Nested Hoare Triples and Frame Rules for Higher-order Store

A.4 RecursionTheorem 35 (Mutually recursive predicates). Suppose that for each i ∈I, F i : Pred I → Pred is a contractive function. Then there exists a uniquep = (p i ) i∈I ∈ Pred I such that F i (p) = p i for all i ∈ I.Proof. Recall that Pred I is an ultrametric space, with distance function d(p, q) =sup i d Pred (p i , q i ) for p = (p i ) i∈I and q = (q i ) i∈I in Pred I . By assumption, eachF i is contractive. In particular, since all distances in Pred are of the form 2 −n ,we have d Pred (F i (p), F i (q)) ≤ 1 2 · d(p, q). Then F : PredI → Pred I given byF (p) = (F i (p)) i∈I is a contractive endofunction on this ultrametric space, whichby the Banach fixed point theorem has a unique fixed point p, satisfying p i =(F (p)) i = F i (p). ⊓⊔