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A finite list prefix followed by bottom records only the constructors known so far. The chain $\perp ,x_1:\perp ,x_1:x_2:\perp ,\dots$ approximates an infinite lazy list. A Scott continuous map is determined on the limit by its values on these approximations. This justifies extending an identity proved on finite partial lists to definable operations on infinite lists; equality merely on finite terminating lists does not provide the same premise.

If $f$ is a strict function and $f(g(a,x))=h(f(a),x)$ for every $a,x$, then f (foldl g a xs) = foldl h (f a) xs. Prove the equation simultaneously for all accumulators by structural induction on the list. The empty case is immediate; at a cons, apply the hypothesis with accumulator g a x and then rewrite its image using the assumed equation. At the bottom value, strictness supplies the required equality. For pure definable functions, equality on finite partial lists extends to infinite lazy lists by finite approximations. For finite terminating spines alone, strictness is unnecessary.

No universally correct sequential Haskell implementation has that guarantee for partial lists. Compare undefined paired with [v], and [v] paired with undefined, where the element type is arbitrary. A generic program cannot fabricate v; it must eventually inspect an input spine to obtain it. Whichever input it first demands can be chosen to be the undefined one, blocking access to the available element in the other input.

In particular, the implementation in part b can emit a defined prefix and then block when alternation reaches an undefined tail. Lazy evaluation postpones demands; it cannot recognize that a demanded computation will never finish. The result is the partial-list interleaving obstruction, not an objection to fairness under the proper-input hypothesis. Additional concurrent primitives would change the computational model.^[ref]

The empty-list case uses $f(a)=b$. At a cons, assuming the equality for the tail, the calculation isThe middle equality is the supplied compatibility condition, instantiated with the folded tail; the last expression is the right fold on the whole list. Structural induction therefore proves fusion for finite total spines.

The strictness condition accounts for a bottom-valued input spine: the inner list fold returns bottom, and $f(\perp )=\perp$ makes the two sides agree. The same induction covers finite partial lists. For pure definable continuous functions, an infinite list is the limit of its finite partial approximations, so equality extends to it. These are the three distinct obligations in the right-fold fusion law; strictness is not needed if the domain is restricted to finite terminating spines.

For finite lists, use structural induction: establish P []; then, for arbitrary x and xs, assume P xs and prove P (x:xs). The induction hypothesis must quantify over any other parameters that need to change in the step.

For all Haskell list values, finite terminating spines are insufficient: also establish P undefined, and allow arbitrary, possibly undefined, heads in the cons step. This proves the equation for finite partial lists ending in the bottom value. An infinite lazy list is the limit of these finite approximations. Equations between pure definable computable functions are preserved by those limits, so the equation extends to infinite inputs. This last argument requires that admissibility property; an arbitrary logical property of lists need not have it. The distinction between [] and the bottom value is essential.^[ref]

Use the same step from part (a):The right fold obeys groupFold [] = [] and groupFold (x:xs) = prependRun x (groupFold xs), exactly the recursive equations already proved. Both implementations give the bottom value for an entirely undefined input spine. Their equal behaviour on finite partial lists, together with definable continuity, establishes equality on infinite lazy lists as well. The helper's irrefutable binding is what lets the first run be produced without demanding that all subsequent runs already exist.

A proper lazy list has a spine exposing either finitely many cons constructors followed by [], or infinitely many cons constructors, without an undefined tail at a finite position. This is a condition on the spine: an element may itself be the bottom value without hiding the list's constructors. A partial list with a finite visible prefix followed by $\perp$ fails this condition. The distinction supports productivity proofs that inspect constructors without evaluating the stored elements.

If $f$ is strict, $f(a)=b$, and $f(g(x,y))=h(x,f(y))$, then $f\circ \mathrm{foldr}(g,a)=\mathrm{foldr}(h,b)$. The empty case uses the base image and the cons case uses compatibility followed by structural induction. Strictness handles a bottom-valued spine. For pure definable continuous functions, equality on finite partial lists extends to infinite lists by finite approximation. For finite total inputs alone, induction does not require the strictness hypothesis.