Recursive Path Orderings Can Also Be Incremental

Termination of Abstract Reduction Systems

“…As Sf s, we have Sf ∈ SN, so t ∈ SN ⊆ gbars {u | u s} SN. Similar arguments hold where (t, s) ∈ σ ⊆ ρ by rule (10), and where (t, s) ∈ τ ⊆ ρ via rule (12).…”

“…This was particularly valuable for §5, where our initial paper proofs turned out to be wrong, as the choice of 2 was particularly difficult to get right. Further, the possible instantiation of a variable by a term headed by a symbol in either F 0 or F 1 complicated matters: see the discussion following [12,Proposition 1]. Also in §4.1 the Isabelle proof confirmed the validity of the rather tricky argument for the soundness of the mutually recursive definitions of SN , ρ and τ .…”

“…We define the relations σ and τ (as inductively defined sets) by rules, as shown below for the combinator S. Then let ρ = ctxt σ ∪ τ . Note that the extra rules (11), (12) and (13) are added only because the proof uses them: for clearly, if ρ ⊇ ρ , and ρ terminates, then so does ρ . These rules are motivated by the relation • of [14, §5]:…”

“…Let (t, s) ∈ ρ and assume ∀s s. s ∈ SN. If (t, s) ∈ ρ via rule (11) (where s = Sf g), we have g ∈ SN, so (t, Sf ) ∈ ρ by rule (12). As Sf s, we have Sf ∈ SN, so t ∈ SN ⊆ gbars {u | u s} SN.…”

“…2 Our rules (11), (12) and (13) were suggested by the definition of • given just below Remark 13 in [14]. As in [14], therefore, there is a resemblance between our proof and the classic reducibility argument: we have, for example, that for S f g to be in SN, it is necessary that for all x ∈ SN, S f g x ∈ SN, which resembles the condition for reducibility in [13, §6.1].…”