I have two questions, about structural induction and the predecessor relation.
Why can't a relation be well-founded if it has an infinite descending chain, provided that it has a maximum element? How does that differ from an ascending chain with a minimum element?
Also, why is the predecessor relation not well-founded? $ \{(m + 1, m) | m ∈ N\}\ on\ N$. Why is this an infinite descending chain?
It’s simply the definition of well-foundedness: a binary relation $R$ on a set $X$ is well-founded if and only if each non-empty $A\subseteq X$ has an $R$-minimal element:
$$\forall A\subseteq X\Big(A\ne\varnothing\to\exists\,a\in A\,\forall\,x\in A\setminus\{a\}\big(\langle x,a\rangle\notin R\big)\Big)\;.$$
An infinite descending chain clearly has no minimal element; whether it has a maximal element is irrelevant.
If $R=\{\langle m+1,m\rangle:m\in\Bbb N\}$, then
$$\ldots\mathrel{R}4\mathrel{R}3\mathrel{R}2\mathrel{R}1\mathrel{R}0$$
is clearly an infinite descending chain: for each $m\in\Bbb N$ we have $m+1\mathrel{R}m$, meaning that $m+1$ is $R$-smaller than $m$.