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POPartial functions versus under-specified total functions
 

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  1. POPartial functions versus under-specified total functions
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    <p>Suppose I have a set <code>A ⊆ nat</code>. I want to model in Isabelle a function <code>f : A ⇒ Y</code>. I could use either:</p> <ol> <li>a partial function, i.e. one of type <code>nat ⇒ Y option</code>, or </li> <li>a total function, i.e. one of type <code>nat ⇒ Y</code> that is unspecified for inputs not in <code>A</code>.</li> </ol> <p>I wonder which is the 'better' option. I see a couple of factors:</p> <ul> <li><p>The "partial function" approach is better because it is easier to compare partial functions for equality. That is, if I want to see if <code>f</code> is equal to another function, <code>g : A ⇒ Y</code>, then I just say <code>f = g</code>. To compare under-specified total functions <code>f</code> and <code>g</code>, I would have to say <code>∀x ∈ A. f x = g x</code>. </p></li> <li><p>The "under-specified total function" approach is better because I don't have to faff with the constructing/deconstructing <code>option</code> types all the time. For instance, if <code>f</code> is an under-specified total function, and <code>x ∈ A</code>, then I can just say <code>f x</code>, but if <code>f</code> is a partial function I would have to say <code>(the ∘ f) x</code>. For another instance, it's trickier to do function composition on partial functions than on total functions.</p></li> </ul> <p>For a concrete instance relevant to this question, consider the following attempt at formalising simple graphs.</p> <pre><code>type_synonym node = nat record 'a graph = V :: "node set" E :: "(node × node) set" label :: "node ⇒ 'a" </code></pre> <p>A graph comprises a set of nodes, an edge relation between them, and a <code>label</code> for each node. We only care about the label of nodes that are in <code>V</code>. So, should <code>label</code> be a partial function <code>node ⇒ 'a option</code> with <code>dom label = V</code>, or should it just be a total function that is unspecified outside of <code>V</code>?</p>
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