Strong Correspondence for HOPLA

We show that the operational semantics for HOPLA is in strong correspondence with its presheaf semantics [2, 1]. The proof is a fairly standard logical relations proof, exploiting the path semantics of the language [3, 4]. Strong correspondence can be proved for full HOPLA by making use of the path semantics to get a logical relations proof off the ground. We’ll use the notation J−K for the path semantics and J−KSet for the presheaf semantics. The proof uses logical predicates AP(p, t), where p : P is a formal path of the path semantics, and t is a closed term of type P. By structural induction on paths, we define: AP→Q(P 7→ q, t) ⇐⇒def ∀u. (AP(P, u) =⇒ AQ(q, t u)) AΣα∈APα(βp, t) ⇐⇒def APβ (p, πβt) A!P(P, t) ⇐⇒def { (JtKSet ∼= ΣdJ!tdK) and (!P : t ! −→ t′ : P =⇒ AP(P, t′)) A μj ~ T .~T (abs p, t) ⇐⇒def ATj [μ~ T .~T/~ T ](p, rep t) (1) Here, the sum index d ranges over derivations of !P : t ! −→ td : P and the logical predicates are extended to sets of paths X ⊆ P by AP(X, t) ⇐⇒def ∀p ∈ X. AP(p, t) . (2) It will be convenient to extend the logical predicates to actions: AP(P, u) A(q, Q : a : P′, P ′) A(P 7→ q, P → Q : u 7→ a : P′, P ′) A(p, Pβ : a : P′, P ′) β ∈ A A(βp,Σα∈APα : βa : P′, P ′) A(P, !P : ! : P, P ) A(p, Tj [μ~ T .~ T/~ T ] : a : P′, P ′) A(abs p, μj ~ T .~ T : abs a : P′, P ′) (3) In a judgement A(p, P : a : P′, P ′) we have that P : a : P′ is an action while p : P and P ′ : !P′ are paths, the latter uniquely determined by p and a; it can be understood intuitively as representing the part of p which has not yet been “realised” by the action a. The following lemma is what makes these judgements useful: