# Types, Polymorphism, and Type Reconstruction

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1 Types, Polymorphism, and Type Reconstruction

2 Sources This material is based on the following sources: Pierce, B.C., Types and Programming Languages. MIT Press, Kanellakis, P.C., Mairson, H.G. and Mitchell, J.C., Unification and ML type reconstruction. In Computational Logic: Essays in Honor of Alan Robinson, ed. J.-L. Lassez and G.D. Plotkin, MIT Press, 1991, pages Damas L., Milner, R., Principal type-schemes for functional programs. Proceedings of the 9th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, ACM, 1982, pages Recommended further reading: Pierce, B.C., Turner D.N., Local type inference. ACM Transactions on Programming Languages and Systems, Volume 22 Issue 1, Jan. 2000, pages 1 44.

3 The Simply Typed Lambda-Calculus

4 The simply typed lambda-calculus The system we are about to define is commonly called the simply typed lambda-calculus, or λ for short. Unlike the untyped lambda-calculus, the pure form of λ (with no primitive values or operations) is not very interesting; to talk about λ, we always begin with some set of base types. So, strictly speaking, there are many variants of λ, depending on the choice of base types. For now, we ll work with a variant constructed over the booleans.

5 Untyped lambda-calculus with booleans t ::= terms x variable λx.t abstraction t t application true constant true false constant false if t then t else t conditional v ::= values λx.t abstraction value true true value false false value

6 Simple Types T ::= types Bool type of booleans T T types of functions What are some examples?

7 Type Annotations We now have a choice to make. Do we... annotate lambda-abstractions with the expected type of the argument λx:t 1. t 2 (as in most mainstream programming languages), or continue to write lambda-abstractions as before λx. t 2 and ask the typing rules to guess an appropriate annotation (as in OCaml)? Both are reasonable choices, but the first makes the job of defining the typing rules simpler. Let s take this choice for now.

8 Typing rules true : Bool false : Bool t 1 : Bool t 2 : T t 3 : T if t 1 then t 2 else t 3 : T (T-True) (T-False) (T-If)

9 Typing rules true : Bool false : Bool t 1 : Bool t 2 : T t 3 : T if t 1 then t 2 else t 3 : T??? λx:t 1.t 2 : T 1 T 2 (T-True) (T-False) (T-If) (T-Abs)

10 Typing rules true : Bool false : Bool t 1 : Bool t 2 : T t 3 : T if t 1 then t 2 else t 3 : T (T-True) (T-False) (T-If) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 x:t Γ Γ x : T (T-Abs) (T-Var)

11 Typing rules Γ true : Bool Γ false : Bool (T-True) (T-False) Γ t 1 : Bool Γ t 2 : T Γ t 3 : T Γ if t 1 then t 2 else t 3 : T (T-If) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 x:t Γ Γ x : T (T-Abs) (T-Var) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App)

12 Typing Derivations What derivations justify the following typing statements? (λx:bool.x) true : Bool f:bool Bool f (if false then true else false) : Bool f:bool Bool λx:bool. f (if x then false else x) : Bool Bool

13 Properties of λ The fundamental property of the type system we have just defined is soundness with respect to the operational semantics. 1. Progress: A closed, well-typed term is not stuck If t : T, then either t is a value or else t t for some t. 2. Preservation: Types are preserved by one-step evaluation If Γ t : T and t t, then Γ t : T.

14 Proving progress inversion lemma for typing relation canonical forms lemma progress theorem

15 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R.

16 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R. 4. If Γ x : R, then

17 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R. 4. If Γ x : R, then x:r Γ.

18 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R. 4. If Γ x : R, then x:r Γ. 5. If Γ λx:t 1.t 2 : R, then

19 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R. 4. If Γ x : R, then x:r Γ. 5. If Γ λx:t 1.t 2 : R, then R = T 1 R 2 for some R 2 with Γ, x:t 1 t 2 : R 2.

20 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R. 4. If Γ x : R, then x:r Γ. 5. If Γ λx:t 1.t 2 : R, then R = T 1 R 2 for some R 2 with Γ, x:t 1 t 2 : R If Γ t 1 t 2 : R, then

21 Inversion Lemma: 1. If Γ true : R, then R = Bool. 2. If Γ false : R, then R = Bool. 3. If Γ if t 1 then t 2 else t 3 : R, then Γ t 1 : Bool and Γ t 2, t 3 : R. 4. If Γ x : R, then x:r Γ. 5. If Γ λx:t 1.t 2 : R, then R = T 1 R 2 for some R 2 with Γ, x:t 1 t 2 : R If Γ t 1 t 2 : R, then there is some type T 11 such that Γ t 1 : T 11 R and Γ t 2 : T 11.

22 Canonical Forms Lemma:

23 Canonical Forms Lemma: 1. If v is a value of type Bool, then

24 Canonical Forms Lemma: 1. If v is a value of type Bool, then v is either true or false.

25 Canonical Forms Lemma: 1. If v is a value of type Bool, then v is either true or false. 2. If v is a value of type T 1 T 2, then

26 Canonical Forms Lemma: 1. If v is a value of type Bool, then v is either true or false. 2. If v is a value of type T 1 T 2, then v has the form λx:t 1.t 2.

27 Progress Theorem: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Proof: By induction

28 Progress Theorem: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Proof: By induction on typing derivations.

29 Progress Theorem: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Proof: By induction on typing derivations. The cases for boolean constants and conditions are the same as before. The variable case is trivial (because t is closed). The abstraction case is immediate, since abstractions are values.

30 Progress Theorem: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Proof: By induction on typing derivations. The cases for boolean constants and conditions are the same as before. The variable case is trivial (because t is closed). The abstraction case is immediate, since abstractions are values. Consider the case for application, where t = t 1 t 2 with t 1 : T 11 T 12 and t 2 : T 11.

31 Progress Theorem: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Proof: By induction on typing derivations. The cases for boolean constants and conditions are the same as before. The variable case is trivial (because t is closed). The abstraction case is immediate, since abstractions are values. Consider the case for application, where t = t 1 t 2 with t 1 : T 11 T 12 and t 2 : T 11. By the induction hypothesis, either t 1 is a value or else it can make a step of evaluation, and likewise t 2.

32 Progress Theorem: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Proof: By induction on typing derivations. The cases for boolean constants and conditions are the same as before. The variable case is trivial (because t is closed). The abstraction case is immediate, since abstractions are values. Consider the case for application, where t = t 1 t 2 with t 1 : T 11 T 12 and t 2 : T 11. By the induction hypothesis, either t 1 is a value or else it can make a step of evaluation, and likewise t 2. If t 1 can take a step, then rule E-App1 applies to t. If t 1 is a value and t 2 can take a step, then rule E-App2 applies. Finally, if both t 1 and t 2 are values, then the canonical forms lemma tells us that t 1 has the form λx:t 11.t 12, and so rule E-AppAbs applies to t.

33 Preservation for STLC

34 Preservation for STLC Theorem: If Γ t : T and t t, then Γ t : T. Proof: By induction

35 Preservation for STLC Theorem: If Γ t : T and t t, then Γ t : T. Proof: By induction on typing derivations. Which case is the hard one??

36 Preservation for STLC Theorem: If Γ t : T and t t, then Γ t : T. Proof: By induction on typing derivations. Case T-App: Given t = t 1 t 2 Γ t 1 : T 11 T 12 Γ t 2 : T 11 T = T 12 Show Γ t : T 12

37 Preservation for STLC Theorem: If Γ t : T and t t, then Γ t : T. Proof: By induction on typing derivations. Case T-App: Given t = t 1 t 2 Γ t 1 : T 11 T 12 Γ t 2 : T 11 T = T 12 Show Γ t : T 12 By the inversion lemma for evaluation, there are three subcases...

38 The Substitution Lemma Lemma: Types are preserved under substitition. That is, if Γ, x:s t : T and Γ s : S, then Γ [x s]t : T.

39 The Substitution Lemma Lemma: Types are preserved under substitition. That is, if Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. Proof:...

40 Weakening and Permutation Two other lemmas will be useful. Weakening tells us that we can add assumptions to the context without losing any true typing statements. Lemma: If Γ t : T and x / dom(γ), then Γ, x:s t : T.

41 Weakening and Permutation Two other lemmas will be useful. Weakening tells us that we can add assumptions to the context without losing any true typing statements. Lemma: If Γ t : T and x / dom(γ), then Γ, x:s t : T. Permutation tells us that the order of assumptions in (the list) Γ does not matter. Lemma: If Γ t : T and is a permutation of Γ, then t : T.

42 Weakening and Permutation Two other lemmas will be useful. Weakening tells us that we can add assumptions to the context without losing any true typing statements. Lemma: If Γ t : T and x / dom(γ), then Γ, x:s t : T. Moreover, the latter derivation has the same depth as the former. Permutation tells us that the order of assumptions in (the list) Γ does not matter. Lemma: If Γ t : T and is a permutation of Γ, then t : T. Moreover, the latter derivation has the same depth as the former.

43 The Substitution Lemma Lemma: If Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. I.e., Types are preserved under substitition.

44 The Substitution Lemma Lemma: If Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. Proof: By induction on the derivation of Γ, x:s t : T. Proceed by cases on the final typing rule used in the derivation.

45 The Substitution Lemma Lemma: If Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. Proof: By induction on the derivation of Γ, x:s t : T. Proceed by cases on the final typing rule used in the derivation.

46 The Substitution Lemma Lemma: If Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. Proof: By induction on the derivation of Γ, x:s t : T. Proceed by cases on the final typing rule used in the derivation. Case T-App: t = t 1 t 2 Γ, x:s t 1 : T 2 T 1 Γ, x:s t 2 : T 2 T = T 1 By the induction hypothesis, Γ [x s]t 1 : T 2 T 1 and Γ [x s]t 2 : T 2. By T-App, Γ [x s]t 1 [x s]t 2 : T, i.e., Γ [x s](t 1 t 2 ) : T.

47 The Substitution Lemma Lemma: If Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. Proof: By induction on the derivation of Γ, x:s t : T. Proceed by cases on the final typing rule used in the derivation. Case T-Var: t = z with z:t (Γ, x:s) There are two sub-cases to consider, depending on whether z is x or another variable. If z = x, then [x s]z = s. The required result is then Γ s : S, which is among the assumptions of the lemma. Otherwise, [x s]z = z, and the desired result is immediate.

48 The Substitution Lemma Lemma: If Γ, x:s t : T and Γ s : S, then Γ [x s]t : T. Proof: By induction on the derivation of Γ, x:s t : T. Proceed by cases on the final typing rule used in the derivation. Case T-Abs: t = λy:t 2.t 1 T = T 2 T 1 Γ, x:s, y:t 2 t 1 : T 1 By our conventions on choice of bound variable names, we may assume x y and y / FV(s). Using permutation on the given subderivation, we obtain Γ, y:t 2, x:s t 1 : T 1. Using weakening on the other given derivation (Γ s : S), we obtain Γ, y:t 2 s : S. Now, by the induction hypothesis, Γ, y:t 2 [x s]t 1 : T 1. By T-Abs, Γ λy:t 2. [x s]t 1 : T 2 T 1, i.e. (by the definition of substitution), Γ [x s]λy:t 2. t 1 : T 2 T 1.

49 Summary: Preservation Theorem: If Γ t : T and t t, then Γ t : T. Lemmas to prove: Weakening Permutation Substitution preserves types Reduction preserves types (i.e., preservation)

50 Review: Type Systems To define and verify a type system, you must: 1. Define types 2. Specify typing rules 3. Prove soundness: progress and preservation

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