  Discrete Fenchel duality is one of the central issues in discrete convex
analysis. The Fenchel-type min-max theorem for a pair of integer-valued
M-natural-convex functions generalizes the min-max formulas for polymatroid
intersection and valuated matroid intersection. In this paper we establish a
Fenchel-type min-max formula for a pair of integer-valued integrally convex and
separable convex functions. Integrally convex functions constitute a
fundamental function class in discrete convex analysis, including both
M-natural-convex functions and L-natural-convex functions, whereas separable
convex functions are characterized as those functions which are both
M-natural-convex and L-natural-convex. The theorem is proved by revealing a
kind of box integrality of subgradients of an integer-valued integrally convex
function. The proof is based on the Fourier-Motzkin elimination.
