Re: Dick Lipton & Ken Regan • (1) • (2)
Let’s go back to the “key lemma” from (2) and try it out on a simple example, just to get a sense of what the terms mean.
Lemma. Let be a Boolean function and let be fixed. Then the Boolean inputs can be partitioned into six sets:
These sets have the following properties:
 The variable is equal to on
 The variable is equal to on
 The union is equal to
 The function is always on
 The function is always on
 Finally and and
Example 1

(1) 
Consider the boolean function pictured in Figure 1 and fix on the variable
The lemma says that the boolean inputs can be partitioned into six sets:
Let’s identify those six sets in the present example.
Back in a flash … in the meantime, exercise for the reader …
Later that day …
I had trouble with the term “Boolean input” in (2). Sometimes people use it to mean one of the input wires to a logic gate, that is, one of the variables Other times people use it to mean one of the coordinate elements It’s always possible that I’m reading things wrong but it looks like the first sense is used to define the “influence” and the related set while the second sense is used to define the six sets of the Lemma. At any rate, I will go with those two senses for now.
On that reading, the six sets named in the Lemma are shown in Figure 2.

(2) 
In other words:
Resources. A few pages on differential logic, which may or may not be useful here.