partial : exn -> ('a -> 'b option) -> 'a -> 'b
Converts a total function to a partial function.
In ML, there are two main ways for a function to signal that it has been called on an element outside of its intended domain of application: exceptions and options. The function partial maps a function returning an element in an option type to one that may raise an exception. Thus, if f x returns NONE, then partial e f x results in the exception e being raised. If f x returns SOME y, then partial e f x returns y.

The function partial has an inverse total. Generally speaking, (partial err o total) f equals f, provided that err is the only exception that f raises. Similarly, (total o partial err) f is equal to f.

When application of the second argument to the third argument returns NONE.
- Int.fromString "foo";
> val it = NONE : int option

- partial (Fail "not convertable") Int.fromString "foo";
! Uncaught exception:
! Fail  "not convertable"

- (total o partial (Fail "not convertable")) Int.fromString "foo";
> val it = NONE : int option

HOL  Kananaskis-14