1. Refining Mechanized Metatheory: Subtyping for LF William Lovas (with Frank Pfenning)

2. LFMTP '07 2 LF: a framework for defining logics (Harper, Honsell, and Plotkin, 1987, 1993) Dependently-typed lambda calculus Encode deductive systems and metatheory, in a machine-checkable way – e.g. a programming language and its type safety theorem Guiding principle: “judgements as types”

3. LFMTP '07 3 Judgements as types On paper: Syntax – e ::= … Judgement – Γ ⊢ e : τ Deduction – D :: Γ ⊢ e : τ Proof checking In LF: Simple type – exp : type. Type family – of : exp →  tp → type. Well-typed term – M : of E T Type checking

4. LFMTP '07 4 Inclusion as subtyping Some judgements have a natural notion of inclusion – all values are expressions – all odd natural numbers are positive More interesting types means more interesting judgements!

5. LFMTP '07 5 Example: natural numbers nat : type. z : nat. s : nat → nat. double : nat → nat → type. % plus rules even : nat → type. % plus some rules… odd : nat → type. % plus some rules… dbl-even : Π X :nat. Π Y :nat. % plus cases double X Y → even Y → type.

6. LFMTP '07 6 Example: nats using refinements nat : type. z : nat. s : nat → nat. double : nat → nat → type. % plus rules even : nat → type. % plus some rules… odd : nat → type. % plus some rules… dbl-even : Π X :nat. Π Y :nat. % plus cases double X Y → even Y → type. nat : type. even ≤ nat. odd ≤ nat. z : even. s : even → odd ∧ odd → even. double : nat → even → type. % plus rules metatheorem checking problem ⇝ typechecking problem!

7. LFMTP '07 7 Example: nats using refinements nat : type. even ≤ nat. odd ≤ nat. z : even. s : even → odd ∧ odd → even. double : nat → even → type. dbl-z : double z z. dbl-s : Π X :nat. Π Y :even. double X Y → double (s X ) (s (s Y )). dbl-s : Π X :nat. Π Y :even. double X Y → double (s X ) (s (s Y )).

8. LFMTP '07 8 Example: the lambda calculus exp : type. val : type. lam : (val → exp) → val. app : exp → exp → exp. $ : val → exp. lam (λ x. app ($ x ) ($ x )) : val Intrinsic values encoding: val ≤ exp. lam (λ x. app x x ) : val

9. LFMTP '07 9 Technology

10. LFMTP '07 10 Adequacy Does this really mean what I think it means? Strategy: exhibit a compositional bijection between mathematical objects and canonical forms following judgements as types. – “canonical forms” are β-normal and η-long.

11. LFMTP '07 11 Canonical forms method Represent only the canonical forms. – β-normal: syntactically – η-long: through typing – Hereditary substitutions contract redexes Simplifies metatheory, emphasizes adequacy Concurrent LF (Watkins, et al, 2003)

12. LFMTP '07 12 LF typing Bidirectional typing Synthesis: Γ ⊢ R ⇒ A – elims: R ::= x | c | R N Checking: Γ ⊢ N ⇐ A – intros: N ::= R | λ x. N

13. LFMTP '07 13 Checking Γ ⊢ R ⇒ P’ P’ = P Γ ⊢ R ⇐ P Γ ⊢ N ⇐ A Key rule: – base type, so atoms fully applied – the only appeal to type equality

14. LFMTP '07 14 Checking… with subtyping! Γ ⊢ R ⇒ P’ P’ ≤ P Γ ⊢ R ⇐ P Γ ⊢ N ⇐ A Easy to adapt! – just change equality to subtyping – subtyping… only at base type?

15. LFMTP '07 15 Intersections Kind of like pairs, but the terms don’t change Γ ⊢ N ⇐ A 1 Γ ⊢ N ⇐ A 2 Γ ⊢ N ⇐ A 1 ∧ A 2 Γ ⊢ N ⇐ ⊤ Γ ⊢ R ⇒ A 1 ∧ A 2 Γ ⊢ R ⇒ A 1 Γ ⊢ R ⇒ A 1 ∧ A 2 Γ ⊢ R ⇒ A 2

16. LFMTP '07 16 Metatheory

17. LFMTP '07 17 LF(R) as a logic Entailment should be reflexive: – A ⊢ A and transitive: – if A ⊢ B and B ⊢ C, then A ⊢ C

18. LFMTP '07 18 LF(R) as a logic Assume x is a proof of A. Is x a proof of A ? – not necessarily! x : A 1 → A 2 ⊬ x ⇐ A 1 → A 2 – have to η-expand: x : A 1 → A 2 ⊢ λ y. x y ⇐ A 1 → A 2

19. LFMTP '07 19 LF(R) as a logic Assume x is a proof of A. Can M ⇐ A stand in for x ? – if substitution is hereditary? [M/ x ] A N not obviously defined…

20. LFMTP '07 20 Important principles Substitution if Γ, x :A ⊢ N ⇐ B and Γ ⊢ M ⇐ A, then Γ ⊢ [M/ x ] A N ⇐ [M/ x ] A B. Identity for all A, Γ, x :A ⊢ η A ( x ) ⇐ A. “Substitution” morally a normalization proof

21. LFMTP '07 21 More about subtyping

22. LFMTP '07 22 Key rule: Bidirectional: subtyping only at mode switch Canonical: mode switch only at base type Subtyping Γ ⊢ R ⇒ P’ P’ ≤ P Γ ⊢ R ⇐ P Γ ⊢ N ⇐ A

23. LFMTP '07 23 Subtyping at higher types? What happened to the structural rules? E.g., Distributivity? A 2 ≤ A 1 B 1 ≤ B 2 A 1 → B 1 ≤ A 2 → B 2 (A → B 1 ) ∧ (A → B 2 ) ≤ A → (B 1 ∧ B 2 )

24. LFMTP '07 24 Subtyping at higher types! Intrinsic subtyping: if A ≤ B and Γ ⊢ N ⇐ A, then Γ ⊢ N ⇐ B. Equivalently: if A ≤ B then x :A ⊢ η A ( x ) ⇐ B. – Just like the Identity principle! – … also the Substitution principle… Usual rules are all sound in this sense.

25. LFMTP '07 25 Subtyping at higher types!? … and also complete! Theorem: if x :A ⊢ η A ( x ) ⇐ B then A ≤ B. Also: if Γ ⊢ N ⇐ A implies Γ ⊢ N ⇐ B, then A ≤ B. There are no new subtyping principles.

26. LFMTP '07 26 Future work Two directions: 1. Extend LFR with Twelf stuff type reconstruction unification proof search 2. Extend LFR with CLF stuff more type constructors subtyping with linearity?

27. LFMTP '07 27 Summary Refinement types are a useful addition to LF. Canonical forms method is up to the task. Concentrating only on canonical forms and bidirectional typing yields new insights into subtyping.

28. LFMTP '07 28 secret slides

29. LFMTP '07 29 Related work Refinement types – Tim Freeman, Rowan Davies, Joshua Dunfield Logical frameworks – Robert Harper, Furio Honsell, Gordon Plotkin – Frank Pfenning Subtyping and dependent types – David Aspinall, Adriana Compagnoni

30. LFMTP '07 30 LF syntax Terms Types R ::= c | x | R N N ::= R | λ x. N P ::= a | P N A, B ::= P | Π x :A.B atomic normal atomic normal no redexes

31. LFMTP '07 31 Hereditary substitution Substitution must contract redexes Example: Indexed by type subscript for termination Sometimes undefined: [(λ x. d x x ) / y ] y z = d z z [M/ x ] A N [(λ x. x x ) / y ] A y y fails by induction on A

32. LFMTP '07 32 Synthesis Γ ⊢ R ⇒ Π x :A.B Γ ⊢ N ⇐ A Γ ⊢ R N ⇒ [N/ x ] A B x :A ∈ Γ Γ ⊢ x ⇒ A c:A ∈ Σ Γ ⊢ c ⇒ A Γ ⊢ R ⇒ A hereditary substitution

33. LFMTP '07 33 Checking Γ ⊢ R ⇒ P’ P’ = P Γ ⊢ R ⇐ P Γ, x :A ⊢ N ⇐ B Γ ⊢ λ x.N ⇐ Π x :A.B Γ ⊢ N ⇐ A

34. LFMTP '07 34 Example: the lambda calculus exp : type. val ⊏ exp. lam : (exp → exp) → exp. app : exp → exp → exp. value : exp → type. lam : (val → exp) → val.

35. LFMTP '07 35 s ⊏ a :: L Γ ⊢ R ⇒ P’ P’ = P Γ ⊢ R ⇐ P Γ ⊢ R ⇒ P’ P’ ≤ P Γ ⊢ R ⇐ P

36. LFMTP '07 36 Subtyping Key rule: Bidirectional: subtyping only at mode switch Canonical: mode switch only at base type subtyping only at base type!

37. LFMTP '07 37 ⇐Γ, x:A ⊢ M ⇐ B ≤ e abcdefghijklmnopqrstuvwxyz ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz ABCDEFGHIJKLMNOPQRSTUVWXYZ Γ ⊢ R ⇒ A