Variants in the Infinitary Unification Wonderland

Inductive Reasoning with Equality Predicates, Contextual Rewriting and Variant-Based Simplification

“…E a constructor ground E 1 -unifier of Γ. But, under the assumptions for B 1 , by the results in [54,70,56] one can compute a complete, finite 8 set Unif Ω E 1 pΓq of constructor E 1 -unifiers of Γ, so that any constructor ground E 1 -unifier of Γ, and therefore up to E-equivalence any ground E-unifier of Γ, is an instance of a unifier in Unif Ω E 1 pΓq. Note, furthermore, that if B 1 does not contain associativity without commutativity axioms, pΩ,B Ω q is an OS-compact theory [54] (resp.…”

“…Note, furthermore, that if B 1 does not contain associativity without commutativity axioms, pΩ,B Ω q is an OS-compact theory [54] (resp. a weakly OS-compact theory if B 1 does contain associativity without commutativity axioms and pΩ,B Ω q satisfies some additional assumptions [56]). Therefore, again by [54] (resp.…”

“…Therefore, again by [54] (resp. [56]) satisfiability (and therefore validity) of any QF Σ 1 -formula in T E 1 , and by T E | Σ 1 T E 1 also in T E , is decidable (resp. has a partial decidability algorithm under some assumptions as explained in [56]).…”

“…[56]) satisfiability (and therefore validity) of any QF Σ 1 -formula in T E 1 , and by T E | Σ 1 T E 1 also in T E , is decidable (resp. has a partial decidability algorithm under some assumptions as explained in [56]). In the 6 An E 1 -variant (or E 1 , B 1 -variant) of a Σ 1 -term t is a pair pv, θq, where θ is a substitution in canonical form, i.e., θ " θ!…”

“…E 1 is FVP iff any such t has a finite set of variants tpu 1 ,α 1 q,...,pu n ,α n qu which are "most general possible" in the precise sense that for any variant pv,θq of t there exist i, 1 ď i ď n, and substitution γ such that: (i) v " B 1 u i γ, and (ii) θ " B 1 α i γ. 7 The notion of an FVP theory has been recently extended in [56] to allow also axioms of associativity without commutativity in B 1 . 8 When B 1 contains associativity without commutativity axioms, the number of unifiers in…”