module plfa.part3.Adequacy where

Introduction

Having proved a preservation property in the last chapter, a natural next step would be to prove progress. That is, to prove a property of the form

If γ ⊢ M ↓ v, then either M is a lambda abstraction or M —→ N for some N.

Such a property would tell us that having a denotation implies either reduction to normal form or divergence. This is indeed true, but we can prove a much stronger property! In fact, having a denotation that is a function value (not ⊥) implies reduction to a lambda abstraction.

This stronger property, reformulated a bit, is known as adequacy. That is, if a term M is denotationally equal to a lambda abstraction, then M reduces to a lambda abstraction.

ℰ M ≃ ℰ (ƛ N)  implies M —↠ ƛ N' for some N'

Recall that ℰ M ≃ ℰ (ƛ N) is equivalent to saying that γ ⊢ M ↓ (v ↦ w) for some v and w. We will show that γ ⊢ M ↓ (v ↦ w) implies multi-step reduction a lambda abstraction. The recursive structure of the derivations for γ ⊢ M ↓ (v ↦ w) are completely different from the structure of multi-step reductions, so a direct proof would be challenging. However, The structure of γ ⊢ M ↓ (v ↦ w) closer to that of BigStep call-by-name evaluation. Further, we already proved that big-step evaluation implies multi-step reduction to a lambda (cbn→reduce). So we shall prove that γ ⊢ M ↓ (v ↦ w) implies that γ' ⊢ M ⇓ c, where c is a closure (a term paired with an environment), γ' is an environment that maps variables to closures, and γ and γ' are appropriate related. The proof will be an induction on the derivation of γ ⊢ M ↓ v, and to strengthen the induction hypothesis, we will relate semantic values to closures using a logical relation 𝕍.

The rest of this chapter is organized as follows.

  • To make the 𝕍 relation down-closed with respect to ⊑, we must loosen the requirement that M result in a function value and instead require that M result in a value that is greater than or equal to a function value. We establish several properties about being ``greater than a function’’.

  • We define the logical relation 𝕍 that relates values and closures, and extend it to a relation on terms 𝔼 and environments 𝔾. We prove several lemmas that culminate in the property that if 𝕍 v c and v′ ⊑ v, then 𝕍 v′ c.

  • We prove the main lemma, that if 𝔾 γ γ' and γ ⊢ M ↓ v, then 𝔼 v (clos M γ').

  • We prove adequacy as a corollary to the main lemma.

Imports

import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; _≢_; refl; trans; sym; cong; cong₂; cong-app)
open import Data.Product using (_×_; Σ; Σ-syntax; ∃; ∃-syntax; proj₁; proj₂)
  renaming (_,_ to ⟨_,_⟩)
open import Data.Sum
open import Relation.Nullary using (¬_)
open import Relation.Nullary.Negation using (contradiction)
open import Data.Empty using (⊥-elim) renaming (⊥ to Bot)
open import Data.Unit
open import Relation.Nullary using (Dec; yes; no)
open import Function using (_∘_)
open import plfa.part2.Untyped
     using (Context; _⊢_; ★; _∋_; ∅; _,_; Z; S_; `_; ƛ_; _·_;
            rename; subst; ext; exts; _[_]; subst-zero;
            _—↠_; _—→⟨_⟩_; _∎; _—→_; ξ₁; ξ₂; β; ζ)
open import plfa.part2.Substitution using (ids; sub-id)
open import plfa.part2.BigStep
     using (Clos; clos; ClosEnv; ∅'; _,'_; _⊢_⇓_; ⇓-var; ⇓-lam; ⇓-app;
            ⇓-determ; cbn→reduce)
open import plfa.part3.Denotational
     using (Value; Env; `∅; _`,_; _↦_; _⊑_; _⊢_↓_; ⊥; all-funs∈; _⊔_; ∈→⊑;
            var; ↦-elim; ↦-intro; ⊔-intro; ⊥-intro; sub; ℰ; _≃_; _iff_;
            ⊑-trans; ⊑-conj-R1; ⊑-conj-R2; ⊑-conj-L; ⊑-refl; ⊑-fun; ⊑-bot;
            ⊑-dist; sub-inv-fun)
open import plfa.part3.Soundness using (soundness)

The property of being greater or equal to a function

We define the following short-hand for saying that a value is greater-than or equal to a function value.

above-fun : Value → Set
above-fun u = Σ[ v ∈ Value ] Σ[ w ∈ Value ] v ↦ w ⊑ u

If a value u is greater than a function, then an even greater value u' is too.

above-fun-⊑ : ∀{u u' : Value}
      → above-fun u → u ⊑ u'
        -------------------
      → above-fun u'
above-fun-⊑ ⟨ v , ⟨ w , lt' ⟩ ⟩ lt = ⟨ v , ⟨ w , ⊑-trans lt' lt ⟩ ⟩

The bottom value ⊥ is not greater than a function.

above-fun⊥ : ¬ above-fun ⊥
above-fun⊥ ⟨ v , ⟨ w , lt ⟩ ⟩
    with sub-inv-fun lt
... | ⟨ Γ , ⟨ f , ⟨ Γ⊆⊥ , ⟨ lt1 , lt2 ⟩ ⟩ ⟩ ⟩
    with all-funs∈ f
... | ⟨ A , ⟨ B , m ⟩ ⟩
    with Γ⊆⊥ m
... | ()

If the join of two values u and u' is greater than a function, then at least one of them is too.

above-fun-⊔ : ∀{u u'}
           → above-fun (u ⊔ u')
           → above-fun u ⊎ above-fun u'
above-fun-⊔{u}{u'} ⟨ v , ⟨ w , v↦w⊑u⊔u' ⟩ ⟩
    with sub-inv-fun v↦w⊑u⊔u'
... | ⟨ Γ , ⟨ f , ⟨ Γ⊆u⊔u' , ⟨ lt1 , lt2 ⟩ ⟩ ⟩ ⟩
    with all-funs∈ f
... | ⟨ A , ⟨ B , m ⟩ ⟩
    with Γ⊆u⊔u' m
... | inj₁ x = inj₁ ⟨ A , ⟨ B , (∈→⊑ x) ⟩ ⟩
... | inj₂ x = inj₂ ⟨ A , ⟨ B , (∈→⊑ x) ⟩ ⟩

On the other hand, if neither of u and u' is greater than a function, then their join is also not greater than a function.

not-above-fun-⊔ : ∀{u u' : Value}
               → ¬ above-fun u → ¬ above-fun u'
               → ¬ above-fun (u ⊔ u')
not-above-fun-⊔ naf1 naf2 af12
    with above-fun-⊔ af12
... | inj₁ af1 = contradiction af1 naf1
... | inj₂ af2 = contradiction af2 naf2

The converse is also true. If the join of two values is not above a function, then neither of them is individually.

not-above-fun-⊔-inv : ∀{u u' : Value} → ¬ above-fun (u ⊔ u')
              → ¬ above-fun u × ¬ above-fun u'
not-above-fun-⊔-inv af = ⟨ f af , g af ⟩
  where
    f : ∀{u u' : Value} → ¬ above-fun (u ⊔ u') → ¬ above-fun u
    f{u}{u'} af12 ⟨ v , ⟨ w , lt ⟩ ⟩ =
        contradiction ⟨ v , ⟨ w , ⊑-conj-R1 lt ⟩ ⟩ af12
    g : ∀{u u' : Value} → ¬ above-fun (u ⊔ u') → ¬ above-fun u'
    g{u}{u'} af12 ⟨ v , ⟨ w , lt ⟩ ⟩ =
        contradiction ⟨ v , ⟨ w , ⊑-conj-R2 lt ⟩ ⟩ af12

The property of being greater than a function value is decidable, as exhibited by the following function.

above-fun? : (v : Value) → Dec (above-fun v)
above-fun? ⊥ = no above-fun⊥
above-fun? (v ↦ w) = yes ⟨ v , ⟨ w , ⊑-refl ⟩ ⟩
above-fun? (u ⊔ u')
    with above-fun? u | above-fun? u'
... | yes ⟨ v , ⟨ w , lt ⟩ ⟩ | _ = yes ⟨ v , ⟨ w , (⊑-conj-R1 lt) ⟩ ⟩
... | no _ | yes ⟨ v , ⟨ w , lt ⟩ ⟩ = yes ⟨ v , ⟨ w , (⊑-conj-R2 lt) ⟩ ⟩
... | no x | no y = no (not-above-fun-⊔ x y)

Relating values to closures

Next we relate semantic values to closures. The relation 𝕍 is for closures whose term is a lambda abstraction, i.e., in weak-head normal form (WHNF). The relation 𝔼 is for any closure. Roughly speaking, 𝔼 v c will hold if, when v is greater than a function value, c evaluates to a closure c' in WHNF and 𝕍 v c'. Regarding 𝕍 v c, it will hold when c is in WHNF, and if v is a function, the body of c evaluates according to v.

𝕍 : Value → Clos → Set
𝔼 : Value → Clos → Set

We define 𝕍 as a function from values and closures to Set and not as a data type because it is mutually recursive with 𝔼 in a negative position (to the left of an implication). We first perform case analysis on the term in the closure. If the term is a variable or application, then 𝕍 is false (Bot). If the term is a lambda abstraction, we define 𝕍 by recursion on the value, which we describe below.

𝕍 v (clos (` x₁) γ) = Bot
𝕍 v (clos (M · M₁) γ) = Bot
𝕍 ⊥ (clos (ƛ M) γ) = ⊤
𝕍 (v ↦ w) (clos (ƛ N) γ) =
    (∀{c : Clos} → 𝔼 v c → above-fun w → Σ[ c' ∈ Clos ]
        (γ ,' c) ⊢ N ⇓ c'  ×  𝕍 w c')
𝕍 (u ⊔ v) (clos (ƛ N) γ) = 𝕍 u (clos (ƛ N) γ) × 𝕍 v (clos (ƛ N) γ)

  • If the value is ⊥, then the result is true (⊤).

  • If the value is a join (u ⊔ v), then the result is the pair (conjunction) of 𝕍 is true for both u and v.

  • The important case is for a function value v ↦ w and closure clos (ƛ N) γ. Given any closure c such that 𝔼 v c, if w is greater than a function, then N evaluates (with γ extended with c) to some closure c' and we have 𝕍 w c'.

The definition of 𝔼 is straightforward. If v is a greater than a function, then M evaluates to a closure related to v.

𝔼 v (clos M γ') = above-fun v → Σ[ c ∈ Clos ] γ' ⊢ M ⇓ c × 𝕍 v c

The proof of the main lemma is by induction on γ ⊢ M ↓ v, so it goes underneath lambda abstractions and must therefore reason about open terms (terms with variables). So we must relate environments of semantic values to environments of closures. In the following, 𝔾 relates γ to γ' if the corresponding values and closures are related by 𝔼.

𝔾 : ∀{Γ} → Env Γ → ClosEnv Γ → Set
𝔾 {Γ} γ γ' = ∀{x : Γ ∋ ★} → 𝔼 (γ x) (γ' x)

𝔾-∅ : 𝔾 `∅ ∅'
𝔾-∅ {()}

𝔾-ext : ∀{Γ}{γ : Env Γ}{γ' : ClosEnv Γ}{v c}
      → 𝔾 γ γ' → 𝔼 v c → 𝔾 (γ `, v) (γ' ,' c)
𝔾-ext {Γ} {γ} {γ'} g e {Z} = e
𝔾-ext {Γ} {γ} {γ'} g e {S x} = g

We need a few properties of the 𝕍 and 𝔼 relations. The first is that a closure in the 𝕍 relation must be in weak-head normal form. We define WHNF has follows.

data WHNF : ∀ {Γ A} → Γ ⊢ A → Set where
  ƛ_ : ∀ {Γ} {N : Γ , ★ ⊢ ★}
     → WHNF (ƛ N)

The proof goes by cases on the term in the closure.

𝕍→WHNF : ∀{Γ}{γ : ClosEnv Γ}{M : Γ ⊢ ★}{v}
       → 𝕍 v (clos M γ) → WHNF M
𝕍→WHNF {M = ` x} {v} ()
𝕍→WHNF {M = ƛ N} {v} vc = ƛ_
𝕍→WHNF {M = L · M} {v} ()

Next we have an introduction rule for 𝕍 that mimics the ⊔-intro rule. If both u and v are related to a closure c, then their join is too.

𝕍⊔-intro : ∀{c u v}
         → 𝕍 u c → 𝕍 v c
           ---------------
         → 𝕍 (u ⊔ v) c
𝕍⊔-intro {clos (` x) γ} () vc
𝕍⊔-intro {clos (ƛ N) γ} uc vc = ⟨ uc , vc ⟩
𝕍⊔-intro {clos (L · M) γ} () vc

In a moment we prove that 𝕍 is preserved when going from a greater value to a lesser value: if 𝕍 v c and v' ⊑ v, then 𝕍 v' c. This property, named 𝕍-sub, is needed by the main lemma in the case for the sub rule.

To prove 𝕍-sub, we in turn need the following property concerning values that are not greater than a function, that is, values that are equivalent to ⊥. In such cases, 𝕍 v (clos (ƛ N) γ') is trivially true.

not-above-fun-𝕍 : ∀{v : Value}{Γ}{γ' : ClosEnv Γ}{N : Γ , ★ ⊢ ★ }
    → ¬ above-fun v
      -------------------
    → 𝕍 v (clos (ƛ N) γ')
not-above-fun-𝕍 {⊥} af = tt
not-above-fun-𝕍 {v ↦ v'} af = contradiction ⟨ v , ⟨ v' , ⊑-refl ⟩ ⟩ af
not-above-fun-𝕍 {v₁ ⊔ v₂} af
    with not-above-fun-⊔-inv af
... | ⟨ af1 , af2 ⟩ = ⟨ not-above-fun-𝕍 af1 , not-above-fun-𝕍 af2 ⟩

The proofs of 𝕍-sub and 𝔼-sub are intertwined.

sub-𝕍 : ∀{c : Clos}{v v'} → 𝕍 v c → v' ⊑ v → 𝕍 v' c
sub-𝔼 : ∀{c : Clos}{v v'} → 𝔼 v c → v' ⊑ v → 𝔼 v' c

We prove 𝕍-sub by case analysis on the closure’s term, to dispatch the cases for variables and application. We then proceed by induction on v' ⊑ v. We describe each case below.

sub-𝕍 {clos (` x) γ} {v} () lt
sub-𝕍 {clos (L · M) γ} () lt
sub-𝕍 {clos (ƛ N) γ} vc ⊑-bot = tt
sub-𝕍 {clos (ƛ N) γ} vc (⊑-conj-L lt1 lt2) = ⟨ (sub-𝕍 vc lt1) , sub-𝕍 vc lt2 ⟩
sub-𝕍 {clos (ƛ N) γ} ⟨ vv1 , vv2 ⟩ (⊑-conj-R1 lt) = sub-𝕍 vv1 lt
sub-𝕍 {clos (ƛ N) γ} ⟨ vv1 , vv2 ⟩ (⊑-conj-R2 lt) = sub-𝕍 vv2 lt
sub-𝕍 {clos (ƛ N) γ} vc (⊑-trans{v = v₂} lt1 lt2) = sub-𝕍 (sub-𝕍 vc lt2) lt1
sub-𝕍 {clos (ƛ N) γ} vc (⊑-fun lt1 lt2) ev1 sf
    with vc (sub-𝔼 ev1 lt1) (above-fun-⊑ sf lt2)
... | ⟨ c , ⟨ Nc , v4 ⟩ ⟩ = ⟨ c , ⟨ Nc , sub-𝕍 v4 lt2 ⟩ ⟩
sub-𝕍 {clos (ƛ N) γ} {v ↦ w ⊔ v ↦ w'} ⟨ vcw , vcw' ⟩ ⊑-dist ev1c sf
    with above-fun? w | above-fun? w'
... | yes af2 | yes af3
    with vcw ev1c af2 | vcw' ev1c af3
... | ⟨ clos L δ , ⟨ L⇓c₂ , 𝕍w ⟩ ⟩
    | ⟨ c₃ , ⟨ L⇓c₃ , 𝕍w' ⟩ ⟩ rewrite ⇓-determ L⇓c₃ L⇓c₂ with 𝕍→WHNF 𝕍w
... | ƛ_ =
      ⟨ clos L δ , ⟨ L⇓c₂ , ⟨ 𝕍w , 𝕍w' ⟩ ⟩ ⟩
sub-𝕍 {c} {v ↦ w ⊔ v ↦ w'} ⟨ vcw , vcw' ⟩  ⊑-dist ev1c sf
    | yes af2 | no naf3
    with vcw ev1c af2
... | ⟨ clos {Γ'} L γ₁ , ⟨ L⇓c2 , 𝕍w ⟩ ⟩
    with 𝕍→WHNF 𝕍w
... | ƛ_ {N = N'} =
      let 𝕍w' = not-above-fun-𝕍{w'}{Γ'}{γ₁}{N'} naf3 in
      ⟨ clos (ƛ N') γ₁ , ⟨ L⇓c2 , 𝕍⊔-intro 𝕍w 𝕍w' ⟩ ⟩
sub-𝕍 {c} {v ↦ w ⊔ v ↦ w'} ⟨ vcw , vcw' ⟩ ⊑-dist ev1c sf
    | no naf2 | yes af3
    with vcw' ev1c af3
... | ⟨ clos {Γ'} L γ₁ , ⟨ L⇓c3 , 𝕍w'c ⟩ ⟩
    with 𝕍→WHNF 𝕍w'c
... | ƛ_ {N = N'} =
      let 𝕍wc = not-above-fun-𝕍{w}{Γ'}{γ₁}{N'} naf2 in
      ⟨ clos (ƛ N') γ₁ , ⟨ L⇓c3 , 𝕍⊔-intro 𝕍wc 𝕍w'c ⟩ ⟩
sub-𝕍 {c} {v ↦ w ⊔ v ↦ w'} ⟨ vcw , vcw' ⟩ ⊑-dist ev1c ⟨ v' , ⟨ w'' , lt ⟩ ⟩
    | no naf2 | no naf3
    with above-fun-⊔ ⟨ v' , ⟨ w'' , lt ⟩ ⟩
... | inj₁ af2 = contradiction af2 naf2
... | inj₂ af3 = contradiction af3 naf3

  • Case ⊑-bot. We immediately have 𝕍 ⊥ (clos (ƛ N) γ).

  • Case ⊑-conj-L.

      v₁' ⊑ v     v₂' ⊑ v
  -------------------
  (v₁' ⊔ v₂') ⊑ v

    The induction hypotheses gives us 𝕍 v₁' (clos (ƛ N) γ) and 𝕍 v₂' (clos (ƛ N) γ), which is all we need for this case.

  • Case ⊑-conj-R1.

      v' ⊑ v₁
  -------------
  v' ⊑ (v₁ ⊔ v₂)

    The induction hypothesis gives us 𝕍 v' (clos (ƛ N) γ).

  • Case ⊑-conj-R2.

      v' ⊑ v₂
  -------------
  v' ⊑ (v₁ ⊔ v₂)

    Again, the induction hypothesis gives us 𝕍 v' (clos (ƛ N) γ).

  • Case ⊑-trans.

      v' ⊑ v₂   v₂ ⊑ v
  -----------------
       v' ⊑ v

    The induction hypothesis for v₂ ⊑ v gives us 𝕍 v₂ (clos (ƛ N) γ). We apply the induction hypothesis for v' ⊑ v₂ to conclude that 𝕍 v' (clos (ƛ N) γ).

  • Case ⊑-dist. This case is the most difficult. We have

      𝕍 (v ↦ w) (clos (ƛ N) γ)
  𝕍 (v ↦ w') (clos (ƛ N) γ)

    and need to show that

      𝕍 (v ↦ (w ⊔ w')) (clos (ƛ N) γ)

    Let c be an arbitrary closure such that 𝔼 v c. Assume w ⊔ w' is greater than a function. Unfortunately, this does not mean that both w and w' are above functions. But thanks to the lemma above-fun-⊔, we know that at least one of them is greater than a function.

    • Suppose both of them are greater than a function. Then we have γ ⊢ N ⇓ clos L δ and 𝕍 w (clos L δ). We also have γ ⊢ N ⇓ c₃ and 𝕍 w' c₃. Because the big-step semantics is deterministic, we have c₃ ≡ clos L δ. Also, from 𝕍 w (clos L δ) we know that L ≡ ƛ N' for some N'. We conclude that 𝕍 (w ⊔ w') (clos (ƛ N') δ).

    • Suppose one of them is greater than a function and the other is not: say above-fun w and ¬ above-fun w'. Then from 𝕍 (v ↦ w) (clos (ƛ N) γ) we have γ ⊢ N ⇓ clos L γ₁ and 𝕍 w (clos L γ₁). From this we have L ≡ ƛ N' for some N'. Meanwhile, from ¬ above-fun w' we have 𝕍 w' (clos L γ₁). We conclude that 𝕍 (w ⊔ w') (clos (ƛ N') γ₁).

The proof of sub-𝔼 is direct and explained below.

sub-𝔼 {clos M γ} {v} {v'} 𝔼v v'⊑v fv'
    with 𝔼v (above-fun-⊑ fv' v'⊑v)
... | ⟨ c , ⟨ M⇓c , 𝕍v ⟩ ⟩ =
      ⟨ c , ⟨ M⇓c , sub-𝕍 𝕍v v'⊑v ⟩ ⟩

From above-fun v' and v' ⊑ v we have above-fun v. Then with 𝔼 v c we obtain a closure c such that γ ⊢ M ⇓ c and 𝕍 v c. We conclude with an application of sub-𝕍 with v' ⊑ v to show 𝕍 v' c.

Programs with function denotation terminate via call-by-name

The main lemma proves that if a term has a denotation that is above a function, then it terminates via call-by-name. More formally, if γ ⊢ M ↓ v and 𝔾 γ γ', then 𝔼 v (clos M γ'). The proof is by induction on the derivation of γ ⊢ M ↓ v we discuss each case below.

The following lemma, kth-x, is used in the case for the var rule.

kth-x : ∀{Γ}{γ' : ClosEnv Γ}{x : Γ ∋ ★}
     → Σ[ Δ ∈ Context ] Σ[ δ ∈ ClosEnv Δ ] Σ[ M ∈ Δ ⊢ ★ ]
                 γ' x ≡ clos M δ
kth-x{γ' = γ'}{x = x} with γ' x
... | clos{Γ = Δ} M δ = ⟨ Δ , ⟨ δ , ⟨ M , refl ⟩ ⟩ ⟩

↓→𝔼 : ∀{Γ}{γ : Env Γ}{γ' : ClosEnv Γ}{M : Γ ⊢ ★ }{v}
            → 𝔾 γ γ' → γ ⊢ M ↓ v → 𝔼 v (clos M γ')
↓→𝔼 {Γ} {γ} {γ'} 𝔾γγ' (var{x = x}) fγx
    with kth-x{Γ}{γ'}{x} | 𝔾γγ'{x = x}
... | ⟨ Δ , ⟨ δ , ⟨ M' , eq ⟩ ⟩ ⟩ | 𝔾γγ'x rewrite eq
    with 𝔾γγ'x fγx
... | ⟨ c , ⟨ M'⇓c , 𝕍γx ⟩ ⟩ =
      ⟨ c , ⟨ (⇓-var eq M'⇓c) , 𝕍γx ⟩ ⟩
↓→𝔼 {Γ} {γ} {γ'} 𝔾γγ' (↦-elim{L = L}{M = M}{v = v₁}{w = v} d₁ d₂) fv
    with ↓→𝔼 𝔾γγ' d₁ ⟨ v₁ , ⟨ v , ⊑-refl ⟩ ⟩
... | ⟨ clos L' δ , ⟨ L⇓L' , 𝕍v₁↦v ⟩ ⟩
    with 𝕍→WHNF 𝕍v₁↦v
... | ƛ_ {N = N}
    with 𝕍v₁↦v {clos M γ'} (↓→𝔼 𝔾γγ' d₂) fv
... | ⟨ c' , ⟨ N⇓c' , 𝕍v ⟩ ⟩ =
    ⟨ c' , ⟨ ⇓-app L⇓L' N⇓c' , 𝕍v ⟩ ⟩
↓→𝔼 {Γ} {γ} {γ'} 𝔾γγ' (↦-intro{N = N}{v = v}{w = w} d) fv↦w =
    ⟨ clos (ƛ N) γ' , ⟨ ⇓-lam , E ⟩ ⟩
    where E : {c : Clos} → 𝔼 v c → above-fun w
            → Σ[ c' ∈ Clos ] (γ' ,' c) ⊢ N ⇓ c'  ×  𝕍 w c'
          E {c} 𝔼vc fw = ↓→𝔼 (λ {x} → 𝔾-ext{Γ}{γ}{γ'} 𝔾γγ' 𝔼vc {x}) d fw
↓→𝔼 𝔾γγ' ⊥-intro f⊥ = ⊥-elim (above-fun⊥ f⊥)
↓→𝔼 𝔾γγ' (⊔-intro{v = v₁}{w = v₂} d₁ d₂) fv12
    with above-fun? v₁ | above-fun? v₂
... | yes fv1 | yes fv2
    with ↓→𝔼 𝔾γγ' d₁ fv1 | ↓→𝔼 𝔾γγ' d₂ fv2
... | ⟨ c₁ , ⟨ M⇓c₁ , 𝕍v₁ ⟩ ⟩ | ⟨ c₂ , ⟨ M⇓c₂ , 𝕍v₂ ⟩ ⟩
    rewrite ⇓-determ M⇓c₂ M⇓c₁ =
    ⟨ c₁ , ⟨ M⇓c₁ , 𝕍⊔-intro 𝕍v₁ 𝕍v₂ ⟩ ⟩
↓→𝔼 𝔾γγ' (⊔-intro{v = v₁}{w = v₂} d₁ d₂) fv12 | yes fv1 | no nfv2
    with ↓→𝔼 𝔾γγ' d₁ fv1
... | ⟨ clos {Γ'} M' γ₁ , ⟨ M⇓c₁ , 𝕍v₁ ⟩ ⟩
    with 𝕍→WHNF 𝕍v₁
... | ƛ_ {N = N} =
    let 𝕍v₂ = not-above-fun-𝕍{v₂}{Γ'}{γ₁}{N} nfv2 in
    ⟨ clos (ƛ N) γ₁ , ⟨ M⇓c₁ , 𝕍⊔-intro 𝕍v₁ 𝕍v₂ ⟩ ⟩
↓→𝔼 𝔾γγ' (⊔-intro{v = v₁}{w = v₂} d₁ d₂) fv12 | no nfv1  | yes fv2
    with ↓→𝔼 𝔾γγ' d₂ fv2
... | ⟨ clos {Γ'} M' γ₁ , ⟨ M'⇓c₂ , 𝕍2c ⟩ ⟩
    with 𝕍→WHNF 𝕍2c
... | ƛ_ {N = N} =
    let 𝕍1c = not-above-fun-𝕍{v₁}{Γ'}{γ₁}{N} nfv1 in
    ⟨ clos (ƛ N) γ₁ , ⟨ M'⇓c₂ , 𝕍⊔-intro 𝕍1c 𝕍2c ⟩ ⟩
↓→𝔼 𝔾γγ' (⊔-intro d₁ d₂) fv12 | no nfv1  | no nfv2
    with above-fun-⊔ fv12
... | inj₁ fv1 = contradiction fv1 nfv1
... | inj₂ fv2 = contradiction fv2 nfv2
↓→𝔼 {Γ} {γ} {γ'} {M} {v'} 𝔾γγ' (sub{v = v} d v'⊑v) fv'
    with ↓→𝔼 {Γ} {γ} {γ'} {M} 𝔾γγ' d (above-fun-⊑ fv' v'⊑v)
... | ⟨ c , ⟨ M⇓c , 𝕍v ⟩ ⟩ =
      ⟨ c , ⟨ M⇓c , sub-𝕍 𝕍v v'⊑v ⟩ ⟩

  • Case var. Looking up x in γ' yields some closure, clos M' δ, and from 𝔾 γ γ' we have 𝔼 (γ x) (clos M' δ). With the premise above-fun (γ x), we obtain a closure c such that δ ⊢ M' ⇓ c and 𝕍 (γ x) c. To conclude γ' ⊢ x ⇓ c via ⇓-var, we need γ' x ≡ clos M' δ, which is obvious, but it requires some Agda shananigans via the kth-x lemma to get our hands on it.

  • Case ↦-elim. We have γ ⊢ L · M ↓ v. The induction hypothesis for γ ⊢ L ↓ v₁ ↦ v gives us γ' ⊢ L ⇓ clos L' δ and 𝕍 v (clos L' δ). Of course, L' ≡ ƛ N for some N. By the induction hypothesis for γ ⊢ M ↓ v₁, we have 𝔼 v₁ (clos M γ'). Together with the premise above-fun v and 𝕍 v (clos L' δ), we obtain a closure c' such that δ ⊢ N ⇓ c' and 𝕍 v c'. We conclude that γ' ⊢ L · M ⇓ c' by rule ⇓-app.

  • Case ↦-intro. We have γ ⊢ ƛ N ↓ v ↦ w. We immediately have γ' ⊢ ƛ M ⇓ clos (ƛ M) γ' by rule ⇓-lam. But we also need to prove 𝕍 (v ↦ w) (clos (ƛ N) γ'). Let c by an arbitrary closure such that 𝔼 v c. Suppose v' is greater than a function value. We need to show that γ' , c ⊢ N ⇓ c' and 𝕍 v' c' for some c'. We prove this by the induction hypothesis for γ , v ⊢ N ↓ v' but we must first show that 𝔾 (γ , v) (γ' , c). We prove that by the lemma 𝔾-ext, using facts 𝔾 γ γ' and 𝔼 v c.

  • Case ⊥-intro. We have the premise above-fun ⊥, but that’s impossible.

  • Case ⊔-intro. We have γ ⊢ M ↓ (v₁ ⊔ v₂) and above-fun (v₁ ⊔ v₂) and need to show γ' ⊢ M ↓ c and 𝕍 (v₁ ⊔ v₂) c for some c. Again, by above-fun-⊔, at least one of v₁ or v₂ is greater than a function.

    • Suppose both v₁ and v₂ are greater than a function value. By the induction hypotheses for γ ⊢ M ↓ v₁ and γ ⊢ M ↓ v₂ we have γ' ⊢ M ⇓ c₁, 𝕍 v₁ c₁, γ' ⊢ M ⇓ c₂, and 𝕍 v₂ c₂ for some c₁ and c₂. Because ⇓ is deterministic, we have c₂ ≡ c₁. Then by 𝕍⊔-intro we conclude that 𝕍 (v₁ ⊔ v₂) c₁.

    • Without loss of generality, suppose v₁ is greater than a function value but v₂ is not. By the induction hypotheses for γ ⊢ M ↓ v₁, and using 𝕍→WHNF, we have γ' ⊢ M ⇓ clos (ƛ N) γ₁ and 𝕍 v₁ (clos (ƛ N) γ₁). Then because v₂ is not greater than a function, we also have 𝕍 v₂ (clos (ƛ N) γ₁). We conclude that 𝕍 (v₁ ⊔ v₂) (clos (ƛ N) γ₁).

  • Case sub. We have γ ⊢ M ↓ v, v' ⊑ v, and above-fun v'. We need to show that γ' ⊢ M ⇓ c and 𝕍 v' c for some c. We have above-fun v by above-fun-⊑, so the induction hypothesis for γ ⊢ M ↓ v gives us a closure c such that γ' ⊢ M ⇓ c and 𝕍 v c. We conclude that 𝕍 v' c by sub-𝕍.

Proof of denotational adequacy

From the main lemma we can directly show that ℰ M ≃ ℰ (ƛ N) implies that M big-steps to a lambda, i.e., ∅ ⊢ M ⇓ clos (ƛ N′) γ.

↓→⇓ : ∀{M : ∅ ⊢ ★}{N : ∅ , ★ ⊢ ★}  →  ℰ M ≃ ℰ (ƛ N)
         →  Σ[ Γ ∈ Context ] Σ[ N′ ∈ (Γ , ★ ⊢ ★) ] Σ[ γ ∈ ClosEnv Γ ]
            ∅' ⊢ M ⇓ clos (ƛ N′) γ
↓→⇓{M}{N} eq
    with ↓→𝔼 𝔾-∅ ((proj₂ (eq `∅ (⊥ ↦ ⊥))) (↦-intro ⊥-intro))
                 ⟨ ⊥ , ⟨ ⊥ , ⊑-refl ⟩ ⟩
... | ⟨ clos {Γ} M′ γ , ⟨ M⇓c , Vc ⟩ ⟩
    with 𝕍→WHNF Vc
... | ƛ_ {N = N′} =
    ⟨ Γ , ⟨ N′ , ⟨ γ , M⇓c ⟩  ⟩ ⟩

The proof goes as follows. We derive ∅ ⊢ ƛ N ↓ ⊥ ↦ ⊥ and then ℰ M ≃ ℰ (ƛ N) gives us ∅ ⊢ M ↓ ⊥ ↦ ⊥. We conclude by applying the main lemma to obtain ∅ ⊢ M ⇓ clos (ƛ N′) γ for some N′ and γ.

Now to prove the adequacy property. We apply the above lemma to obtain ∅ ⊢ M ⇓ clos (ƛ N′) γ and then apply cbn→reduce to conclude.

adequacy : ∀{M : ∅ ⊢ ★}{N : ∅ , ★ ⊢ ★}
   →  ℰ M ≃ ℰ (ƛ N)
   → Σ[ N′ ∈ (∅ , ★ ⊢ ★) ]
     (M —↠ ƛ N′)
adequacy{M}{N} eq
    with ↓→⇓ eq
... | ⟨ Γ , ⟨ N′ , ⟨ γ , M⇓ ⟩ ⟩ ⟩ =
    cbn→reduce M⇓

Call-by-name is equivalent to beta reduction

As promised, we return to the question of whether call-by-name evaluation is equivalent to beta reduction. In chapter BigStep we established the forward direction: that if call-by-name produces a result, then the program beta reduces to a lambda abstraction (cbn→reduce). We now prove the backward direction of the if-and-only-if, leveraging our results about the denotational semantics.

reduce→cbn : ∀ {M : ∅ ⊢ ★} {N : ∅ , ★ ⊢ ★}
           → M —↠ ƛ N
           → Σ[ Δ ∈ Context ] Σ[ N′ ∈ Δ , ★ ⊢ ★ ] Σ[ δ ∈ ClosEnv Δ ]
             ∅' ⊢ M ⇓ clos (ƛ N′) δ
reduce→cbn M—↠ƛN = ↓→⇓ (soundness M—↠ƛN)

Suppose M —↠ ƛ N. Soundness of the denotational semantics gives us ℰ M ≃ ℰ (ƛ N). Then by ↓→⇓ we conclude that ∅' ⊢ M ⇓ clos (ƛ N′) δ for some N′ and δ.

Putting the two directions of the if-and-only-if together, we establish that call-by-name evaluation is equivalent to beta reduction in the following sense.

cbn↔reduce : ∀ {M : ∅ ⊢ ★}
           → (Σ[ N ∈ ∅ , ★ ⊢ ★ ] (M —↠ ƛ N))
             iff
             (Σ[ Δ ∈ Context ] Σ[ N′ ∈ Δ , ★ ⊢ ★ ] Σ[ δ ∈ ClosEnv Δ ]
               ∅' ⊢ M ⇓ clos (ƛ N′) δ)
cbn↔reduce {M} = ⟨ (λ x → reduce→cbn (proj₂ x)) ,
                   (λ x → cbn→reduce (proj₂ (proj₂ (proj₂ x)))) ⟩

Unicode

This chapter uses the following unicode:

𝔼  U+1D53C  MATHEMATICAL DOUBLE-STRUCK CAPITAL E (\bE)
𝔾  U+1D53E  MATHEMATICAL DOUBLE-STRUCK CAPITAL G (\bG)
𝕍  U+1D53E  MATHEMATICAL DOUBLE-STRUCK CAPITAL V (\bV)