Theorems used by proofs generated by TORPA H. Zantema Department of Computer Science, TU Eindhoven P.O. Box 513, 5600 MB, Eindhoven, The Netherlands e-mail h.zantema@tue.nl This document lists all theorems used by proofs generated by TORPA version 1.6, prepared for the Termination Competition 2006. They are identified by letters in square brackets; the output of TORPA 1.6 uses the same identification. [A] Monotone Algebras Theorem. Let R, S , R ′ and S ′ be SRSs satisfying • R ∪ S = R ′ ∪ S ′ and R ∩ S = R ′ ∩ S ′ = ∅, and • SN(R ′ /S ′ ) and SN((R ∩ S ′ )/(S ∩ S ′ )). Then SN(R/S ). Proof: [3], Theorem 1. ✷ Theorem. Let A be a non-empty set and let > be a well-founded order on A. Let f a : A → A be strictly monotone for every a ∈ Σ, i.e., f a (x) >f a (y ) for every x, y ∈ A satisfying x>y . Let R and S be disjoint SRSs over Σ such that f ℓ (x) >f r (x) for all x ∈ A and ℓ → r ∈ R, and f ℓ (x) ≥ f r (x) for all x ∈ A and ℓ → r ∈ S . Then SN(R/S ). Proof: [3], Theorem 4. ✷ These theorems are applied as follows: if SN(R/S ) has to be proved then an interpretation is chosen for which f ℓ (x) ≥ f r (x) for all x ∈ A and ℓ → r ∈ R ∪ S . Then R ′ is defined to consist of the rules ℓ → r of R ∪ S satisfying f ℓ (x) >f r (x) for all x ∈ A, and S ′ =(R ∪ S ) \ R ′ . Then SN(R ′ /S ′ ) holds by the second theorem, and by the first theorem the remaining proof obligation is SN((R ∩ S ′ )/(S ∩ S ′ )). 1