Type inference with simple subtypes

ion clause, we use A -{x: σ} to denote the set difference, i.e., the type assignment defined by removing x: σ from A. G(x) = { s⊆t }, { x:s} ⊃ x:t G(MN) = let C1, A1 ⊃ M:σ = G(M) C2, A2 ⊃ N:τ = G(N), with type variables renamed to be disjoint from those in G(M) S = UNIFY({ α=β | x:α∈A1 and x:β∈A2} ∪ { σ=τ→t}) where t is a fresh type variable in SC1∪SC2∪{St⊆u}, SA1∪SA2 ⊃ MN:u where u is a fresh type variable G(λx.M) = let C, A ⊃ M:τ = G(M) in if x:σ∈A for someσ then C∪{ σ→τ⊆u}, (A -{x: σ}) ⊃ λx.M:u else C∪{s→τ⊆u}, A ⊃ λx.M:u, where s, u are fresh type variables The algorithm could concei vably fail in the application case if the call to UNIFY fails. Howev er, we will see that this does not happen. It is not to hard to pro ve that if G(M) succeeds, then it produces a pro vable typing for M. THEOREM 13. If G(M) = C,A ⊃ M:σ, then− C,A⊃ M:σ. It follows, by Lemma 11, that e v ry instance of G(M) is a pro vable typing for M. Conversely, every provable typing for M is an instance of G(M). THEOREM 14. Suppose− C,A ⊃ M:σ. Then G(M) succeeds and produces a typing with C,A⊃ M:σ as an instance. Both theorems are pro ved below. From Lemma 2, we kno w that every term has a pro vable CC typing. Therefore, Theorem 14 implies G(M) al ways succeeds.