Sideward Merge is Merge M_{S,S'} where at least one of S, S' is an accessible term of a
component of the workspace, other than in Internal Merge. Marcolli, Chomsky and Berwick
distinguish three cases (their §1.4.1): in 2(b) S is a component and S' an accessible term
of another component; in 3(a) both are accessible terms of one component; in 3(b) they are
accessible terms of two different components. Lemmas 1.4.4 and 1.4.5 show that the Merge
operator of Definition 1.3.4 realizes all three, so Merge alone does not rule them out.
Each case evaluates the Merge formula mergeOpG_apply_of' on the workspace. On a single tree the
matching cuts are the cuts with crown {S, S'}, since the tree itself is a one-element crown. On
two trees the hypotheses force each matching forest cut to take S from the first tree and S'
from the second, so the sum factors through the two one-tree sums. A spectator workspace passes
through unchanged (mergeOpG_of'_mul).
Main results
Implementation notes
The book states each lemma for the term that extracts the matching objects. With several
occurrences Merge sums over them, as (1.3.11) does, so the results are sums over the cuts with
the matching crown, and a workspace without one is sent to 0.
The lemmas hold over enumerations admitting Merge (IsMergeCuts), which the deletion cuts of the
book's mergeOp are not; on objects without unary vertices
mergeOp_eq_rename_contractUnary_comp carries them to mergeOp, contracting the trunks.
References