FabULous Interoperability for ML and a Linear Language

If dom(γ 1) = Γ1, dom(γ 2) = Γ2, and Γ = Γ1 Γ2, then γ 1 γ 2 is de ned when for any variable x ∈ Γ1 ∩ Γ2, γ 1 (x ) = γ 2 (x ) and is de ned as γ (x ) = γ 1 (x ) if x ∈ Γ1 and γ (x ) = γ 2 (x ) if x ∈ Γ2. (R,σ1,σ2) is well formed if σ1,σ2 are closed types and R ∈ Rel[σ1,σ2]. The relationsV JρKj , E JρKj , . . . below are only de ned for closed relation types ρ, ρ. Substitution is extended to ρ, ρ by considering (R,σ1,σ2) to be closed. Every ρ has two associated types, the types of terms that it relates, which we denote (ρ)1, (ρ)2. It is de ned as follows: