A simple theorem of glance functions

THIS IS NAIVELY WRONG BUT MAY CONTAIN AN IDEA

A little Theorem about glance functions [described in my Hypothesis Testing... article].

Let h_a be the glance function of a simple interval of length a.

If H is the glance function for a collections of intervals separated by gaps, and a new interval of length a is added after a gap of length b, then the glance function of the new collection of intervals is

(*)                                      H -> H – H^b  + H^a + h_b + h_a  - h_a+b

Where the superscript on a function H indicates a term-wise shift of the independent variable by adding that amount, so  H^a(x)= H(a+x).

Corollary: the value of the glance function after the last step equals the number of intervals being glanced.

Proof: It is true for one interval, since h_a equals 1 at it's last step. If true for H, with N final steps, then we note the terms in the above (*) has final step values: +N , -N, +N,  +1, +1, -1. This totals to N+1. Hence the corollary is true by induction.

Update: The correct formula is

(**)                     H à H – H^b  + H^a+b + (h_a + h_b – h_a+b)

The induction argument is the same. We can name the first part the "shifted H" part. The rest is the "self-contained" info about the added segment.

Note: this encourages thinking about re-constructing the original segments, starting from their glance function. Because we know how many segments are involved and the maximum length from first to last end points of the segments.