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  1. PO
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    <p>You probably want to define the following function (even if you annotate it properly you [le_S m n x] does not have the type you want) :</p> <pre><code> Fixpoint lesseq (m n : nat) : option (m &lt;= n) := match n with | 0 =&gt; match m with | 0 =&gt; Some (le_n 0) | S m0 =&gt; None end | S p =&gt; match lesseq m p with | Some l =&gt; Some (le_S m p l) | None =&gt; None end end. </code></pre> <p>But as you've noticed, the typechecker is not clever enough to guess the new context when you destruct a variable appearing in the type of the result. You have to annotate the match in the following way : </p> <pre><code> Fixpoint lesseq (m n : nat) : option (m &lt;= n) := match n return (option (m &lt;= n)) with | 0 =&gt; match m return (option (m &lt;= 0)) with | 0 =&gt; Some (le_n 0) | S m0 =&gt; None end | S p =&gt; match lesseq m p with | Some l =&gt; Some (le_S m p l) | None =&gt; None end end. </code></pre> <p>See the reference manual if you really want to understand how pattern matching works with dependent types. If you don't feel brave enough for that, you'd rather use tactics mechanisms to build your proof (the "refine" tactic is a great tool for that). </p> <pre><code> Definition lesseq m n : option (m &lt;= n). refine (fix lesseq (m : nat) (n : nat) {struct n} := _). destruct n. destruct m. apply Some; apply le_n. apply None. destruct (lesseq m n). apply Some. apply le_S. assumption. apply None. Defined. </code></pre> <p>By the way, I don't think your function will be really helpfull (even if it is a good idea), because you will need to prove the following lemma : Lemma lesseq_complete: forall m n, lesseq m n &lt;> None -> m > n. </p> <p>This is why people use Coq.Arith.Compare_dec. Have fun. </p>
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    1. COBtw, if you follow this approach I greatly advice using the Program feature. It allows you to write your program in the style of the 1st listing, but leave any number of "holes" (_), which you can then "fill" using tactics (style of the second listing). Hence it allows you to nicely separate programming & proving.
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    2. COThank you both for the help. I have much to learn, didn't realise you could use tactics for defining functions (although it makes sense). I now agree that a sum type (a <= b \/ a > b) would be great, because I managed to implement the wrong lesseq by returning None when I want Some; this wouldn't be possible with a more specific type. I'll definitely take another look through the standard library, too.
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