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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 guardedness 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 (* begin hide *)
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96 Definition Acc_intro' := Acc_intro.
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97 (* end hide *)
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98
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99 Print Acc.
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100 (** %\vspace{-.15in}% [[
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101 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop :=
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102 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x
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103 ]]
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104
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105 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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106
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107 CoInductive isChain A (R : A -> A -> Prop) : stream A -> Prop :=
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108 | ChainCons : forall x y s, isChain R (Cons y s)
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109 -> R y x
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110 -> isChain R (Cons x (Cons y s)).
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111
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112 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *)
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113
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114 (* begin thide *)
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115 Lemma noChains' : forall A (R : A -> A -> Prop) x, Acc R x
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116 -> forall s, ~isChain R (Cons x s).
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117 induction 1; crush;
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118 match goal with
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119 | [ H : isChain _ _ |- _ ] => inversion H; eauto
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120 end.
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121 Qed.
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122
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123 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *)
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124
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125 Theorem noChains : forall A (R : A -> A -> Prop), well_founded R
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126 -> forall s, ~isChain R s.
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127 destruct s; apply noChains'; auto.
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128 Qed.
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129 (* end thide *)
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130
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131 (** 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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132
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133 Check Fix.
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134 (** %\vspace{-.15in}%[[
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135 Fix
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136 : forall (A : Type) (R : A -> A -> Prop),
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137 well_founded R ->
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138 forall P : A -> Type,
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139 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
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140 forall x : A, P x
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141 ]]
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142
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143 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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144 [[
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145 forall x : A, (forall y : A, R y x -> P y) -> P x
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146 ]]
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147
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148 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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149
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150 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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151
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152 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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153
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154 Definition lengthOrder (ls1 ls2 : list A) :=
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155 length ls1 < length ls2.
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156
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157 (** 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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158
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159 Hint Constructors Acc.
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160
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161 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
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162 unfold lengthOrder; induction len; crush.
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163 Defined.
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164
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165 Theorem lengthOrder_wf : well_founded lengthOrder.
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166 red; intro; eapply lengthOrder_wf'; eauto.
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167 Defined.
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168
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169 (** 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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170
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171 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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172
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173 Lemma partition_wf : forall len ls, 2 <= length ls <= len
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174 -> let (ls1, ls2) := partition ls in
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175 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
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176 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
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177 destruct (le_lt_dec 2 (length ls));
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178 repeat (match goal with
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179 | [ _ : length ?E < 2 |- _ ] => destruct E
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180 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
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181 | [ IH : _ |- context[partition ?L] ] =>
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182 specialize (IH L); destruct (partition L); destruct IH
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183 end; crush).
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184 Defined.
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185
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186 Ltac partition := intros ls ?; intros; generalize (@partition_wf (length ls) ls);
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187 destruct (partition ls); destruct 1; crush.
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188
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189 Lemma partition_wf1 : forall ls, 2 <= length ls
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190 -> lengthOrder (fst (partition ls)) ls.
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191 partition.
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192 Defined.
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193
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194 Lemma partition_wf2 : forall ls, 2 <= length ls
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195 -> lengthOrder (snd (partition ls)) ls.
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196 partition.
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197 Defined.
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198
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199 Hint Resolve partition_wf1 partition_wf2.
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200
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201 (** 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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202
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203 Definition mergeSort : list A -> list A.
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204 (* begin thide *)
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205 refine (Fix lengthOrder_wf (fun _ => list A)
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206 (fun (ls : list A)
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207 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
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208 if le_lt_dec 2 (length ls)
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209 then let lss := partition ls in
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210 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
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211 else ls)); subst lss; eauto.
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212 Defined.
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213 (* end thide *)
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214 End mergeSort.
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215
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216 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
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217
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218 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
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219 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
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220
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221 (** 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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222
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223 (* begin thide *)
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224 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
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225 mergeSort le ls = if le_lt_dec 2 (length ls)
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226 then let lss := partition ls in
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227 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
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228 else ls.
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229 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
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230
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231 (** 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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232
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233 Check Fix_eq.
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234 (** %\vspace{-.15in}%[[
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235 Fix_eq
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236 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
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237 (P : A -> Type)
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238 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
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239 (forall (x : A) (f g : forall y : A, R y x -> P y),
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240 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
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241 forall x : A,
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242 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
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243 ]]
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244
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245 Most such obligations are dischargable with straightforward proof automation, and this example is no exception. *)
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246
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247 match goal with
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248 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
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249 end; simpl; f_equal; auto.
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250 Qed.
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251 (* end thide *)
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252
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253 (** As a final test of our definition's suitability, we can extract to OCaml. *)
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254
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255 Extraction mergeSort.
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256
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257 (** <<
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258 let rec mergeSort le x =
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259 match le_lt_dec (S (S O)) (length x) with
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260 | Left ->
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261 let lss = partition x in
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262 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
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263 | Right -> x
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264 >>
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265
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266 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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267
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268 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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269
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270 Check well_founded_induction.
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271 (** %\vspace{-.15in}%[[
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272 well_founded_induction
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273 : forall (A : Type) (R : A -> A -> Prop),
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274 well_founded R ->
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275 forall P : A -> Set,
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276 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
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277 forall a : A, P a
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278 ]]
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279
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280 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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281
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282
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adam@354
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283 (** * A Non-Termination Monad Inspired by Domain Theory *)
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284
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adam@424
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285 (** 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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286
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287 Consider this definition of a type of computations. *)
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288
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289 Section computation.
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adam@352
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290 Variable A : Type.
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adam@355
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291 (** The type [A] describes the result a computation will yield, if it terminates.
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292
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293 We give a rich dependent type to computations themselves: *)
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294
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295 Definition computation :=
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296 {f : nat -> option A
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297 | forall (n : nat) (v : A),
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298 f n = Some v
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299 -> forall (n' : nat), n' >= n
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300 -> f n' = Some v}.
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301
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302 (** 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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303
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304 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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305
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306 Definition runTo (m : computation) (n : nat) (v : A) :=
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307 proj1_sig m n = Some v.
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308
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adam@355
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309 (** 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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310
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311 Definition run (m : computation) (v : A) :=
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312 exists n, runTo m n v.
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313 End computation.
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314
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315 (** 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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316 (* begin hide *)
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317
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adam@352
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318 Hint Unfold runTo.
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319
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320 Ltac run' := unfold run, runTo in *; try red; crush;
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321 repeat (match goal with
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adam@352
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322 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
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323 match goal with
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adam@352
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324 | [ x : _ |- _ ] =>
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adam@352
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325 match x with
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326 | E => destruct E
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adam@352
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327 end
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adam@352
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328 end
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329 | [ |- context[match ?M with exist _ _ => _ end] ] => let Heq := fresh "Heq" in
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330 case_eq M; intros ? ? Heq; try rewrite Heq in *; try subst
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331 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
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332 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
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333 | [ H : forall n v, ?E n = Some v -> _,
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334 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
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335 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
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336 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
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337 end; simpl in *); eauto 7.
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338
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adam@352
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339 Ltac run := run'; repeat (match goal with
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adam@352
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340 | [ H : forall n v, ?E n = Some v -> _
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adam@352
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341 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
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adam@352
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342 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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343 end; run').
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344
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adam@352
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345 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
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adam@352
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346 exists 0; auto.
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adam@352
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347 Qed.
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adam@352
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348
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adam@352
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349 Hint Resolve ex_irrelevant.
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adam@352
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350
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adam@352
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351 Require Import Max.
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adam@352
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352
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adam@380
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353 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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354 induction n; destruct m; simpl; intuition;
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adam@380
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355 specialize (IHn m); intuition.
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adam@380
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356 Qed.
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adam@380
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357
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adam@352
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358 Ltac max := intros n m; generalize (max_spec_le n m); crush.
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adam@352
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359
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adam@352
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360 Lemma max_1 : forall n m, max n m >= n.
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adam@352
|
361 max.
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adam@352
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362 Qed.
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adam@352
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363
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adam@352
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364 Lemma max_2 : forall n m, max n m >= m.
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adam@352
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365 max.
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adam@352
|
366 Qed.
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adam@352
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367
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adam@352
|
368 Hint Resolve max_1 max_2.
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adam@352
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369
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adam@352
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370 Lemma ge_refl : forall n, n >= n.
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adam@352
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371 crush.
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adam@352
|
372 Qed.
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adam@352
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373
|
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adam@352
|
374 Hint Resolve ge_refl.
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adam@352
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375
|
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adam@352
|
376 Hint Extern 1 => match goal with
|
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adam@352
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377 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
|
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adam@352
|
378 end.
|
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adam@355
|
379 (* end hide *)
|
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adam@355
|
380 (** remove printing exists *)
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adam@355
|
381
|
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adam@357
|
382 (** 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
|
383
|
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adam@352
|
384 Section Bottom.
|
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adam@352
|
385 Variable A : Type.
|
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adam@352
|
386
|
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adam@352
|
387 Definition Bottom : computation A.
|
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adam@352
|
388 exists (fun _ : nat => @None A); abstract run.
|
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adam@352
|
389 Defined.
|
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adam@352
|
390
|
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adam@352
|
391 Theorem run_Bottom : forall v, ~run Bottom v.
|
|
adam@352
|
392 run.
|
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adam@352
|
393 Qed.
|
|
adam@352
|
394 End Bottom.
|
|
adam@352
|
395
|
|
adam@355
|
396 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *)
|
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adam@355
|
397
|
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adam@352
|
398 Section Return.
|
|
adam@352
|
399 Variable A : Type.
|
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adam@352
|
400 Variable v : A.
|
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adam@352
|
401
|
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adam@352
|
402 Definition Return : computation A.
|
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adam@352
|
403 intros; exists (fun _ : nat => Some v); abstract run.
|
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adam@352
|
404 Defined.
|
|
adam@352
|
405
|
|
adam@352
|
406 Theorem run_Return : run Return v.
|
|
adam@352
|
407 run.
|
|
adam@352
|
408 Qed.
|
|
adam@352
|
409 End Return.
|
|
adam@352
|
410
|
|
adam@356
|
411 (** 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
|
412
|
|
adam@352
|
413 Section Bind.
|
|
adam@352
|
414 Variables A B : Type.
|
|
adam@352
|
415 Variable m1 : computation A.
|
|
adam@352
|
416 Variable m2 : A -> computation B.
|
|
adam@352
|
417
|
|
adam@352
|
418 Definition Bind : computation B.
|
|
adam@352
|
419 exists (fun n =>
|
|
adam@357
|
420 let (f1, _) := m1 in
|
|
adam@352
|
421 match f1 n with
|
|
adam@352
|
422 | None => None
|
|
adam@352
|
423 | Some v =>
|
|
adam@357
|
424 let (f2, _) := m2 v in
|
|
adam@352
|
425 f2 n
|
|
adam@352
|
426 end); abstract run.
|
|
adam@352
|
427 Defined.
|
|
adam@352
|
428
|
|
adam@352
|
429 Theorem run_Bind : forall (v1 : A) (v2 : B),
|
|
adam@352
|
430 run m1 v1
|
|
adam@352
|
431 -> run (m2 v1) v2
|
|
adam@352
|
432 -> run Bind v2.
|
|
adam@352
|
433 run; match goal with
|
|
adam@352
|
434 | [ x : nat, y : nat |- _ ] => exists (max x y)
|
|
adam@352
|
435 end; run.
|
|
adam@352
|
436 Qed.
|
|
adam@352
|
437 End Bind.
|
|
adam@352
|
438
|
|
adam@355
|
439 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *)
|
|
adam@352
|
440
|
|
adam@352
|
441 Notation "x <- m1 ; m2" :=
|
|
adam@352
|
442 (Bind m1 (fun x => m2)) (right associativity, at level 70).
|
|
adam@352
|
443
|
|
adam@424
|
444 (** 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
|
445
|
|
adam@352
|
446 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
|
|
adam@352
|
447
|
|
adam@352
|
448 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
|
|
adam@352
|
449 meq (Bind (Return a) f) (f a).
|
|
adam@352
|
450 run.
|
|
adam@352
|
451 Qed.
|
|
adam@352
|
452
|
|
adam@352
|
453 Theorem right_identity : forall A (m : computation A),
|
|
adam@352
|
454 meq (Bind m (@Return _)) m.
|
|
adam@352
|
455 run.
|
|
adam@352
|
456 Qed.
|
|
adam@352
|
457
|
|
adam@357
|
458 Theorem associativity : forall A B C (m : computation A)
|
|
adam@357
|
459 (f : A -> computation B) (g : B -> computation C),
|
|
adam@352
|
460 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
|
|
adam@352
|
461 run.
|
|
adam@352
|
462 Qed.
|
|
adam@352
|
463
|
|
adam@398
|
464 (** 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
|
465
|
|
adam@352
|
466 Section lattice.
|
|
adam@352
|
467 Variable A : Type.
|
|
adam@352
|
468
|
|
adam@352
|
469 Definition leq (x y : option A) :=
|
|
adam@352
|
470 forall v, x = Some v -> y = Some v.
|
|
adam@352
|
471 End lattice.
|
|
adam@352
|
472
|
|
adam@355
|
473 (** 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
|
474
|
|
adam@352
|
475 Section Fix.
|
|
adam@355
|
476 (** First, we have the function domain and range types. *)
|
|
adam@355
|
477
|
|
adam@352
|
478 Variables A B : Type.
|
|
adam@355
|
479
|
|
adam@355
|
480 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *)
|
|
adam@355
|
481
|
|
adam@352
|
482 Variable f : (A -> computation B) -> (A -> computation B).
|
|
adam@352
|
483
|
|
adam@355
|
484 (** 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
|
485
|
|
adam@352
|
486 Hypothesis f_continuous : forall n v v1 x,
|
|
adam@352
|
487 runTo (f v1 x) n v
|
|
adam@352
|
488 -> forall (v2 : A -> computation B),
|
|
adam@352
|
489 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
|
|
adam@352
|
490 -> runTo (f v2 x) n v.
|
|
adam@352
|
491
|
|
adam@355
|
492 (** 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
|
493
|
|
adam@352
|
494 Fixpoint Fix' (n : nat) (x : A) : computation B :=
|
|
adam@352
|
495 match n with
|
|
adam@352
|
496 | O => Bottom _
|
|
adam@352
|
497 | S n' => f (Fix' n') x
|
|
adam@352
|
498 end.
|
|
adam@352
|
499
|
|
adam@355
|
500 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *)
|
|
adam@355
|
501
|
|
adam@352
|
502 Hint Extern 1 (_ >= _) => omega.
|
|
adam@352
|
503 Hint Unfold leq.
|
|
adam@352
|
504
|
|
adam@352
|
505 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
|
|
adam@352
|
506 -> forall n', n' >= n
|
|
adam@352
|
507 -> proj1_sig (Fix' n' x) steps = Some v.
|
|
adam@352
|
508 unfold runTo in *; induction n; crush;
|
|
adam@352
|
509 match goal with
|
|
adam@352
|
510 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
|
|
adam@352
|
511 end.
|
|
adam@352
|
512 Qed.
|
|
adam@352
|
513
|
|
adam@352
|
514 Hint Resolve Fix'_ok.
|
|
adam@352
|
515
|
|
adam@352
|
516 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
|
|
adam@352
|
517 match goal with
|
|
adam@352
|
518 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
|
|
adam@352
|
519 end.
|
|
adam@352
|
520
|
|
adam@352
|
521 Definition Fix : A -> computation B.
|
|
adam@352
|
522 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
|
|
adam@352
|
523 Defined.
|
|
adam@352
|
524
|
|
adam@355
|
525 (** Finally, we can prove that [Fix] obeys the expected computation rule. *)
|
|
adam@352
|
526
|
|
adam@352
|
527 Theorem run_Fix : forall x v,
|
|
adam@352
|
528 run (f Fix x) v
|
|
adam@352
|
529 -> run (Fix x) v.
|
|
adam@352
|
530 run; match goal with
|
|
adam@352
|
531 | [ n : nat |- _ ] => exists (S n); eauto
|
|
adam@352
|
532 end.
|
|
adam@352
|
533 Qed.
|
|
adam@352
|
534 End Fix.
|
|
adam@352
|
535
|
|
adam@355
|
536 (* begin hide *)
|
|
adam@352
|
537 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
|
|
adam@352
|
538 -> x = y.
|
|
adam@352
|
539 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
|
|
adam@352
|
540 Qed.
|
|
adam@352
|
541
|
|
adam@352
|
542 Lemma leq_None : forall A (x y : A), leq (Some x) None
|
|
adam@352
|
543 -> False.
|
|
adam@352
|
544 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
|
|
adam@352
|
545 Qed.
|
|
adam@352
|
546
|
|
adam@355
|
547 Ltac mergeSort' := run;
|
|
adam@355
|
548 repeat (match goal with
|
|
adam@355
|
549 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
|
|
adam@355
|
550 end; run);
|
|
adam@355
|
551 repeat match goal with
|
|
adam@355
|
552 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
|
|
adam@355
|
553 match goal with
|
|
adam@355
|
554 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
|
|
adam@355
|
555 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
|
|
adam@355
|
556 end
|
|
adam@355
|
557 end; run; repeat match goal with
|
|
adam@355
|
558 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
|
|
adam@355
|
559 end; auto.
|
|
adam@355
|
560 (* end hide *)
|
|
adam@355
|
561
|
|
adam@355
|
562 (** 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
|
563
|
|
adam@352
|
564 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
|
|
adam@352
|
565 refine (fun A le => Fix
|
|
adam@352
|
566 (fun (mergeSort : list A -> computation (list A))
|
|
adam@352
|
567 (ls : list A) =>
|
|
adam@352
|
568 if le_lt_dec 2 (length ls)
|
|
adam@352
|
569 then let lss := partition ls in
|
|
adam@352
|
570 ls1 <- mergeSort (fst lss);
|
|
adam@352
|
571 ls2 <- mergeSort (snd lss);
|
|
adam@352
|
572 Return (merge le ls1 ls2)
|
|
adam@355
|
573 else Return ls) _); abstract mergeSort'.
|
|
adam@352
|
574 Defined.
|
|
adam@352
|
575
|
|
adam@424
|
576 (** 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
|
577
|
|
adam@352
|
578 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
|
|
adam@352
|
579 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
|
|
adam@352
|
580 exists 4; reflexivity.
|
|
adam@352
|
581 Qed.
|
|
adam@352
|
582
|
|
adam@355
|
583 (** 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
|
584
|
|
adam@355
|
585 (* begin hide *)
|
|
adam@355
|
586 Ltac looper := unfold leq in *; run;
|
|
adam@355
|
587 repeat match goal with
|
|
adam@355
|
588 | [ x : unit |- _ ] => destruct x
|
|
adam@355
|
589 | [ x : bool |- _ ] => destruct x
|
|
adam@355
|
590 end; auto.
|
|
adam@355
|
591 (* end hide *)
|
|
adam@355
|
592
|
|
adam@352
|
593 Definition looper : bool -> computation unit.
|
|
adam@352
|
594 refine (Fix (fun looper (b : bool) =>
|
|
adam@355
|
595 if b then Return tt else looper b) _); abstract looper.
|
|
adam@352
|
596 Defined.
|
|
adam@352
|
597
|
|
adam@352
|
598 Lemma test_looper : run (looper true) tt.
|
|
adam@352
|
599 exists 1; reflexivity.
|
|
adam@352
|
600 Qed.
|
|
adam@354
|
601
|
|
adam@355
|
602 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level.
|
|
adam@355
|
603
|
|
adam@424
|
604 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
|
605
|
|
adam@354
|
606
|
|
adam@354
|
607 (** * Co-Inductive Non-Termination Monads *)
|
|
adam@354
|
608
|
|
adam@356
|
609 (** 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
|
610
|
|
adam@356
|
611 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
|
612
|
|
adam@354
|
613 CoInductive thunk (A : Type) : Type :=
|
|
adam@354
|
614 | Answer : A -> thunk A
|
|
adam@354
|
615 | Think : thunk A -> thunk A.
|
|
adam@354
|
616
|
|
adam@356
|
617 (** 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
|
618
|
|
adam@424
|
619 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
|
620
|
|
adam@356
|
621 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
|
|
adam@354
|
622 match m1 with
|
|
adam@354
|
623 | Answer x => m2 x
|
|
adam@354
|
624 | Think m1' => Think (TBind m1' m2)
|
|
adam@354
|
625 end.
|
|
adam@354
|
626
|
|
adam@356
|
627 (** 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
|
628
|
|
adam@356
|
629 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
|
630
|
|
adam@356
|
631 (* begin hide *)
|
|
adam@354
|
632 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
|
|
adam@354
|
633 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
|
|
adam@354
|
634 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
|
|
adam@354
|
635 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
|
|
adam@354
|
636
|
|
adam@354
|
637 Section thunk_eq_coind.
|
|
adam@354
|
638 Variable A : Type.
|
|
adam@354
|
639 Variable P : thunk A -> thunk A -> Prop.
|
|
adam@354
|
640
|
|
adam@354
|
641 Hypothesis H : forall m1 m2, P m1 m2
|
|
adam@354
|
642 -> match m1, m2 with
|
|
adam@354
|
643 | Answer x1, Answer x2 => x1 = x2
|
|
adam@354
|
644 | Think m1', Think m2' => P m1' m2'
|
|
adam@354
|
645 | Think m1', _ => P m1' m2
|
|
adam@354
|
646 | _, Think m2' => P m1 m2'
|
|
adam@354
|
647 end.
|
|
adam@354
|
648
|
|
adam@354
|
649 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
|
|
adam@354
|
650 cofix; intros;
|
|
adam@354
|
651 match goal with
|
|
adam@354
|
652 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
|
|
adam@354
|
653 end; destruct m1; destruct m2; subst; repeat constructor; auto.
|
|
adam@354
|
654 Qed.
|
|
adam@354
|
655 End thunk_eq_coind.
|
|
adam@356
|
656 (* end hide *)
|
|
adam@356
|
657
|
|
adam@356
|
658 (** 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
|
659
|
|
adam@354
|
660 Definition frob A (m : thunk A) : thunk A :=
|
|
adam@354
|
661 match m with
|
|
adam@354
|
662 | Answer x => Answer x
|
|
adam@354
|
663 | Think m' => Think m'
|
|
adam@354
|
664 end.
|
|
adam@354
|
665
|
|
adam@354
|
666 Theorem frob_eq : forall A (m : thunk A), frob m = m.
|
|
adam@354
|
667 destruct m; reflexivity.
|
|
adam@354
|
668 Qed.
|
|
adam@354
|
669
|
|
adam@356
|
670 (* begin hide *)
|
|
adam@354
|
671 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
|
|
adam@354
|
672 thunk_eq (frob m1) (frob m2)
|
|
adam@354
|
673 -> thunk_eq m1 m2.
|
|
adam@354
|
674 intros; repeat rewrite frob_eq in *; auto.
|
|
adam@354
|
675 Qed.
|
|
adam@354
|
676
|
|
adam@354
|
677 Ltac findDestr := match goal with
|
|
adam@354
|
678 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
|
|
adam@354
|
679 match E with
|
|
adam@354
|
680 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
|
|
adam@354
|
681 | _ => destruct E
|
|
adam@354
|
682 end
|
|
adam@354
|
683 end.
|
|
adam@354
|
684
|
|
adam@354
|
685 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
|
|
adam@354
|
686 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
|
|
adam@354
|
687 Qed.
|
|
adam@354
|
688
|
|
adam@354
|
689 Hint Resolve thunk_eq_refl.
|
|
adam@354
|
690
|
|
adam@354
|
691 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
|
|
adam@354
|
692 thunk_eq (TBind (Answer a) f) (f a).
|
|
adam@354
|
693 intros; apply thunk_eq_frob; crush.
|
|
adam@354
|
694 Qed.
|
|
adam@354
|
695
|
|
adam@354
|
696 Theorem tright_identity : forall A (m : thunk A),
|
|
adam@354
|
697 thunk_eq (TBind m (@Answer _)) m.
|
|
adam@354
|
698 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
|
|
adam@354
|
699 findDestr; reflexivity.
|
|
adam@354
|
700 Qed.
|
|
adam@354
|
701
|
|
adam@354
|
702 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
|
|
adam@354
|
703 TBind (Answer v) m2 = m2 v.
|
|
adam@354
|
704 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
|
|
adam@354
|
705 simpl; findDestr; reflexivity.
|
|
adam@354
|
706 Qed.
|
|
adam@354
|
707
|
|
adam@375
|
708 Hint Rewrite TBind_Answer.
|
|
adam@354
|
709
|
|
adam@355
|
710 (** printing exists $\exists$ *)
|
|
adam@355
|
711
|
|
adam@354
|
712 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
|
|
adam@354
|
713 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
|
|
adam@354
|
714 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
|
|
adam@354
|
715 m1 = TBind (TBind m f) g
|
|
adam@354
|
716 /\ m2 = TBind m (fun x => TBind (f x) g))
|
|
adam@354
|
717 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
|
|
adam@354
|
718 Qed.
|
|
adam@356
|
719 (* end hide *)
|
|
adam@356
|
720
|
|
adam@356
|
721 (** As a simple example, here is how we might define a tail-recursive factorial function. *)
|
|
adam@354
|
722
|
|
adam@354
|
723 CoFixpoint fact (n acc : nat) : thunk nat :=
|
|
adam@354
|
724 match n with
|
|
adam@354
|
725 | O => Answer acc
|
|
adam@354
|
726 | S n' => Think (fact n' (S n' * acc))
|
|
adam@354
|
727 end.
|
|
adam@354
|
728
|
|
adam@356
|
729 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *)
|
|
adam@356
|
730
|
|
adam@354
|
731 Inductive eval A : thunk A -> A -> Prop :=
|
|
adam@354
|
732 | EvalAnswer : forall x, eval (Answer x) x
|
|
adam@354
|
733 | EvalThink : forall m x, eval m x -> eval (Think m) x.
|
|
adam@354
|
734
|
|
adam@375
|
735 Hint Rewrite frob_eq.
|
|
adam@354
|
736
|
|
adam@354
|
737 Lemma eval_frob : forall A (c : thunk A) x,
|
|
adam@354
|
738 eval (frob c) x
|
|
adam@354
|
739 -> eval c x.
|
|
adam@354
|
740 crush.
|
|
adam@354
|
741 Qed.
|
|
adam@354
|
742
|
|
adam@354
|
743 Theorem eval_fact : eval (fact 5 1) 120.
|
|
adam@354
|
744 repeat (apply eval_frob; simpl; constructor).
|
|
adam@354
|
745 Qed.
|
|
adam@354
|
746
|
|
adam@356
|
747 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs.
|
|
adam@356
|
748
|
|
adam@357
|
749 Now consider another very similar definition, this time of a Fibonacci number funtion. *)
|
|
adam@357
|
750
|
|
adam@357
|
751 Notation "x <- m1 ; m2" :=
|
|
adam@357
|
752 (TBind m1 (fun x => m2)) (right associativity, at level 70).
|
|
adam@357
|
753
|
|
adam@404
|
754 (* begin hide *)
|
|
adam@424
|
755 Definition fib := pred.
|
|
adam@404
|
756 (* end hide *)
|
|
adam@404
|
757
|
|
adam@357
|
758 (** %\vspace{-.15in}%[[
|
|
adam@354
|
759 CoFixpoint fib (n : nat) : thunk nat :=
|
|
adam@354
|
760 match n with
|
|
adam@354
|
761 | 0 => Answer 1
|
|
adam@354
|
762 | 1 => Answer 1
|
|
adam@357
|
763 | _ => n1 <- fib (pred n);
|
|
adam@357
|
764 n2 <- fib (pred (pred n));
|
|
adam@357
|
765 Answer (n1 + n2)
|
|
adam@354
|
766 end.
|
|
adam@354
|
767 ]]
|
|
adam@354
|
768
|
|
adam@356
|
769 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
|
770
|
|
adam@356
|
771 %\medskip%
|
|
adam@356
|
772
|
|
adam@424
|
773 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
|
774
|
|
adam@354
|
775 CoInductive comp (A : Type) : Type :=
|
|
adam@354
|
776 | Ret : A -> comp A
|
|
adam@354
|
777 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
|
|
adam@354
|
778
|
|
adam@404
|
779 (** 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
|
780
|
|
adam@356
|
781 It is easy to define the semantics of terminating [comp] computations. *)
|
|
adam@356
|
782
|
|
adam@354
|
783 Inductive exec A : comp A -> A -> Prop :=
|
|
adam@354
|
784 | ExecRet : forall x, exec (Ret x) x
|
|
adam@354
|
785 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
|
|
adam@354
|
786 -> exec (f x1) x2
|
|
adam@354
|
787 -> exec (Bnd c f) x2.
|
|
adam@354
|
788
|
|
adam@356
|
789 (** 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
|
790
|
|
adam@356
|
791 (* begin hide *)
|
|
adam@354
|
792 Hint Constructors exec.
|
|
adam@354
|
793
|
|
adam@354
|
794 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
|
|
adam@354
|
795
|
|
adam@354
|
796 Ltac inverter := repeat match goal with
|
|
adam@354
|
797 | [ H : exec _ _ |- _ ] => inversion H; []; crush
|
|
adam@354
|
798 end.
|
|
adam@354
|
799
|
|
adam@354
|
800 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
|
|
adam@354
|
801 comp_eq (Bnd (Ret a) f) (f a).
|
|
adam@354
|
802 red; crush; inverter; eauto.
|
|
adam@354
|
803 Qed.
|
|
adam@354
|
804
|
|
adam@354
|
805 Theorem cright_identity : forall A (m : comp A),
|
|
adam@354
|
806 comp_eq (Bnd m (@Ret _)) m.
|
|
adam@354
|
807 red; crush; inverter; eauto.
|
|
adam@354
|
808 Qed.
|
|
adam@354
|
809
|
|
adam@354
|
810 Lemma cassociativity1 : 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 (Bnd m f) g
|
|
adam@354
|
813 -> exec (Bnd m (fun x => Bnd (f x) 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 A.
|
|
adam@354
|
819 generalize dependent B0.
|
|
adam@354
|
820 do 2 intro.
|
|
adam@354
|
821 subst.
|
|
adam@354
|
822 crush.
|
|
adam@354
|
823 inversion H; clear H; crush.
|
|
adam@354
|
824 eauto.
|
|
adam@354
|
825 Qed.
|
|
adam@354
|
826
|
|
adam@354
|
827 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
|
|
adam@354
|
828 exec c r
|
|
adam@354
|
829 -> forall m, c = Bnd m (fun x => Bnd (f x) g)
|
|
adam@354
|
830 -> exec (Bnd (Bnd m f) g) r.
|
|
adam@354
|
831 induction 1; crush.
|
|
adam@354
|
832 match goal with
|
|
adam@354
|
833 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
|
|
adam@354
|
834 end.
|
|
adam@354
|
835 move H3 after B.
|
|
adam@354
|
836 generalize dependent B0.
|
|
adam@354
|
837 do 2 intro.
|
|
adam@354
|
838 subst.
|
|
adam@354
|
839 crush.
|
|
adam@354
|
840 inversion H0; clear H0; crush.
|
|
adam@354
|
841 eauto.
|
|
adam@354
|
842 Qed.
|
|
adam@354
|
843
|
|
adam@354
|
844 Hint Resolve cassociativity1 cassociativity2.
|
|
adam@354
|
845
|
|
adam@354
|
846 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
|
|
adam@354
|
847 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
|
|
adam@354
|
848 red; crush; eauto.
|
|
adam@354
|
849 Qed.
|
|
adam@356
|
850 (* end hide *)
|
|
adam@356
|
851
|
|
adam@424
|
852 (** 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
|
853
|
|
adam@356
|
854 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)).
|
|
adam@354
|
855
|
|
adam@354
|
856 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
|
|
adam@354
|
857 if le_lt_dec 2 (length ls)
|
|
adam@354
|
858 then let lss := partition ls in
|
|
adam@356
|
859 ls1 <- mergeSort'' le (fst lss);
|
|
adam@356
|
860 ls2 <- mergeSort'' le (snd lss);
|
|
adam@356
|
861 Ret (merge le ls1 ls2)
|
|
adam@354
|
862 else Ret ls.
|
|
adam@354
|
863
|
|
adam@356
|
864 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *)
|
|
adam@356
|
865
|
|
adam@354
|
866 Definition frob' A (c : comp A) :=
|
|
adam@354
|
867 match c with
|
|
adam@354
|
868 | Ret x => Ret x
|
|
adam@354
|
869 | Bnd _ c' f => Bnd c' f
|
|
adam@354
|
870 end.
|
|
adam@354
|
871
|
|
adam@354
|
872 Lemma exec_frob : forall A (c : comp A) x,
|
|
adam@354
|
873 exec (frob' c) x
|
|
adam@354
|
874 -> exec c x.
|
|
adam@356
|
875 destruct c; crush.
|
|
adam@354
|
876 Qed.
|
|
adam@354
|
877
|
|
adam@356
|
878 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *)
|
|
adam@356
|
879
|
|
adam@354
|
880 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
|
|
adam@354
|
881 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
|
|
adam@354
|
882 repeat (apply exec_frob; simpl; econstructor).
|
|
adam@354
|
883 Qed.
|
|
adam@354
|
884
|
|
adam@356
|
885 (** 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
|
886
|
|
adam@354
|
887 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
|
|
adam@354
|
888
|
|
adam@356
|
889 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way.
|
|
adam@356
|
890 [[
|
|
adam@356
|
891 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3).
|
|
adam@354
|
892 ]]
|
|
adam@354
|
893
|
|
adam@354
|
894 <<
|
|
adam@354
|
895 Error: Universe inconsistency.
|
|
adam@354
|
896 >>
|
|
adam@356
|
897
|
|
adam@356
|
898 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
|
899
|
|
adam@354
|
900
|
|
adam@357
|
901 (** * Comparing the Alternatives *)
|
|
adam@354
|
902
|
|
adam@356
|
903 (** 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
|
904
|
|
adam@356
|
905 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
|
906
|
|
adam@356
|
907 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
|
908
|
|
adam@356
|
909 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
|
910
|
|
adam@356
|
911 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
|
912
|
|
adam@356
|
913 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
|
914
|
|
adam@356
|
915 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
|
916
|
|
adam@354
|
917 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
|
|
adam@354
|
918
|
|
adam@356
|
919 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3).
|
|
adam@356
|
920
|
|
adam@357
|
921 (** 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
|
922
|
|
adam@354
|
923 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
|
|
adam@354
|
924 match ls with
|
|
adam@354
|
925 | nil => Return nil
|
|
adam@354
|
926 | x :: ls' => Bind (f x) (fun x' =>
|
|
adam@354
|
927 Bind (map f ls') (fun ls'' =>
|
|
adam@354
|
928 Return (x' :: ls'')))
|
|
adam@354
|
929 end.
|
|
adam@354
|
930
|
|
adam@355
|
931 (** remove printing exists *)
|
|
adam@356
|
932 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil))
|
|
adam@356
|
933 (2 :: 3 :: 4 :: nil).
|
|
adam@354
|
934 exists 1; reflexivity.
|
|
adam@354
|
935 Qed.
|
|
adam@356
|
936
|
|
adam@356
|
937 (** 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
|
938
|
|
adam@357
|
939 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. *)
|
