Polarized Subtyping for Sized Types

ion. LEQ-λ Γ, X :pκ ` F ≤ F ′ : κ′ Γ ` λXF ≤ λXF ′ : pκ→ κ′ Application. There are two kinds of congruence rules for application: one kind states that if functions F and F ′ are in the subtyping relation, so are their values F G and F ′G at a certain argument G. LEQ-FUN Γ ` F ≤ F ′ : pκ→ κ′ p−1Γ ` G : κ Γ ` F G ≤ F ′G : κ′ The other kind of rules concern the opposite case: If F is a function and two arguments G and G′ are in a subtyping relation, so are the values F G and F G′ of the function at these arguments. However, such a relation can only exist if F is covariant or contravariant. LEQ-ARG+ Γ ` F : +κ→ κ′ Γ ` G ≤ G′ : κ Γ ` F G ≤ F G′ : κ′ LEQ-ARG− Γ ` F : −κ→ κ ′ −Γ ` G′ ≤ G : κ Γ ` F G ≤ F G′ : κ′ Note that we only included a subsumption rule for equality (EQ-SUB), not for subtyping. Subsumption for subtyping is admissible, as item 2 of the following lemma states: Lemma 2.7 (Admissible rules for equality and subtyping I). LetR ∈ {=,≤}. 1 Validity: If D :: Γ ` F R F ′ : κ then Γ ` F : κ and Γ ` F ′ : κ. 2 Weakening: If D :: Γ ` F R F ′ : κ and both Γ′ ≤ Γ and κ ≤ κ′, then Γ′ ` F R F ′ : κ′. 3 Strengthening: If D :: Γ, X :pκ ` F R F ′ : κ′ and X 6∈ FV(F, F ′), then Γ ` F R F ′ : κ′. Proof. Each by induction on D. Lemma 2.8 (Admissible rules for equality and subtyping II). See Fig. 2. 3. Algorithmic Polarized Subtyping In this section, we present an algorithm for deciding whether two well-kinded constructors are equal or related by subtyping. The algorithm is an adaption of Coquand’s βη-equality test (Coquand, 1991) to the needs of subtyping and polarities. The idea is to first weak-head normalize Polarized Subtyping 9 EQ-REFL Γ ` F : κ Γ ` F = F : κ LEQ-APP◦ Γ ` F ≤ F ′ : ◦κ→ κ′ ◦−1Γ ` G = G′ : κ Γ ` F G ≤ F ′G′ : κ′ LEQ-APP+ Γ ` F ≤ F ′ : +κ→ κ′ Γ ` G ≤ G′ : κ Γ ` F G ≤ F ′G′ : κ′ LEQ-APP− Γ ` F ≤ F ′ : −κ→ κ′ −Γ ` G′ ≤ G : κ Γ ` F G ≤ F ′G′ : κ′ LEQ-APP> Γ ` F ≤ F ′ : >κ→ κ′ >−1Γ ` G : κ >−1Γ ` G′ : κ Γ ` F G ≤ F ′G′ : κ′ EQ-APP-> Γ ` F = F ′ : >κ→ κ′ >−1Γ ` G : κ >−1Γ ` G′ : κ Γ ` F G = F ′G′ : κ′ Fig. 2. Admissible rules for equality and subtyping. the constructors under consideration and then compare their head symbols. If they are related, one continues to recursively compare the subcomponents, otherwise subtyping fails. Weak head normal forms W ∈ Val are given by the grammar: Ne 3 N ::= C | X | N G neutral constructors Val 3 V,W ::= N | λXF weak head values Weak head evaluation F ↘ W , a big-step call-by-name operational semantics, is defined inductively by the following rules: EVAL-C C ↘ C EVAL-VAR X ↘ X EVAL-LAM λXF ↘ λXF EVAL-APP-NE F ↘ N F G↘ N G EVAL-APP-β F ↘ λXF ′ [G/X]F ′ ↘W F G↘W Lemma 3.1 (Properties of weak head evaluation). 1 If F X ↘ W then either F ↘ λXF ′ and F ′ ↘ W , or F ↘ N and W = N X . If X 6∈ FV(F ) then X 6∈ FV(N). 2 If F ↘W and [G/X]W ↘ V then [G/X]F ↘ V . 3 If F ↘W andW G↘ V then F G↘ V . Lemma 3.2 (Soundness of weak head evaluation). If Γ ` F : κ and F ↘ W then Γ ` F = W : κ.