Do the ranges overlap?

Given two ranges:

$R_{1} = {n | b_{1} \leq n < b_{1} + s_{1}}$

$R_{2} = {n | b_{2} \leq n < b_{2} + s_{2}}$

determine whether they overlap.

The ranges are specified using two values: an inclusive beginning b and a size s.

Solution

Tuple comparison is defined canonically (first differing element decides):

$(b_{1}, s_{1}) > (b_{2}, s_{2})$ iff $(b_{1} > b_{2}) \lor (b_{1} = b_{2} \land s_{1} > s_{2})$

There is an exhaustive list of cases how two ranges may overlap:

0	1	2	3	4	5	6	7	8	9	name	blue		orange		first		second		left comparison value	right comparison value	comparison result	overlap observed	comparison matches observation
0	1	2	3	4	5	6	7	8	9	name	beginning	size	beginning	size	beginning	size	beginning	size	left comparison value	right comparison value	comparison result	overlap observed	comparison matches observation
										(base range)
										after	3	4	0	2	0	2	3	4	2	3	⊥	⊥	⊤
										start touching	3	4	0	3	0	3	3	4	3	3	⊥	⊥	⊤
										start inside	3	4	0	4	0	4	3	4	4	3	⊤	⊤	⊤
										inside start touching	3	4	3	7	3	4	3	7	7	3	⊤	⊤	⊤
										enclosing start touching	3	4	3	3	3	3	3	4	6	3	⊤	⊤	⊤
										enclosing	3	4	4	2	3	4	4	2	7	4	⊤	⊤	⊤
										enclosing end touching	3	4	4	3	3	4	4	3	7	4	⊤	⊤	⊤
										exact match	3	4	3	4	3	4	3	4	7	3	⊤	⊤	⊤
										inside	3	4	0	10	0	10	3	4	10	3	⊤	⊤	⊤
										inside end touching	3	4	0	7	0	7	3	4	7	3	⊤	⊤	⊤
										end inside	3	4	6	4	3	4	6	4	7	6	⊤	⊤	⊤
										end touching	3	4	7	3	3	4	7	3	7	7	⊥	⊥	⊤
										before	3	4	8	2	3	4	8	2	7	8	⊥	⊥	⊤