Theorem 9.4

Theorem 9.4 (Opaque-element staging). Let a strict purely functional sequence operation:

1. use only finite constructor tests and a worst-case bounded number of fixed-arity constructor, projection, and pointer steps;

2. treat elements of its base type as opaque atoms, with no equality, hashing, pattern match, identity test, or traversal through a base element, while permitting inspection of the representation’s own pairs, buffers, and recursive child structures; and

3. return a fixed finite tuple of roots and removed elements.

For fixed old roots and symbolic new elements, its control branch and fresh structural allocation graph are determined independently of the symbolic values. If one symbol denotes a fresh owner that stores a resulting sequence root, the owner and structural graph form one bounded simultaneous batch and compile by Theorem 9.2.