Zero-knowledge integer comparison
Statement $a &gt; c$
could be proved as $a - c - 1 = b_1^2 + b_2^2 + b_3^2 + b_4^2$.
According to Lagrange theorem,
a 4-tuple of integers exists for any non-negative integer such that it equals to the sum of squares.
This kind of proof was suggested by Helger Lipmaa (On diophantine complexity and statistical zero-knowledge arguments. ASIACRYPT 2003).
Proving statements about integers (not finite field elements) committed with Okamoto-Fujisaki commitment scheme, is doable
with a group of order hidden from Prover.

Scenario described previously looks somewhat like a Zcash JoinSplit (section 1.2 High-level Overview).