I have just learnt double induction and I want to make sure that my interpretation is correct.
Theorem
Let, $P(m,n)$ be a statement for $m,n\in \mathbb N$.Suppose,
$(1)$ $P(1,n)$ is true for all $n\in \mathbb N$ and $P(m,1)$ is true for all $m\in \mathbb N$.
$(2)$ For $m_0,n_0>1$, $P(m_0-1,n_0)$ and $P(m_0,n_0-1)$ are true $\implies$ $P(m_0,n_0)$ is true.
Then $P(m,n)$ is true for all $m,n\in \mathbb N$.
I want to give a visual interpretation of why this should be true.Consider the following diagram:
I want to conclude that for any $m_0,n_0\in \mathbb N$ the statement $P(m_0,n_0)$ is true.So,let us take an example when $(m_0,n_0)=(3,4)$,we claim that $P(3,4)$ is true.$P(3,4)$ is true if its left one and bottom one are true and they are true if their left and bottom ones are true which ultimately boils down to $(3,1),(2,1)$ and $(1,2),(1,3),(1,4)$ being true as the arrows suggest in the diagram.But these are indeed true because $P(m,1)$ and $P(1,n)$ are true for all $m,n\in \mathbb N$.Similarly for any $(m,n)$ in this array,we can reach $(m_0,n_0)$ by checking if the immediate left one and bottom one are true which ultimately lead to $P(1,n)$ and $P(,1)$ being true for all $m,n\in \mathbb N$.Is this explanation correct?