
adam@354
|
816 do 2 intro.
|
|
adam@354
|
817 subst.
|
|
adam@354
|
818 crush.
|
|
adam@354
|
819 inversion H; clear H; crush.
|
|
adam@354
|
820 eauto.
|
|
adam@354
|
821 Qed.
|
|
adam@354
|
822
|
|
adam@354
|
823 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
|
|
adam@354
|
824 exec c r
|
|
adam@354
|
825 -> forall m, c = Bnd m (fun x => Bnd (f x) g)
|
|
adam@354
|
826 -> exec (Bnd (Bnd m f) g) r.
|
|
adam@354
|
827 induction 1; crush.
|
|
adam@354
|
828 match goal with
|
|
adam@354
|
829 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
|
|
adam@354
|
830 end.
|
|
adam@354
|
831 move H3 after B.
|
|
adam@354
|
832 generalize dependent B0.
|
|
adam@354
|
833 do 2 intro.
|
|
adam@354
|
834 subst.
|
|
adam@354
|
835 crush.
|
|
adam@354
|
836 inversion H0; clear H0; crush.
|
|
adam@354
|
837 eauto.
|
|
adam@354
|
838 Qed.
|
|
adam@354
|
839
|
|
adam@354
|
840 Hint Resolve cassociativity1 cassociativity2.
|
|
adam@354
|
841
|
|
adam@354
|
842 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
|
|
adam@354
|
843 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
|
|
adam@354
|
844 red; crush; eauto.
|
|
adam@354
|
845 Qed.
|
|
adam@356
|
846 (* end hide *)
|
|
adam@356
|
847
|
|
adam@356
|
848 (** 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
|
849
|
|
adam@356
|
850 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)).
|
|
adam@354
|
851
|
|
adam@354
|
852 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
|
|
adam@354
|
853 if le_lt_dec 2 (length ls)
|
|
adam@354
|
854 then let lss := partition ls in
|
|
adam@356
|
855 ls1 <- mergeSort'' le (fst lss);
|
|
adam@356
|
856 ls2 <- mergeSort'' le (snd lss);
|
|
adam@356
|
857 Ret (merge le ls1 ls2)
|
|
adam@354
|
858 else Ret ls.
|
|
adam@354
|
859
|
|
adam@356
|
860 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *)
|
|
adam@356
|
861
|
|
adam@354
|
862 Definition frob' A (c : comp A) :=
|
|
adam@354
|
863 match c with
|
|
adam@354
|
864 | Ret x => Ret x
|
|
adam@354
|
865 | Bnd _ c' f => Bnd c' f
|
|
adam@354
|
866 end.
|
|
adam@354
|
867
|
|
adam@354
|
868 Lemma exec_frob : forall A (c : comp A) x,
|
|
adam@354
|
869 exec (frob' c) x
|
|
adam@354
|
870 -> exec c x.
|
|
adam@356
|
871 destruct c; crush.
|
|
adam@354
|
872 Qed.
|
|
adam@354
|
873
|
|
adam@356
|
874 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *)
|
|
adam@356
|
875
|
|
adam@354
|
876 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
|
|
adam@354
|
877 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
|
|
adam@354
|
878 repeat (apply exec_frob; simpl; econstructor).
|
|
adam@354
|
879 Qed.
|
|
adam@354
|
880
|
|
adam@356
|
881 (** 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
|
882
|
|
adam@354
|
883 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
|
|
adam@354
|
884
|
|
adam@356
|
885 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way.
|
|
adam@356
|
886 [[
|
|
adam@356
|
887 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3).
|
|
adam@354
|
888 ]]
|
|
adam@354
|
889
|
|
adam@354
|
890 <<
|
|
adam@354
|
891 Error: Universe inconsistency.
|
|
adam@354
|
892 >>
|
|
adam@356
|
893
|
|
adam@356
|
894 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
|
895
|
|
adam@354
|
896
|
|
adam@357
|
897 (** * Comparing the Alternatives *)
|
|
adam@354
|
898
|
|
adam@356
|
899 (** 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
|
900
|
|
adam@356
|
901 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
|
902
|
|
adam@356
|
903 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
|
904
|
|
adam@356
|
905 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
|
906
|
|
adam@356
|
907 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
|
908
|
|
adam@356
|
909 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
|
910
|
|
adam@356
|
911 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
|
912
|
|
adam@354
|
913 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
|
|
adam@354
|
914
|
|
adam@356
|
915 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3).
|
|
adam@356
|
916
|
|
adam@357
|
917 (** The same techniques also apply to more interesting higher-order functions like list map, and, as in all four techniques, we can mix primitive and general recursion, preferring the former when possible to avoid proof obligations. *)
|
|
adam@354
|
918
|
|
adam@354
|
919 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
|
|
adam@354
|
920 match ls with
|
|
adam@354
|
921 | nil => Return nil
|
|
adam@354
|
922 | x :: ls' => Bind (f x) (fun x' =>
|
|
adam@354
|
923 Bind (map f ls') (fun ls'' =>
|
|
adam@354
|
924 Return (x' :: ls'')))
|
|
adam@354
|
925 end.
|
|
adam@354
|
926
|
|
adam@355
|
927 (** remove printing exists *)
|
|
adam@356
|
928 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil))
|
|
adam@356
|
929 (2 :: 3 :: 4 :: nil).
|
|
adam@354
|
930 exists 1; reflexivity.
|
|
adam@354
|
931 Qed.
|
|
adam@356
|
932
|
|
adam@356
|
933 (** 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
|
934
|
|
adam@357
|
935 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. *)
|
