How to use complex patterns in functions?

问题

Here is a sample function:

fun divide :: "enat option ⇒ enat option ⇒ real option" where
  "divide (Some ∞) _ = None"
| "divide _ (Some ∞) = None"
| "divide _ (Some 0) = None"
| "divide (Some a) (Some b) = Some (a / b)"
| "divide _ _ = None"

Isabelle HOL shows me the following error:

Malformed definition:
Non-constructor pattern not allowed in sequential mode.
⋀uw_. divide uw_ (Some 0) = None

Why pattern-matching works fine for Some ∞ and doesn't work for Some 0? ∞ is a constant for class infinity and 0 is a constant for class zero. What is the difference between these constants?

回答1:

Pattern matching with fun only works for constructors, which are typically generated using datatype and codatatype commands. (In fact, it suffices if they are registered as free constructors using free_constructors.) The extended naturals enat as defined in ~~/src/HOL/Library/Extended_Nat have two such constructors registered: ∞ and enat :: nat ⇒ enat. So 0 is not a constructor of enat, but of the ordinary naturals nat. So if you write

| "divide _ (Some (enat 0)) = None"

instead, it will work because there are only registered constructors in the patterns.

Conversely, if your theory imports Coinductive_Nat from the APF entry Coinductive, then enat is registered to have the constructors 0 and eSuc, i.e., as if it were a codatatype. Then you can pattern-match on 0, but you can no longer pattern match on ∞.

来源：https://stackoverflow.com/questions/45932256/how-to-use-complex-patterns-in-functions