Proving that $\{ a^i b^j c^{\max(i,j)} \}$ is not context-free

Prove that $$L$$ is not a Context-free language, where $$L = \{ a^{i} b^{j}c^{h}\mid i,j,h\in \mathbb{N} \wedge h = \max(i,j)\}.$$

I have an idea: It can be divided into two situations:

1. When $$i < j$$, $$w = a^{i} b^{j} c^{i}$$

2. When $$i > j$$, $$w = a^{i} b^{j} c^{j}$$

Then with the help of the pumping lemma，but I will only use special examples to prove it. I always feel not rigorous. How should I write it more rigorously?

My writing is:

When $$i < j$$, $$w = a^{i}b^{j} c^{i}$$, $$i = 4$$, $$j = 3$$. $$w = aaaa bbb cccc$$, then use $$uvxyz$$ to prove step by step.

• How should I write it more rigorously? Try to emulate known examples: ones that you've seen in class, ones that you've seen in textbooks, or ones that you've seen on Computer Science. Jul 13 '21 at 9:48
• Just don't use specific values. For example, my process is to let i=4 and j=3. But I don’t think this is rigorous. Jul 13 '21 at 10:07
• providing a counter-example to a statement is rigorous Jul 13 '21 at 19:43