Optimal Inference of Fields in Row- Polymorphic Records (Proof Appendix)

This report contains the derivation details of the paper of the same name [7]. A. Additional Preliminaries This section contains proves and more on the derivation process. It is an extension of [6]. Without loss of generality we assume that all letand λ-bound variables x ∈ X are pair-wise different. Let X ⊆ X denote the set of variables that are λ-bound. Let x ≺X y iff x is in scope of y. We write environments as [x1 7→ t1, . . . xn 7→ tn] where xi ≺X xi+1. Define args ⊆ P(X) such that m ∈ args contains a function f ∈ X \ X and m \ {f} are the λ-bound arguments of f . For instance, args = {{f, s}, {s′}, {v}} for the introductory example. We give a formal definition of how to compute args from a program e ∈ E. Let ξ ∈ X \ X be a function symbol that represents the owner of outermost λ-variables. Each function symbol and its owned λ-bound variables form a so-called monomorphic group, a notion we use to specify what variables to instantiate when accessing letand λ-bound variables. DEFINITION 1. A variable set V ⊆ X forms a monomorphic group in program P iff V ∈ args(P ) where args(P ) := {V ⊆ X | ∃f . {f} = (V \ X) ∧ ∀x ∈ (V ∩ X) . 〈f, x〉 ∈ own(ξ, P )}. As informally described above, we let args denote the set of all monomorphic groups of the current program. Observe that args is a partitioning of λ-bound variables: PROPOSITION 1. Let {V1, . . . Vn} = args(P ) and X the be the set of variables in P . Then Vi ∩ Vj = ∅ for all i 6= j and X ⊆ ⋃n i=1 Vi. Proof. Let ex denote the expression ex ≡ λx . e′ that binds x. Show i 6= j implies Vi ∩ Vj = ∅. By construction, exactly one function symbol f ∈ X \ X exists in Vi. Let x ∈ Vi ∩ Vj ∩ X and thus 〈f, x〉 ∈ own(ξ, P ). Then there exists a function symbol g ∈ Vj with 〈g, x〉 ∈ own(ξ, P ). Since all variable bindings feature different variables, the same expression ex must have been visited by own, once with own(f, ex) and once with own(g, ex) which is a contradiction since own visits every sub-expression of P exactly once. Now show X ⊆ ⋃n i=1 Vi. By definition of own, for each x ∈ X, ex is visited exactly once, returning 〈f, x〉. Thus there exists Vi with f, x ∈ Vi as required. Intuitively, each monomorphic group Vi contains one let-bound function f and its arguments Vi\{f}. Since args(P ) partitions X, each λ-bound variable is argument to only one function. B. Concrete Semantics In this section we present the concrete semantic of the language in Fig. 1 in [7] by closely following a definition by Milner [5] but adding the operations on records. Let S⊥ := S ∪ {⊥S} where ⊥S / ∈ S denotes an undefined value or a non-terminating evaluation. The denotational standard semantics evaluates e ∈ E to a program value U⊥ where U is a sum of constructors terms A, records R, functions F and the wrong value ω, that denotes a type error. A record r ∈ R is a partial map from names to values. They are constructed from the empty map [] and extended using r[N 7→ v]. Define values U as follows: