module plfa.part2.Inference where

So far in our development, type derivations for the corresponding term have been provided by fiat. In Chapter Lambda type derivations are extrinsic to the term, while in Chapter DeBruijn type derivations are intrinsic to the term, but in both we have written out the type derivations in full.

In practice, one often writes down a term with a few decorations and applies an algorithm to infer the corresponding type derivation. Indeed, this is exactly what happens in Agda: we specify the types for top-level function declarations, and type information for everything else is inferred from what has been given. The style of inference Agda uses is based on a technique called bidirectional type inference, which will be presented in this chapter.

This chapter ties our previous developments together. We begin with a term with some type annotations, close to the raw terms of Chapter Lambda, and from it we compute an intrinsically-typed term, in the style of Chapter DeBruijn.

Introduction: Inference rules as algorithms

In the calculus we have considered so far, a term may have more than one type. For example,

(ƛ x ⇒ x) ⦂ (A ⇒ A)

holds for every type A. We start by considering a small language for lambda terms where every term has a unique type. All we need do is decorate each abstraction term with the type of its argument. This gives us the grammar:

L, M, N ::=                         decorated terms
  x                                   variable
  ƛ x ⦂ A ⇒ N                         abstraction (decorated)
  L · M                               application

Each of the associated type rules can be read as an algorithm for type checking. For each typing judgment, we label each position as either an input or an output.

For the judgment

Γ ∋ x ⦂ A

we take the context Γ and the variable x as inputs, and the type A as output. Consider the rules:

----------------- Z
Γ , x ⦂ A ∋ x ⦂ A

Γ ∋ x ⦂ A
----------------- S
Γ , y ⦂ B ∋ x ⦂ A

From the inputs we can determine which rule applies: if the last variable in the context matches the given variable then the first rule applies, else the second. (For de Bruijn indices, it is even easier: zero matches the first rule and successor the second.) For the first rule, the output type can be read off as the last type in the input context. For the second rule, the inputs of the conclusion determine the inputs of the hypothesis, and the output of the hypothesis determines the output of the conclusion.

For the judgment

Γ ⊢ M ⦂ A

we take the context Γ and term M as inputs, and the type A as output. Consider the rules:

Γ ∋ x ⦂ A
-----------
Γ ⊢ ` x ⦂ A

Γ , x ⦂ A ⊢ N ⦂ B
---------------------------
Γ ⊢ (ƛ x ⦂ A ⇒ N) ⦂ (A ⇒ B)

Γ ⊢ L ⦂ A ⇒ B
Γ ⊢ M ⦂ A′
A ≡ A′
-------------
Γ ⊢ L · M ⦂ B

The input term determines which rule applies: variables use the first rule, abstractions the second, and applications the third. We say such rules are syntax directed. For the variable rule, the inputs of the conclusion determine the inputs of the hypothesis, and the output of the hypothesis determines the output of the conclusion. Same for the abstraction rule — the bound variable and argument are carried from the term of the conclusion into the context of the hypothesis; this works because we added the argument type to the abstraction. For the application rule, we add a third hypothesis to check whether the domain of the function matches the type of the argument; this judgment is decidable when both types are given as inputs. The inputs of the conclusion determine the inputs of the first two hypotheses, the outputs of the first two hypotheses determine the inputs of the third hypothesis, and the output of the first hypothesis determines the output of the conclusion.

Converting the above to an algorithm is straightforward, as is adding naturals and fixpoint. We omit the details. Instead, we consider a detailed description of an approach that requires less obtrusive decoration. The idea is to break the normal typing judgment into two judgments, one that produces the type as an output (as above), and another that takes it as an input.

Synthesising and inheriting types

In addition to the lookup judgment for variables, which will remain as before, we now have two judgments for the type of the term:

Γ ⊢ M ↑ A
Γ ⊢ M ↓ A

The first of these synthesises the type of a term, as before, while the second inherits the type. In the first, the context and term are inputs and the type is an output; while in the second, all three of the context, term, and type are inputs.

Which terms use synthesis and which inheritance? Our approach will be that the main term in a deconstructor is typed via synthesis while constructors are typed via inheritance. For instance, the function in an application is typed via synthesis, but an abstraction is typed via inheritance. The inherited type in an abstraction term serves the same purpose as the argument type decoration of the previous section.

Terms that deconstruct a value of a type always have a main term (supplying an argument of the required type) and often have side-terms. For application, the main term supplies the function and the side term supplies the argument. For case terms, the main term supplies a natural and the side terms are the two branches. In a deconstructor, the main term will be typed using synthesis but the side terms will be typed using inheritance. As we will see, this leads naturally to an application as a whole being typed by synthesis, while a case term as a whole will be typed by inheritance. Variables are naturally typed by synthesis, since we can look up the type in the input context. Fixed points will be naturally typed by inheritance.

In order to get a syntax-directed type system we break terms into two kinds, Term⁺ and Term⁻, which are typed by synthesis and inheritance, respectively. A subterm that is typed by synthesis may appear in a context where it is typed by inheritance, or vice-versa, and this gives rise to two new term forms.

For instance, we said above that the argument of an application is typed by inheritance and that variables are typed by synthesis, giving a mismatch if the argument of an application is a variable. Hence, we need a way to treat a synthesized term as if it is inherited. We introduce a new term form, M ↑ for this purpose. The typing judgment checks that the inherited and synthesised types match.

Similarly, we said above that the function of an application is typed by synthesis and that abstractions are typed by inheritance, giving a mismatch if the function of an application is an abstraction. Hence, we need a way to treat an inherited term as if it is synthesised. We introduce a new term form M ↓ A for this purpose. The typing judgment returns A as the synthesized type of the term as a whole, as well as using it as the inherited type for M.

The term form M ↓ A represents the only place terms need to be decorated with types. It only appears when switching from synthesis to inheritance, that is, when a term that deconstructs a value of a type contains as its main term a term that constructs a value of a type, in other words, a place where a β-reduction will occur. Typically, we will find that decorations are only required on top level declarations.

We can extract the grammar for terms from the above:

L⁺, M⁺, N⁺ ::=                      terms with synthesized type
  x                                   variable
  L⁺ · M⁻                             application
  M⁻ ↓ A                              switch to inherited

L⁻, M⁻, N⁻ ::=                      terms with inherited type
  ƛ x ⇒ N⁻                            abstraction
  `zero                               zero
  `suc M⁻                             successor
  case L⁺ [zero⇒ M⁻ |suc x ⇒ N⁻ ]     case
  μ x ⇒ N⁻                            fixpoint
  M⁺ ↑                                switch to synthesized

We will formalise the above shortly.

Soundness and completeness

What we intend to show is that the typing judgments are decidable:

synthesize : ∀ (Γ : Context) (M : Term⁺)
    ------------------------------------
  → Dec (∃[ A ] Γ ⊢ M ↑ A)

inherit : ∀ (Γ : Context) (M : Term⁻) (A : Type)
          --------------------------------------
        → Dec (Γ ⊢ M ↓ A)

Given context Γ and synthesised term M, we must decide whether there exists a type A such that Γ ⊢ M ↑ A holds, or its negation. Similarly, given context Γ, inherited term M, and type A, we must decide whether Γ ⊢ M ↓ A holds, or its negation.

Our proof is constructive. In the synthesised case, it will either deliver a pair of a type A and evidence that Γ ⊢ M ↓ A, or a function that given such a pair produces evidence of a contradiction. In the inherited case, it will either deliver evidence that Γ ⊢ M ↑ A, or a function that given such evidence produces evidence of a contradiction. The positive case is referred to as soundness — synthesis and inheritance succeed only if the corresponding relation holds. The negative case is referred to as completeness — synthesis and inheritance fail only when they cannot possibly succeed.

Another approach might be to return a derivation if synthesis or inheritance succeeds, and an error message otherwise — for instance, see the section of the Agda user manual discussing syntactic sugar. Such an approach demonstrates soundness, but not completeness. If it returns a derivation, we know it is correct; but there is nothing to prevent us from writing a function that always returns an error, even when there exists a correct derivation. Demonstrating both soundness and completeness is significantly stronger than demonstrating soundness alone. The negative proof can be thought of as a semantically verified error message, although in practice it may be less readable than a well-crafted error message.

We are now ready to begin the formal development.

Imports

import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong; cong₂; _≢_)
open import Data.Empty using (⊥)
open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_)
open import Data.String using (String; _≟_)
open import Data.Product.Base using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Relation.Nullary.Negation using (¬_; contradiction)
open import Relation.Nullary.Decidable using ( Dec; yes; no; False; toWitnessFalse)

Once we have a type derivation, it will be easy to construct from it the intrinsically-typed representation. In order that we can compare with our previous development, we import module plfa.part2.More:

import plfa.part2.More as DB

The phrase as DB allows us to refer to definitions from that module as, for instance, DB._⊢_, which is invoked as Γ DB.⊢ A, where Γ has type DB.Context and A has type DB.Type.

Syntax

First, we get all our infix declarations out of the way. We list separately operators for judgments and terms:

infix   4  _∋_⦂_
infix   4  _⊢_↑_
infix   4  _⊢_↓_
infixl  5  _,_⦂_

infixr  7  _⇒_

infix   5  ƛ_⇒_
infix   5  μ_⇒_
infix   6  _↑
infix   6  _↓_
infixl  7  _·_
infix   8  `suc_
infix   9  `_
Identifiers, types, and contexts are as before:
Id : Set
Id = String

data Type : Set where
  `ℕ    : Type
  _⇒_   : Type → Type → Type

data Context : Set where
  ∅     : Context
  _,_⦂_ : Context → Id → Type → Context
The syntax of terms is defined by mutual recursion. We use Term⁺ and Term⁻ for terms with synthesized and inherited types, respectively. Note the inclusion of the switching forms, M ↓ A and M ↑:
data Term⁺ : Set
data Term⁻ : Set

data Term⁺ where
  `_                       : Id → Term⁺
  _·_                      : Term⁺ → Term⁻ → Term⁺
  _↓_                      : Term⁻ → Type → Term⁺

data Term⁻ where
  ƛ_⇒_                     : Id → Term⁻ → Term⁻
  `zero                    : Term⁻
  `suc_                    : Term⁻ → Term⁻
  `case_[zero⇒_|suc_⇒_]    : Term⁺ → Term⁻ → Id → Term⁻ → Term⁻
  μ_⇒_                     : Id → Term⁻ → Term⁻
  _↑                       : Term⁺ → Term⁻

The choice as to whether each term is synthesized or inherited follows the discussion above, and can be read off from the informal grammar presented earlier. Main terms in deconstructors synthesise, constructors and side terms in deconstructors inherit.

Example terms

We can recreate the examples from preceding chapters. First, computing two plus two on naturals:
two : Term⁻
two = `suc (`suc `zero)

plus : Term⁺
plus = (μ "p" ⇒ ƛ "m" ⇒ ƛ "n" ⇒
          `case (` "m") [zero⇒ ` "n" ↑
                        |suc "m" ⇒ `suc (` "p" · (` "m" ↑) · (` "n" ↑) ↑) ])
            ↓ (`ℕ ⇒ `ℕ ⇒ `ℕ)

2+2 : Term⁺
2+2 = plus · two · two

The only change is to decorate with down and up arrows as required. The only type decoration required is for plus.

Next, computing two plus two with Church numerals:
Ch : Type
Ch = (`ℕ ⇒ `ℕ) ⇒ `ℕ ⇒ `ℕ

twoᶜ : Term⁻
twoᶜ = (ƛ "s" ⇒ ƛ "z" ⇒ ` "s" · (` "s" · (` "z" ↑) ↑) ↑)

plusᶜ : Term⁺
plusᶜ = (ƛ "m" ⇒ ƛ "n" ⇒ ƛ "s" ⇒ ƛ "z" ⇒
           ` "m" · (` "s" ↑) · (` "n" · (` "s" ↑) · (` "z" ↑) ↑) ↑)
             ↓ (Ch ⇒ Ch ⇒ Ch)

sucᶜ : Term⁻
sucᶜ = ƛ "x" ⇒ `suc (` "x" ↑)

2+2ᶜ : Term⁺
2+2ᶜ = plusᶜ · twoᶜ · twoᶜ · sucᶜ · `zero

The only type decoration required is for plusᶜ. One is not even required for sucᶜ, which inherits its type as an argument of plusᶜ.

Bidirectional type checking

The typing rules for variables are as in Lambda:
data _∋_⦂_ : Context → Id → Type → Set where

  Z : ∀ {Γ x A}
      -----------------
    → Γ , x ⦂ A ∋ x ⦂ A

  S : ∀ {Γ x y A B}
    → x ≢ y
    → Γ ∋ x ⦂ A
      -----------------
    → Γ , y ⦂ B ∋ x ⦂ A
As with syntax, the judgments for synthesizing and inheriting types are mutually recursive:
data _⊢_↑_ : Context → Term⁺ → Type → Set
data _⊢_↓_ : Context → Term⁻ → Type → Set

data _⊢_↑_ where

  ⊢` : ∀ {Γ A x}
    → Γ ∋ x ⦂ A
      -----------
    → Γ ⊢ ` x ↑ A

  _·_ : ∀ {Γ L M A B}
    → Γ ⊢ L ↑ A ⇒ B
    → Γ ⊢ M ↓ A
      -------------
    → Γ ⊢ L · M ↑ B

  ⊢↓ : ∀ {Γ M A}
    → Γ ⊢ M ↓ A
      ---------------
    → Γ ⊢ (M ↓ A) ↑ A

data _⊢_↓_ where

  ⊢ƛ : ∀ {Γ x N A B}
    → Γ , x ⦂ A ⊢ N ↓ B
      -------------------
    → Γ ⊢ ƛ x ⇒ N ↓ A ⇒ B

  ⊢zero : ∀ {Γ}
      --------------
    → Γ ⊢ `zero ↓ `ℕ

  ⊢suc : ∀ {Γ M}
    → Γ ⊢ M ↓ `ℕ
      ---------------
    → Γ ⊢ `suc M ↓ `ℕ

  ⊢case : ∀ {Γ L M x N A}
    → Γ ⊢ L ↑ `ℕ
    → Γ ⊢ M ↓ A
    → Γ , x ⦂ `ℕ ⊢ N ↓ A
      -------------------------------------
    → Γ ⊢ `case L [zero⇒ M |suc x ⇒ N ] ↓ A

  ⊢μ : ∀ {Γ x N A}
    → Γ , x ⦂ A ⊢ N ↓ A
      -----------------
    → Γ ⊢ μ x ⇒ N ↓ A

  ⊢↑ : ∀ {Γ M A B}
    → Γ ⊢ M ↑ A
    → A ≡ B
      -------------
    → Γ ⊢ (M ↑) ↓ B

We follow the same convention as Chapter Lambda, prefacing the constructor with ⊢ to derive the name of the corresponding type rule.

The rules are similar to those in Chapter Lambda, modified to support synthesised and inherited types. The two new rules are those for ⊢↓ and ⊢↑. The former both passes the type decoration as the inherited type and returns it as the synthesised type. The latter takes the synthesised type and the inherited type and confirms they are identical — it should remind you of the equality test in the application rule in the first section.

Exercise bidirectional-mul (recommended)

Rewrite your definition of multiplication from Chapter Lambda, decorated to support inference.

-- Your code goes here

Exercise bidirectional-products (recommended)

Extend the bidirectional type rules to include products from Chapter More.

-- Your code goes here

Exercise bidirectional-rest (stretch)

Extend the bidirectional type rules to include the rest of the constructs from Chapter More.

-- Your code goes here

Prerequisites

The rule for M ↑ requires the ability to decide whether two types are equal. It is straightforward to code:
_≟Tp_ : (A B : Type) → Dec (A ≡ B)
`ℕ      ≟Tp `ℕ              =  yes refl
`ℕ      ≟Tp (A ⇒ B)         =  no λ()
(A ⇒ B) ≟Tp `ℕ              =  no λ()
(A ⇒ B) ≟Tp (A′ ⇒ B′)
  with A ≟Tp A′ | B ≟Tp B′
...  | no A≢    | _         =  no λ{refl → A≢ refl}
...  | yes _    | no B≢     =  no λ{refl → B≢ refl}
...  | yes refl | yes refl  =  yes refl
We will also need a couple of obvious lemmas; the domain and range of equal function types are equal:
dom≡ : ∀ {A A′ B B′} → A ⇒ B ≡ A′ ⇒ B′ → A ≡ A′
dom≡ refl = refl

rng≡ : ∀ {A A′ B B′} → A ⇒ B ≡ A′ ⇒ B′ → B ≡ B′
rng≡ refl = refl
We will also need to know that the types `ℕ and A ⇒ B are not equal:
ℕ≢⇒ : ∀ {A B} → `ℕ ≢ A ⇒ B
ℕ≢⇒ ()

Unique types

Looking up a type in the context is unique. Given two derivations, one showing Γ ∋ x ⦂ A and one showing Γ ∋ x ⦂ B, it follows that A and B must be identical:
uniq-∋ : ∀ {Γ x A B} → Γ ∋ x ⦂ A → Γ ∋ x ⦂ B → A ≡ B
uniq-∋ Z Z                 =  refl
uniq-∋ Z (S x≢y _)         =  contradiction refl x≢y
uniq-∋ (S x≢y _) Z         =  contradiction refl x≢y
uniq-∋ (S _ ∋x) (S _ ∋x′)  =  uniq-∋ ∋x ∋x′

If both derivations are by rule Z then uniqueness follows immediately, while if both derivations are by rule S then uniqueness follows by induction. It is a contradiction if one derivation is by rule Z and one by rule S, since rule Z requires the variable we are looking for is the final one in the context, while rule S requires it is not.

Synthesizing a type is also unique. Given two derivations, one showing Γ ⊢ M ↑ A and one showing Γ ⊢ M ↑ B, it follows that A and B must be identical:
uniq-↑ : ∀ {Γ M A B} → Γ ⊢ M ↑ A → Γ ⊢ M ↑ B → A ≡ B
uniq-↑ (⊢` ∋x) (⊢` ∋x′)       =  uniq-∋ ∋x ∋x′
uniq-↑ (⊢L · ⊢M) (⊢L′ · ⊢M′)  =  rng≡ (uniq-↑ ⊢L ⊢L′)
uniq-↑ (⊢↓ ⊢M) (⊢↓ ⊢M′)       =  refl

There are three possibilities for the term. If it is a variable, uniqueness of synthesis follows from uniqueness of lookup. If it is an application, uniqueness follows by induction on the function in the application, since the range of equal types are equal. If it is a switch expression, uniqueness follows since both terms are decorated with the same type.

Lookup type of a variable in the context

Given Γ and two distinct variables x and y, if there is no type A such that Γ ∋ x ⦂ A holds, then there is also no type A such that Γ , y ⦂ B ∋ x ⦂ A holds:
ext∋ : ∀ {Γ B x y}
  → x ≢ y
  → ¬ (∃[ A ] Γ ∋ x ⦂ A)
    ----------------------------
  → ¬ (∃[ A ] Γ , y ⦂ B ∋ x ⦂ A)
ext∋ x≢y _  ⟨ A , Z ⟩       =  x≢y refl
ext∋ _   ¬∃ ⟨ A , S _ ∋x ⟩  =  ¬∃ ⟨ A , ∋x ⟩

Given a type A and evidence that Γ , y ⦂ B ∋ x ⦂ A holds, we must demonstrate a contradiction. If the judgment holds by Z, then we must have that x and y are the same, which contradicts the first assumption. If the judgment holds by S _ ∋x then ∋x provides evidence that Γ ∋ x ⦂ A, which contradicts the second assumption.

Given a context Γ and a variable x, we decide whether there exists a type A such that Γ ∋ x ⦂ A holds, or its negation:
lookup : ∀ (Γ : Context) (x : Id)
         ------------------------
       → Dec (∃[ A ] Γ ∋ x ⦂ A)
lookup ∅ x                        =  no  (λ ())
lookup (Γ , y ⦂ B) x with x ≟ y
... | yes refl                    =  yes ⟨ B , Z ⟩
... | no x≢y with lookup Γ x
...             | no  ¬∃          =  no  (ext∋ x≢y ¬∃)
...             | yes ⟨ A , ∋x ⟩  =  yes ⟨ A , S x≢y ∋x ⟩

Consider the context:

  • If it is empty, then trivially there is no possible derivation.

  • If it is non-empty, compare the given variable to the most recent binding:

    • If they are identical, we have succeeded, with Z as the appropriate derivation.

    • If they differ, we recurse:

      • If lookup fails, we apply ext∋ to convert the proof there is no derivation from the contained context to the extended context.

      • If lookup succeeds, we extend the derivation with S.

Promoting negations

For each possible term form, we need to show that if one of its components fails to type, then the whole fails to type. Most of these results are easy to demonstrate inline, but we provide auxiliary functions for a couple of the trickier cases.

If Γ ⊢ L ↑ A ⇒ B holds but Γ ⊢ M ↓ A does not hold, then there is no type B′ such that Γ ⊢ L · M ↑ B′ holds:
¬arg : ∀ {Γ A B L M}
  → Γ ⊢ L ↑ A ⇒ B
  → ¬ Γ ⊢ M ↓ A
    --------------------------
  → ¬ (∃[ B′ ] Γ ⊢ L · M ↑ B′)
¬arg ⊢L ¬⊢M ⟨ B′ , ⊢L′ · ⊢M′ ⟩ rewrite dom≡ (uniq-↑ ⊢L ⊢L′) = ¬⊢M ⊢M′

Let ⊢L be evidence that Γ ⊢ L ↑ A ⇒ B holds and ¬⊢M be evidence that Γ ⊢ M ↓ A does not hold. Given a type B′ and evidence that Γ ⊢ L · M ↑ B′ holds, we must demonstrate a contradiction. The evidence must take the form ⊢L′ · ⊢M′, where ⊢L′ is evidence that Γ ⊢ L ↑ A′ ⇒ B′ and ⊢M′ is evidence that Γ ⊢ M ↓ A′. By uniq-↑ applied to ⊢L and ⊢L′, we know that A ⇒ B ≡ A′ ⇒ B′, and hence that A ≡ A′, which means that ¬⊢M and ⊢M′ yield a contradiction. Without the rewrite clause, Agda would not allow us to derive a contradiction between ¬⊢M and ⊢M′, since one concerns type A and the other type A′.

If Γ ⊢ M ↑ A holds and A ≢ B, then Γ ⊢ (M ↑) ↓ B does not hold:
¬switch : ∀ {Γ M A B}
  → Γ ⊢ M ↑ A
  → A ≢ B
    ---------------
  → ¬ Γ ⊢ (M ↑) ↓ B
¬switch ⊢M A≢B (⊢↑ ⊢M′ A′≡B) rewrite uniq-↑ ⊢M ⊢M′ = A≢B A′≡B

Let ⊢M be evidence that Γ ⊢ M ↑ A holds, and A≢B be evidence that A ≢ B. Given evidence that Γ ⊢ (M ↑) ↓ B holds, we must demonstrate a contradiction. The evidence must take the form ⊢↑ ⊢M′ A′≡B, where ⊢M′ is evidence that Γ ⊢ M ↑ A′ and A′≡B is evidence that A′ ≡ B. By uniq-↑ applied to ⊢M and ⊢M′ we know that A ≡ A′, which means that A≢B and A′≡B yield a contradiction. Without the rewrite clause, Agda would not allow us to derive a contradiction between A≢B and A′≡B, since one concerns type A and the other type A′.

Synthesize and inherit types

The table has been set and we are ready for the main course. We define two mutually recursive functions, one for synthesis and one for inheritance. Synthesis is given a context Γ and a synthesis term M and either returns a type A and evidence that Γ ⊢ M ↑ A, or its negation. Inheritance is given a context Γ, an inheritance term M, and a type A and either returns evidence that Γ ⊢ M ↓ A, or its negation:
synthesize : ∀ (Γ : Context) (M : Term⁺)
             ---------------------------
           → Dec (∃[ A ] Γ ⊢ M ↑ A )

inherit : ∀ (Γ : Context) (M : Term⁻) (A : Type)
    ---------------
  → Dec (Γ ⊢ M ↓ A)
We first consider the code for synthesis:
synthesize Γ (` x) with lookup Γ x
... | no  ¬∃              =  no  (λ{ ⟨ A , ⊢` ∋x ⟩ → ¬∃ ⟨ A , ∋x ⟩ })
... | yes ⟨ A , ∋x ⟩      =  yes ⟨ A , ⊢` ∋x ⟩
synthesize Γ (L · M) with synthesize Γ L
... | no  ¬∃              =  no  (λ{ ⟨ B , ⊢L  · _  ⟩  →  ¬∃ ⟨ _ ⇒ B , ⊢L ⟩ })
... | yes ⟨ `ℕ ,    ⊢L ⟩  =  no  (λ{ ⟨ _ , ⊢L′ · _  ⟩  →  ℕ≢⇒ (uniq-↑ ⊢L ⊢L′) })
... | yes ⟨ A ⇒ B , ⊢L ⟩ with inherit Γ M A
...    | no  ¬⊢M          =  no  (¬arg ⊢L ¬⊢M)
...    | yes ⊢M           =  yes ⟨ B , ⊢L · ⊢M ⟩
synthesize Γ (M ↓ A) with inherit Γ M A
... | no  ¬⊢M             =  no  (λ{ ⟨ _ , ⊢↓ ⊢M ⟩  →  ¬⊢M ⊢M })
... | yes ⊢M              =  yes ⟨ A , ⊢↓ ⊢M ⟩

There are three cases:

  • If the term is a variable ` x, we use lookup as defined above:

    • If it fails, then ¬∃ is evidence that there is no A such that Γ ∋ x ⦂ A holds. Evidence that Γ ⊢ ` x ↑ A holds must have the form ⊢` ∋x, where ∋x is evidence that Γ ∋ x ⦂ A, which yields a contradiction.

    • If it succeeds, then ∋x is evidence that Γ ∋ x ⦂ A, and hence ⊢` ∋x is evidence that Γ ⊢ ` x ↑ A.

  • If the term is an application L · M, we recurse on the function L:

    • If it fails, then ¬∃ is evidence that there is no type such that Γ ⊢ L ↑ _ holds. Evidence that Γ ⊢ L · M ↑ _ holds must have the form ⊢L · _, where ⊢L is evidence that Γ ⊢ L ↑ _, which yields a contradiction.

    • If it succeeds, there are two possibilities:

      • One is that ⊢L is evidence that Γ ⊢ L ⦂ `ℕ. Evidence that Γ ⊢ L · M ↑ _ holds must have the form ⊢L′ · _ where ⊢L′ is evidence that Γ ⊢ L ↑ A ⇒ B for some types A and B. Applying uniq-↑ to ⊢L and ⊢L′ yields a contradiction, since `ℕ cannot equal A ⇒ B.

      • The other is that ⊢L is evidence that Γ ⊢ L ↑ A ⇒ B, in which case we recurse on the argument M:

        • If it fails, then ¬⊢M is evidence that Γ ⊢ M ↓ A does not hold. By ¬arg applied to ⊢L and ¬⊢M, it follows that Γ ⊢ L · M ↑ B cannot hold.

        • If it succeeds, then ⊢M is evidence that Γ ⊢ M ↓ A, and ⊢L · ⊢M provides evidence that Γ ⊢ L · M ↑ B.

  • If the term is a switch M ↓ A from synthesised to inherited, we recurse on the subterm M, supplying type A by inheritance:

    • If it fails, then ¬⊢M is evidence that Γ ⊢ M ↓ A does not hold. Evidence that Γ ⊢ (M ↓ A) ↑ A holds must have the form ⊢↓ ⊢M where ⊢M is evidence that Γ ⊢ M ↓ A holds, which yields a contradiction.

    • If it succeeds, then ⊢M is evidence that Γ ⊢ M ↓ A, and ⊢↓ ⊢M provides evidence that Γ ⊢ (M ↓ A) ↑ A.

We next consider the code for inheritance:
inherit Γ (ƛ x ⇒ N) `ℕ      =  no  (λ())
inherit Γ (ƛ x ⇒ N) (A ⇒ B) with inherit (Γ , x ⦂ A) N B
... | no ¬⊢N                =  no  (λ{ (⊢ƛ ⊢N)  →  ¬⊢N ⊢N })
... | yes ⊢N                =  yes (⊢ƛ ⊢N)
inherit Γ `zero `ℕ          =  yes ⊢zero
inherit Γ `zero (A ⇒ B)     =  no  (λ())
inherit Γ (`suc M) `ℕ with inherit Γ M `ℕ
... | no ¬⊢M                =  no  (λ{ (⊢suc ⊢M)  →  ¬⊢M ⊢M })
... | yes ⊢M                =  yes (⊢suc ⊢M)
inherit Γ (`suc M) (A ⇒ B)  =  no  (λ())
inherit Γ (`case L [zero⇒ M |suc x ⇒ N ]) A with synthesize Γ L
... | no ¬∃                 =  no  (λ{ (⊢case ⊢L  _ _) → ¬∃ ⟨ `ℕ , ⊢L ⟩})
... | yes ⟨ _ ⇒ _ , ⊢L ⟩    =  no  (λ{ (⊢case ⊢L′ _ _) → ℕ≢⇒ (uniq-↑ ⊢L′ ⊢L) })
... | yes ⟨ `ℕ ,    ⊢L ⟩ with inherit Γ M A
...    | no ¬⊢M             =  no  (λ{ (⊢case _ ⊢M _) → ¬⊢M ⊢M })
...    | yes ⊢M with inherit (Γ , x ⦂ `ℕ) N A
...       | no ¬⊢N          =  no  (λ{ (⊢case _ _ ⊢N) → ¬⊢N ⊢N })
...       | yes ⊢N          =  yes (⊢case ⊢L ⊢M ⊢N)
inherit Γ (μ x ⇒ N) A with inherit (Γ , x ⦂ A) N A
... | no ¬⊢N                =  no  (λ{ (⊢μ ⊢N) → ¬⊢N ⊢N })
... | yes ⊢N                =  yes (⊢μ ⊢N)
inherit Γ (M ↑) B with synthesize Γ M
... | no  ¬∃                =  no  (λ{ (⊢↑ ⊢M _) → ¬∃ ⟨ _ , ⊢M ⟩ })
... | yes ⟨ A , ⊢M ⟩ with A ≟Tp B
...   | no  A≢B             =  no  (¬switch ⊢M A≢B)
...   | yes A≡B             =  yes (⊢↑ ⊢M A≡B)

We consider only the cases for abstraction and and for switching from inherited to synthesized:

  • If the term is an abstraction ƛ x ⇒ N and the inherited type is `ℕ, then it is trivial that Γ ⊢ (ƛ x ⇒ N) ↓ `ℕ cannot hold.

  • If the term is an abstraction ƛ x ⇒ N and the inherited type is A ⇒ B, then we recurse with context Γ , x ⦂ A on subterm N inheriting type B:

    • If it fails, then ¬⊢N is evidence that Γ , x ⦂ A ⊢ N ↓ B does not hold. Evidence that Γ ⊢ (ƛ x ⇒ N) ↓ A ⇒ B holds must have the form ⊢ƛ ⊢N where ⊢N is evidence that Γ , x ⦂ A ⊢ N ↓ B, which yields a contradiction.

    • If it succeeds, then ⊢N is evidence that Γ , x ⦂ A ⊢ N ↓ B holds, and ⊢ƛ ⊢N provides evidence that Γ ⊢ (ƛ x ⇒ N) ↓ A ⇒ B.

  • If the term is a switch M ↑ from inherited to synthesised, we recurse on the subterm M:

    • If it fails, then ¬∃ is evidence there is no A such that Γ ⊢ M ↑ A holds. Evidence that Γ ⊢ (M ↑) ↓ B holds must have the form ⊢↑ ⊢M _ where ⊢M is evidence that Γ ⊢ M ↑ _, which yields a contradiction.

    • If it succeeds, then ⊢M is evidence that Γ ⊢ M ↑ A holds. We apply _≟Tp_ do decide whether A and B are equal:

      • If it fails, then A≢B is evidence that A ≢ B. By ¬switch applied to ⊢M and A≢B it follow that Γ ⊢ (M ↑) ↓ B cannot hold.

      • If it succeeds, then A≡B is evidence that A ≡ B, and ⊢↑ ⊢M A≡B provides evidence that Γ ⊢ (M ↑) ↓ B.

The remaining cases are similar, and their code can pretty much be read directly from the corresponding typing rules.

Testing the example terms

First, we copy the smart constructor S′ introduced earlier that makes it easy to access a variable in a context:
S′ : ∀ {Γ x y A B}
   → {x≢y : False (x ≟ y)}
   → Γ ∋ x ⦂ A
     ------------------
   → Γ , y ⦂ B ∋ x ⦂ A

S′ {x≢y = x≢y} x = S (toWitnessFalse x≢y) x
Here is the result of typing two plus two on naturals:
⊢2+2 : ∅ ⊢ 2+2 ↑ `ℕ
⊢2+2 =
  (⊢↓
   (⊢μ
    (⊢ƛ
     (⊢ƛ
      (⊢case (⊢` (S′ Z)) (⊢↑ (⊢` Z) refl)
       (⊢suc
        (⊢↑
         (⊢`
          (S′
           (S′
            (S′ Z)))
          · ⊢↑ (⊢` Z) refl
          · ⊢↑ (⊢` (S′ Z)) refl)
         refl))))))
   · ⊢suc (⊢suc ⊢zero)
   · ⊢suc (⊢suc ⊢zero))
We confirm that synthesis on the relevant term returns natural as the type and the above derivation:
_ : synthesize ∅ 2+2 ≡ yes ⟨ `ℕ , ⊢2+2 ⟩
_ = refl

Indeed, the above derivation was computed by evaluating the term on the left, with minor editing of the result. The only editing required was to use the smart constructor S′ to obtain the evidence that two variable names (as strings) are unequal (which it cannot print nor read).

Here is the result of typing two plus two with Church numerals:
⊢2+2ᶜ : ∅ ⊢ 2+2ᶜ ↑ `ℕ
⊢2+2ᶜ =
  ⊢↓
  (⊢ƛ
   (⊢ƛ
    (⊢ƛ
     (⊢ƛ
      (⊢↑
       (⊢`
        (S′
         (S′
          (S′ Z)))
        · ⊢↑ (⊢` (S′ Z)) refl
        ·
        ⊢↑
        (⊢`
         (S′
          (S′ Z))
         · ⊢↑ (⊢` (S′ Z)) refl
         · ⊢↑ (⊢` Z) refl)
        refl)
       refl)))))
  ·
  ⊢ƛ
  (⊢ƛ
   (⊢↑
    (⊢` (S′ Z) ·
     ⊢↑ (⊢` (S′ Z) · ⊢↑ (⊢` Z) refl)
     refl)
    refl))
  ·
  ⊢ƛ
  (⊢ƛ
   (⊢↑
    (⊢` (S′ Z) ·
     ⊢↑ (⊢` (S′ Z) · ⊢↑ (⊢` Z) refl)
     refl)
    refl))
  · ⊢ƛ (⊢suc (⊢↑ (⊢` Z) refl))
  · ⊢zero
We confirm that synthesis on the relevant term returns natural as the type and the above derivation:
_ : synthesize ∅ 2+2ᶜ ≡ yes ⟨ `ℕ , ⊢2+2ᶜ ⟩
_ = refl

Again, the above derivation was computed by evaluating the term on the left and editing.

Testing the error cases

It is important not just to check that code works as intended, but also that it fails as intended. Here are checks for several possible errors:

Unbound variable:
_ : synthesize ∅ ((ƛ "x" ⇒ ` "y" ↑) ↓ (`ℕ ⇒ `ℕ)) ≡ no _
_ = refl
Argument in application is ill typed:
_ : synthesize ∅ (plus · sucᶜ) ≡ no _
_ = refl
Function in application is ill typed:
_ : synthesize ∅ (plus · sucᶜ · two) ≡ no _
_ = refl
Function in application has type natural:
_ : synthesize ∅ ((two ↓ `ℕ) · two) ≡ no _
_ = refl
Abstraction inherits type natural:
_ : synthesize ∅ (twoᶜ ↓ `ℕ) ≡ no _
_ = refl
Zero inherits a function type:
_ : synthesize ∅ (`zero ↓ `ℕ ⇒ `ℕ) ≡ no _
_ = refl
Successor inherits a function type:
_ : synthesize ∅ (two ↓ `ℕ ⇒ `ℕ) ≡ no _
_ = refl
Successor of an ill-typed term:
_ : synthesize ∅ (`suc twoᶜ ↓ `ℕ) ≡ no _
_ = refl
Case of a term with a function type:
_ : synthesize ∅
      ((`case (twoᶜ ↓ Ch) [zero⇒ `zero |suc "x" ⇒ ` "x" ↑ ] ↓ `ℕ) ) ≡ no _
_ = refl
Case of an ill-typed term:
_ : synthesize ∅
      ((`case (twoᶜ ↓ `ℕ) [zero⇒ `zero |suc "x" ⇒ ` "x" ↑ ] ↓ `ℕ) ) ≡ no _
_ = refl
Inherited and synthesised types disagree in a switch:
_ : synthesize ∅ (((ƛ "x" ⇒ ` "x" ↑) ↓ `ℕ ⇒ (`ℕ ⇒ `ℕ))) ≡ no _
_ = refl

Erasure

From the evidence that a decorated term has the correct type it is easy to extract the corresponding intrinsically-typed term. We use the name DB to refer to the code in Chapter DeBruijn. It is easy to define an erasure function that takes an extrinsic type judgment into the corresponding intrinsically-typed term.

First, we give code to erase a type:
∥_∥Tp : Type → DB.Type
∥ `ℕ ∥Tp             =  DB.`ℕ
∥ A ⇒ B ∥Tp          =  ∥ A ∥Tp DB.⇒ ∥ B ∥Tp

It simply renames to the corresponding constructors in module DB.

Next, we give the code to erase a context:
∥_∥Cx : Context → DB.Context
∥ ∅ ∥Cx              =  DB.∅
∥ Γ , x ⦂ A ∥Cx      =  ∥ Γ ∥Cx DB., ∥ A ∥Tp

It simply drops the variable names.

Next, we give the code to erase a lookup judgment:
∥_∥∋ : ∀ {Γ x A} → Γ ∋ x ⦂ A → ∥ Γ ∥Cx DB.∋ ∥ A ∥Tp
∥ Z ∥∋               =  DB.Z
∥ S x≢ ∋x ∥∋         =  DB.S ∥ ∋x ∥∋

It simply drops the evidence that variable names are distinct.

Finally, we give the code to erase a typing judgment. Just as there are two mutually recursive typing judgments, there are two mutually recursive erasure functions:
∥_∥⁺ : ∀ {Γ M A} → Γ ⊢ M ↑ A → ∥ Γ ∥Cx DB.⊢ ∥ A ∥Tp
∥_∥⁻ : ∀ {Γ M A} → Γ ⊢ M ↓ A → ∥ Γ ∥Cx DB.⊢ ∥ A ∥Tp

∥ ⊢` ⊢x ∥⁺           =  DB.` ∥ ⊢x ∥∋
∥ ⊢L · ⊢M ∥⁺         =  ∥ ⊢L ∥⁺ DB.· ∥ ⊢M ∥⁻
∥ ⊢↓ ⊢M ∥⁺           =  ∥ ⊢M ∥⁻

∥ ⊢ƛ ⊢N ∥⁻           =  DB.ƛ ∥ ⊢N ∥⁻
∥ ⊢zero ∥⁻           =  DB.`zero
∥ ⊢suc ⊢M ∥⁻         =  DB.`suc ∥ ⊢M ∥⁻
∥ ⊢case ⊢L ⊢M ⊢N ∥⁻  =  DB.case ∥ ⊢L ∥⁺ ∥ ⊢M ∥⁻ ∥ ⊢N ∥⁻
∥ ⊢μ ⊢M ∥⁻           =  DB.μ ∥ ⊢M ∥⁻
∥ ⊢↑ ⊢M refl ∥⁻      =  ∥ ⊢M ∥⁺

Erasure replaces constructors for each typing judgment by the corresponding term constructor from DB. The constructors that correspond to switching from synthesized to inherited or vice versa are dropped.

We confirm that the erasure of the type derivations in this chapter yield the corresponding intrinsically-typed terms from the earlier chapter:
_ : ∥ ⊢2+2 ∥⁺ ≡ DB.2+2
_ = refl

_ : ∥ ⊢2+2ᶜ ∥⁺ ≡ DB.2+2ᶜ
_ = refl

Thus, we have confirmed that bidirectional type inference converts decorated versions of the lambda terms from Chapter Lambda to the intrinsically-typed terms of Chapter DeBruijn.

Apply inference to your decorated definition of multiplication from exercise bidirectional-mul, and show that erasure of the inferred typing yields your definition of multiplication from Chapter DeBruijn.

-- Your code goes here

Using your rules from exercise bidirectional-products, extend bidirectional inference to include products. Also extend erasure.

-- Your code goes here

Exercise inference-rest (stretch)

Using your rules from exercise bidirectional-rest, extend bidirectional inference to include the rest of the constructs from Chapter More. Also extend erasure.

-- Your code goes here

Bidirectional inference in Agda

Agda itself uses bidirectional inference. This explains why constructors can be overloaded while other defined names cannot — here by overloaded we mean that the same name can be used for constructors of different types. Constructors are typed by inheritance, and so the type is available when resolving the constructor, whereas variables are typed by synthesis, and so each variable must have a unique type.

Most top-level definitions in Agda are of functions, which are typed by inheritance, which is why Agda requires a type declaration for those definitions. A definition with a right-hand side that is a term typed by synthesis, such as an application, does not require a type declaration.
answer = 6 * 7

Unicode

This chapter uses the following unicode:

↓  U+2193:  DOWNWARDS ARROW (\d)
↑  U+2191:  UPWARDS ARROW (\u)
∥  U+2225:  PARALLEL TO (\||)