Context-free grammar for $ \{a^lb^n c^m |l, n, m ∈ \mathcal{N}^+, l \geq \min(n,m)\}$
Let us start by noticing that $X \to a^+$, $Y \to b^+$, $Z \to c^+$. Furthermore, $A \to a^n X b^n$ and $B \to a^m Y c^m$. Therefore $S$ generates words of the following two forms:
$$
a^na^+b^nbc^+ \\
a^+a^mb^+c^mc
$$
You take it from here.