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Let $P(zss)$ mean concat (map f zss) = f (concat zss). We use structural induction on lazy lists, including the bottom value and allowing arbitrary, possibly partial, inner lists.

For [], the left side is [] and the right side is f [] = []. For the bottom value $\perp$, both map f and concat are strict functions in the outer spine, and the supplied strictness of $f$ makes the right side $f\perp =\perp$ too.

For the cons case, assume $P(zss)$ and write the new head as xs. The defining equations and the induction hypothesis giveThe third equality is exactly the supplied list concatenation preservation law. No inspection of xs was required, so the argument includes partial inner lists and undefined elements.

This proves the identity for every finite outer spine ending in either [] or $\perp$. In the usual pure Haskell semantics, these functions are Scott continuous maps. An infinite outer list is obtained as a limit by finite approximation of a lazy list; both composite functions preserve that limit. Equality on all its finite approximations therefore gives equality on the infinite outer list. Equivalently, $P$ is an admissible predicate on lazy lists. ConsequentlyThe continuity assumption is the usual semantic assumption for definable functions here; an extension from finite to infinite inputs cannot be justified by ordinary finite induction alone.

For finite, defined lists, structural induction requires $P([])$ and $P(xs)\Rightarrow P(x:xs)$ for every head $x$ and tail $xs$. This proves $P$ for every finite list.

An infinite list cannot be reached by finitely many constructor steps. For structural induction on lazy lists, also check $P(\perp )$, where $\perp$ is the bottom value, and require an admissible predicate on lazy lists: if increasingly defined finite approximations all satisfy $P$, their limit does too. Together with the empty-list and constructor cases this covers finite, partial and infinite lists. Merely asserting the two finite-list cases is insufficient; “the list is finite” itself satisfies those cases but fails on infinite lists.

For an admissible predicate on lazy lists $P$, prove $P(\perp )$ and $P([])$, and prove $P(xs)\Rightarrow P(x:xs)$ for every possibly partial head $x$. Constructor induction then establishes $P$ on every finite partial approximation, and admissibility extends it to limits, including infinite lists. Without the bottom value case and the limit condition, ordinary finite structural induction does not justify a claim about infinite data.^[ref]