For overloading tables on infix operators, we need to break down interval_sets and interval_maps to the specific class templates

choosing Si and Mi as placeholders.

The indices i of Si and Mi represent a property called segmentational fineness or short fineness, which is a type trait on interval containers.

Segmentational fineness represents the fact, that for interval containers holding the same elements, a splitting interval container may contain more segments as a separating container which in turn may contain more segments than a joining one. So for an

operator >

where

x > y   // means that
x is_finer_than y

// we have

finer                                             coarser
split_interval_set                           interval_set
                   > separate_interval_set >  
split_interval_map                           interval_map

This relation is needed to resolve the instantiation of infix operators e.g. T operator + (P, Q) for two interval container types P and Q. If both P and Q are candidates for the result type T, one of them must be chosen by the compiler. We choose the type that is segmentational finer as result type T. This way we do not loose the segment information that is stored in the finer one of the container types P and Q.

// overload tables for
T operator + (T, const P&)
T operator + (const P&, T)

element containers:     interval containers:
+  | e b s m            +  | e  i  b  p  S1 S2 S3 M1 M3     
---+--------            ---+---------------------------
e  |     s              e  |             S1 S2 S3 
b  |       m            i  |             S1 S2 S3 
s  | s   s              b  |                      M1 M3  
m  |   m   m            p  |                      M1 M3
                        S1 | S1 S1       S1 S2 S3 
                        S2 | S2 S2       S2 S2 S3 
                        S3 | S3 S3       S3 S3 S3 
                        M1 |       M1 M1          M1 M3
                        M3 |       M3 M3          M3 M3

So looking up a type combination for e.g. T operator + (interval_map, split_interval_map) which is equivalent to T operator + (M1, M3) we find for row type M1 and column type M3 that M3 will be assigned as result type, because M3 is finer than M1. So this type combination will result in choosing this

split_interval_map operator + (const interval_map&, split_interval_map)

implementation by the compiler.