Untyped: Untyped lambda calculus with full normalisation
module plfa.part2.Untyped where
In this chapter we play with variations on a theme:

Previous chapters consider intrinsicallytyped calculi; here we consider one that is untyped but intrinsically scoped.

Previous chapters consider callbyvalue calculi; here we consider callbyname.

Previous chapters consider weak head normal form, where reduction stops at a lambda abstraction; here we consider full normalisation, where reduction continues underneath a lambda.

Previous chapters consider deterministic reduction, where there is at most one redex in a given term; here we consider nondeterministic reduction where a term may contain many redexes and any one of them may reduce.

Previous chapters consider reduction of closed terms, those with no free variables; here we consider open terms, those which may have free variables.

Previous chapters consider lambda calculus extended with natural numbers and fixpoints; here we consider a tiny calculus with just variables, abstraction, and application, in which the other constructs may be encoded.
In general, one may mix and match these features, save that full normalisation requires open terms and encoding naturals and fixpoints requires being untyped. The aim of this chapter is to give some appreciation for the range of different lambda calculi one may encounter.
Imports
import Relation.Binary.PropositionalEquality as Eq open Eq using (_≡_; refl; sym; trans; cong) open import Data.Empty using (⊥; ⊥elim) open import Data.Nat using (ℕ; zero; suc; _+_; _∸_) open import Data.Product using (_×_) renaming (_,_ to ⟨_,_⟩) open import Data.Unit using (⊤; tt) open import Function using (_∘_) open import Function.Equivalence using (_⇔_; equivalence) open import Relation.Nullary using (¬_; Dec; yes; no) open import Relation.Nullary.Decidable using (map) open import Relation.Nullary.Negation using (contraposition) open import Relation.Nullary.Product using (_×dec_)
Untyped is Unityped
Our development will be close to that in
Chapter DeBruijn,
save that every term will have exactly the same type, written ★
and pronounced “any”.
This matches a slogan introduced by Dana Scott
and echoed by Robert Harper: “Untyped is Unityped”.
One consequence of this approach is that constructs which previously
had to be given separately (such as natural numbers and fixpoints)
can now be defined in the language itself.
Syntax
First, we get all our infix declarations out of the way:
infix 4 _⊢_ infix 4 _∋_ infixl 5 _,_ infix 6 ƛ_ infix 6 ′_ infixl 7 _·_
Types
We have just one type:
data Type : Set where ★ : Type
Exercise (Type≃⊤
) (practice)
Show that Type
is isomorphic to ⊤
, the unit type.
 Your code goes here
Contexts
As before, a context is a list of types, with the type of the most recently bound variable on the right:
data Context : Set where ∅ : Context _,_ : Context → Type → Context
We let Γ
and Δ
range over contexts.
Exercise (Context≃ℕ
) (practice)
Show that Context
is isomorphic to ℕ
.
 Your code goes here
Variables and the lookup judgment
Intrinsicallyscoped variables correspond to the lookup judgment. The rules are as before:
data _∋_ : Context → Type → Set where Z : ∀ {Γ A}  → Γ , A ∋ A S_ : ∀ {Γ A B} → Γ ∋ A  → Γ , B ∋ A
We could write the rules with all instances of A
and B
replaced by ★
, but arguably it is clearer not to do so.
Because ★
is the only type, the judgment doesn’t guarantee anything
useful about types. But it does ensure that all variables are in
scope. For instance, we cannot use S S Z
in a context that only
binds two variables.
Terms and the scoping judgment
Intrinsicallyscoped terms correspond to the typing judgment, but with
★
as the only type. The result is that we check that terms are
well scoped — that is, that all variables they mention are in scope —
but not that they are well typed:
data _⊢_ : Context → Type → Set where `_ : ∀ {Γ A} → Γ ∋ A  → Γ ⊢ A ƛ_ : ∀ {Γ} → Γ , ★ ⊢ ★  → Γ ⊢ ★ _·_ : ∀ {Γ} → Γ ⊢ ★ → Γ ⊢ ★  → Γ ⊢ ★
Now we have a tiny calculus, with only variables, abstraction, and application. Below we will see how to encode naturals and fixpoints into this calculus.
Writing variables as numerals
As before, we can convert a natural to the corresponding de Bruijn index. We no longer need to lookup the type in the context, since every variable has the same type:
count : ∀ {Γ} → ℕ → Γ ∋ ★ count {Γ , ★} zero = Z count {Γ , ★} (suc n) = S (count n) count {∅} _ = ⊥elim impossible where postulate impossible : ⊥
We can then introduce a convenient abbreviation for variables:
#_ : ∀ {Γ} → ℕ → Γ ⊢ ★ # n = ` count n
Test examples
Our only example is computing two plus two on Church numerals:
twoᶜ : ∀ {Γ} → Γ ⊢ ★ twoᶜ = ƛ ƛ (# 1 · (# 1 · # 0)) fourᶜ : ∀ {Γ} → Γ ⊢ ★ fourᶜ = ƛ ƛ (# 1 · (# 1 · (# 1 · (# 1 · # 0)))) plusᶜ : ∀ {Γ} → Γ ⊢ ★ plusᶜ = ƛ ƛ ƛ ƛ (# 3 · # 1 · (# 2 · # 1 · # 0)) 2+2ᶜ : ∅ ⊢ ★ 2+2ᶜ = plusᶜ · twoᶜ · twoᶜ
Before, reduction stopped when we reached a lambda term, so we had to
compute plusᶜ · twoᶜ · twoᶜ · sucᶜ · `zero
to ensure we reduced
to a representation of the natural four. Now, reduction continues
under lambda, so we don’t need the extra arguments. It is convenient
to define a term to represent four as a Church numeral, as well as
two.
Renaming
Our definition of renaming is as before. First, we need an extension lemma:
ext : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ∋ A)  → (∀ {A B} → Γ , B ∋ A → Δ , B ∋ A) ext ρ Z = Z ext ρ (S x) = S (ρ x)
We could replace all instances of A
and B
by ★
, but arguably it is
clearer not to do so.
Now it is straightforward to define renaming:
rename : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ∋ A)  → (∀ {A} → Γ ⊢ A → Δ ⊢ A) rename ρ (` x) = ` (ρ x) rename ρ (ƛ N) = ƛ (rename (ext ρ) N) rename ρ (L · M) = (rename ρ L) · (rename ρ M)
This is exactly as before, save that there are fewer term forms.
Simultaneous substitution
Our definition of substitution is also exactly as before. First we need an extension lemma:
exts : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ⊢ A)  → (∀ {A B} → Γ , B ∋ A → Δ , B ⊢ A) exts σ Z = ` Z exts σ (S x) = rename S_ (σ x)
Again, we could replace all instances of A
and B
by ★
.
Now it is straightforward to define substitution:
subst : ∀ {Γ Δ} → (∀ {A} → Γ ∋ A → Δ ⊢ A)  → (∀ {A} → Γ ⊢ A → Δ ⊢ A) subst σ (` k) = σ k subst σ (ƛ N) = ƛ (subst (exts σ) N) subst σ (L · M) = (subst σ L) · (subst σ M)
Again, this is exactly as before, save that there are fewer term forms.
Single substitution
It is easy to define the special case of substitution for one free variable:
substzero : ∀ {Γ B} → (Γ ⊢ B) → ∀ {A} → (Γ , B ∋ A) → (Γ ⊢ A) substzero M Z = M substzero M (S x) = ` x _[_] : ∀ {Γ A B} → Γ , B ⊢ A → Γ ⊢ B  → Γ ⊢ A _[_] {Γ} {A} {B} N M = subst {Γ , B} {Γ} (substzero M) {A} N
Neutral and normal terms
Reduction continues until a term is fully normalised. Hence, instead of values, we are now interested in normal forms. Terms in normal form are defined by mutual recursion with neutral terms:
data Neutral : ∀ {Γ A} → Γ ⊢ A → Set data Normal : ∀ {Γ A} → Γ ⊢ A → Set
Neutral terms arise because we now consider reduction of open terms, which may contain free variables. A term is neutral if it is a variable or a neutral term applied to a normal term:
data Neutral where `_ : ∀ {Γ A} (x : Γ ∋ A)  → Neutral (` x) _·_ : ∀ {Γ} {L M : Γ ⊢ ★} → Neutral L → Normal M  → Neutral (L · M)
A term is a normal form if it is neutral or an abstraction where the
body is a normal form. We use ′_
to label neutral terms.
Like `_
, it is unobtrusive:
data Normal where ′_ : ∀ {Γ A} {M : Γ ⊢ A} → Neutral M  → Normal M ƛ_ : ∀ {Γ} {N : Γ , ★ ⊢ ★} → Normal N  → Normal (ƛ N)
We introduce a convenient abbreviation for evidence that a variable is neutral:
#′_ : ∀ {Γ} (n : ℕ) → Neutral {Γ} (# n) #′ n = ` count n
For example, here is the evidence that the Church numeral two is in normal form:
_ : Normal (twoᶜ {∅}) _ = ƛ ƛ (′ #′ 1 · (′ #′ 1 · (′ #′ 0)))
The evidence that a term is in normal form is almost identical to
the term itself, decorated with some additional primes to indicate
neutral terms, and using #′
in place of #
Reduction step
The reduction rules are altered to switch from callbyvalue to callbyname and to enable full normalisation:

The rule
ξ₁
remains the same as it was for the simplytyped lambda calculus. 
In rule
ξ₂
, the requirement that the termL
is a value is dropped. So this rule can overlap withξ₁
and reduction is nondeterministic. One can choose to reduce a term inside eitherL
orM
. 
In rule
β
, the requirement that the argument is a value is dropped, corresponding to callbyname evaluation. This introduces further nondeterminism, asβ
overlaps withξ₂
when there are redexes in the argument. 
A new rule
ζ
is added, to enable reduction underneath a lambda.
Here are the formalised rules:
infix 2 _—→_ data _—→_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where ξ₁ : ∀ {Γ} {L L′ M : Γ ⊢ ★} → L —→ L′  → L · M —→ L′ · M ξ₂ : ∀ {Γ} {L M M′ : Γ ⊢ ★} → M —→ M′  → L · M —→ L · M′ β : ∀ {Γ} {N : Γ , ★ ⊢ ★} {M : Γ ⊢ ★}  → (ƛ N) · M —→ N [ M ] ζ : ∀ {Γ} {N N′ : Γ , ★ ⊢ ★} → N —→ N′  → ƛ N —→ ƛ N′
Exercise (variant1
) (practice)
How would the rules change if we want callbyvalue where terms
normalise completely? Assume that β
should not permit reduction
unless both terms are in normal form.
 Your code goes here
Exercise (variant2
) (practice)
How would the rules change if we want callbyvalue where terms
do not reduce underneath lambda? Assume that β
permits reduction when both terms are values (that is, lambda
abstractions). What would 2+2ᶜ
reduce to in this case?
 Your code goes here
Reflexive and transitive closure
We cutandpaste the previous definition:
infix 2 _—↠_ infix 1 begin_ infixr 2 _—→⟨_⟩_ infix 3 _∎ data _—↠_ : ∀ {Γ A} → (Γ ⊢ A) → (Γ ⊢ A) → Set where _∎ : ∀ {Γ A} (M : Γ ⊢ A)  → M —↠ M _—→⟨_⟩_ : ∀ {Γ A} (L : Γ ⊢ A) {M N : Γ ⊢ A} → L —→ M → M —↠ N  → L —↠ N begin_ : ∀ {Γ} {A} {M N : Γ ⊢ A} → M —↠ N  → M —↠ N begin M—↠N = M—↠N
Example reduction sequence
Here is the demonstration that two plus two is four:
_ : 2+2ᶜ —↠ fourᶜ _ = begin plusᶜ · twoᶜ · twoᶜ —→⟨ ξ₁ β ⟩ (ƛ ƛ ƛ twoᶜ · # 1 · (# 2 · # 1 · # 0)) · twoᶜ —→⟨ β ⟩ ƛ ƛ twoᶜ · # 1 · (twoᶜ · # 1 · # 0) —→⟨ ζ (ζ (ξ₁ β)) ⟩ ƛ ƛ ((ƛ # 2 · (# 2 · # 0)) · (twoᶜ · # 1 · # 0)) —→⟨ ζ (ζ β) ⟩ ƛ ƛ # 1 · (# 1 · (twoᶜ · # 1 · # 0)) —→⟨ ζ (ζ (ξ₂ (ξ₂ (ξ₁ β)))) ⟩ ƛ ƛ # 1 · (# 1 · ((ƛ # 2 · (# 2 · # 0)) · # 0)) —→⟨ ζ (ζ (ξ₂ (ξ₂ β))) ⟩ ƛ (ƛ # 1 · (# 1 · (# 1 · (# 1 · # 0)))) ∎
After just two steps the toplevel term is an abstraction,
and ζ
rules drive the rest of the normalisation.
Progress
Progress adapts. Instead of claiming that every term either is a value or takes a reduction step, we claim that every term is either in normal form or takes a reduction step.
Previously, progress only applied to closed, welltyped terms. We had
to rule out terms where we apply something other than a function (such
as `zero
) or terms with a free variable. Now we can demonstrate
it for open, wellscoped terms. The definition of normal form permits
free variables, and we have no terms that are not functions.
A term makes progress if it can take a step or is in normal form:
data Progress {Γ A} (M : Γ ⊢ A) : Set where step : ∀ {N : Γ ⊢ A} → M —→ N  → Progress M done : Normal M  → Progress M
If a term is well scoped then it satisfies progress:
progress : ∀ {Γ A} → (M : Γ ⊢ A) → Progress M progress (` x) = done (′ ` x) progress (ƛ N) with progress N ...  step N—→N′ = step (ζ N—→N′) ...  done NrmN = done (ƛ NrmN) progress (` x · M) with progress M ...  step M—→M′ = step (ξ₂ M—→M′) ...  done NrmM = done (′ (` x) · NrmM) progress ((ƛ N) · M) = step β progress (L@(_ · _) · M) with progress L ...  step L—→L′ = step (ξ₁ L—→L′) ...  done (′ NeuL) with progress M ...  step M—→M′ = step (ξ₂ M—→M′) ...  done NrmM = done (′ NeuL · NrmM)
We induct on the evidence that the term is well scoped:
 If the term is a variable, then it is in normal form. (This contrasts with previous proofs, where the variable case was ruled out by the restriction to closed terms.)
 If the term is an abstraction, recursively invoke progress on the body.
(This contrast with previous proofs, where an abstraction is
immediately a value.):
 If it steps, then the whole term steps via
ζ
.  If it is in normal form, then so is the whole term.
 If it steps, then the whole term steps via
 If the term is an application, consider the function subterm:
 If it is a variable, recursively invoke progress on the argument:
 If it steps, then the whole term steps via
ξ₂
;  If it is normal, then so is the whole term.
 If it steps, then the whole term steps via
 If it is an abstraction, then the whole term steps via
β
.  If it is an application, recursively apply progress to the function subterm:
 If it steps, then the whole term steps via
ξ₁
.  If it is normal, recursively apply progress to the argument subterm:
 If it steps, then the whole term steps via
ξ₂
.  If it is normal, then so is the whole term.
 If it steps, then the whole term steps via
 If it steps, then the whole term steps via
 If it is a variable, recursively invoke progress on the argument:
The final equation for progress uses an at pattern of the form P@Q
,
which matches only if both pattern P
and pattern Q
match. Character
@
is one of the few that Agda doesn’t allow in names, so spaces are not
required around it. In this case, the pattern ensures that L
is an
application.
Evaluation
As previously, progress immediately yields an evaluator.
Gas is specified by a natural number:
data Gas : Set where gas : ℕ → Gas
When our evaluator returns a term N
, it will either give evidence that
N
is normal or indicate that it ran out of gas:
data Finished {Γ A} (N : Γ ⊢ A) : Set where done : Normal N  → Finished N outofgas :  Finished N
Given a term L
of type A
, the evaluator will, for some N
, return
a reduction sequence from L
to N
and an indication of whether
reduction finished:
data Steps : ∀ {Γ A} → Γ ⊢ A → Set where steps : ∀ {Γ A} {L N : Γ ⊢ A} → L —↠ N → Finished N  → Steps L
The evaluator takes gas and a term and returns the corresponding steps:
eval : ∀ {Γ A} → Gas → (L : Γ ⊢ A)  → Steps L eval (gas zero) L = steps (L ∎) outofgas eval (gas (suc m)) L with progress L ...  done NrmL = steps (L ∎) (done NrmL) ...  step {M} L—→M with eval (gas m) M ...  steps M—↠N fin = steps (L —→⟨ L—→M ⟩ M—↠N) fin
The definition is as before, save that the empty context ∅
generalises to an arbitrary context Γ
.
Example
We reiterate our previous example. Two plus two is four, with Church numerals:
_ : eval (gas 100) 2+2ᶜ ≡ steps ((ƛ (ƛ (ƛ (ƛ (` (S (S (S Z)))) · (` (S Z)) · ((` (S (S Z))) · (` (S Z)) · (` Z)))))) · (ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) · (ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) —→⟨ ξ₁ β ⟩ (ƛ (ƛ (ƛ (ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) · (` (S Z)) · ((` (S (S Z))) · (` (S Z)) · (` Z))))) · (ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) —→⟨ β ⟩ ƛ (ƛ (ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) · (` (S Z)) · ((ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) · (` (S Z)) · (` Z))) —→⟨ ζ (ζ (ξ₁ β)) ⟩ ƛ (ƛ (ƛ (` (S (S Z))) · ((` (S (S Z))) · (` Z))) · ((ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) · (` (S Z)) · (` Z))) —→⟨ ζ (ζ β) ⟩ ƛ (ƛ (` (S Z)) · ((` (S Z)) · ((ƛ (ƛ (` (S Z)) · ((` (S Z)) · (` Z)))) · (` (S Z)) · (` Z)))) —→⟨ ζ (ζ (ξ₂ (ξ₂ (ξ₁ β)))) ⟩ ƛ (ƛ (` (S Z)) · ((` (S Z)) · ((ƛ (` (S (S Z))) · ((` (S (S Z))) · (` Z))) · (` Z)))) —→⟨ ζ (ζ (ξ₂ (ξ₂ β))) ⟩ ƛ (ƛ (` (S Z)) · ((` (S Z)) · ((` (S Z)) · ((` (S Z)) · (` Z))))) ∎) (done (ƛ (ƛ (′ (` (S Z)) · (′ (` (S Z)) · (′ (` (S Z)) · (′ (` (S Z)) · (′ (` Z))))))))) _ = refl
Naturals and fixpoint
We could simulate naturals using Church numerals, but computing predecessor is tricky and expensive. Instead, we use a different representation, called Scott numerals, where a number is essentially defined by the expression that corresponds to its own case statement.
Recall that Church numerals apply a given function for the corresponding number of times. Using named terms, we represent the first three Church numerals as follows:
zero = ƛ s ⇒ ƛ z ⇒ z
one = ƛ s ⇒ ƛ z ⇒ s · z
two = ƛ s ⇒ ƛ z ⇒ s · (s · z)
In contrast, for Scott numerals, we represent the first three naturals as follows:
zero = ƛ s ⇒ ƛ z ⇒ z
one = ƛ s ⇒ ƛ z ⇒ s · zero
two = ƛ s ⇒ ƛ z ⇒ s · one
Each representation expects two arguments, one corresponding to the successor branch of the case (it expects an additional argument, the predecessor of the current argument) and one corresponding to the zero branch of the case. (The cases could be in either order. We put the successor case first to ease comparison with Church numerals.)
Here is the representation of naturals encoded with de Bruijn indexes:
`zero : ∀ {Γ} → (Γ ⊢ ★) `zero = ƛ ƛ (# 0) `suc_ : ∀ {Γ} → (Γ ⊢ ★) → (Γ ⊢ ★) `suc_ M = (ƛ ƛ ƛ (# 1 · # 2)) · M case : ∀ {Γ} → (Γ ⊢ ★) → (Γ ⊢ ★) → (Γ , ★ ⊢ ★) → (Γ ⊢ ★) case L M N = L · (ƛ N) · M
Here we have been careful to retain the exact form of our previous definitions. The successor branch expects an additional variable to be in scope (as indicated by its type), so it is converted to an ordinary term using lambda abstraction.
We can also define fixpoint. Using named terms, we define:
μ f = (ƛ x ⇒ f · (x · x)) · (ƛ x ⇒ f · (x · x))
This works because:
μ f
≡
(ƛ x ⇒ f · (x · x)) · (ƛ x ⇒ f · (x · x))
—→
f · ((ƛ x ⇒ f · (x · x)) · (ƛ x ⇒ f · (x · x)))
≡
f · (μ f)
With de Bruijn indices, we have the following:
μ_ : ∀ {Γ} → (Γ , ★ ⊢ ★) → (Γ ⊢ ★) μ N = (ƛ ((ƛ (# 1 · (# 0 · # 0))) · (ƛ (# 1 · (# 0 · # 0))))) · (ƛ N)
The argument to fixpoint is treated similarly to the successor branch of case.
We can now define two plus two exactly as before:
infix 5 μ_ two : ∀ {Γ} → Γ ⊢ ★ two = `suc `suc `zero four : ∀ {Γ} → Γ ⊢ ★ four = `suc `suc `suc `suc `zero plus : ∀ {Γ} → Γ ⊢ ★ plus = μ ƛ ƛ (case (# 1) (# 0) (`suc (# 3 · # 0 · # 1)))
Because `suc
is now a defined term rather than primitive,
it is no longer the case that plus · two · two
reduces to four
,
but they do both reduce to the same normal term.
Exercise pluseval
(practice)
Use the evaluator to confirm that plus · two · two
and four
normalise to the same term.
 Your code goes here
Exercise multiplicationuntyped
(recommended)
Use the encodings above to translate your definition of multiplication from previous chapters with the Scott representation and the encoding of the fixpoint operator. Confirm that two times two is four.
 Your code goes here
Exercise encodemore
(stretch)
Along the lines above, encode all of the constructs of Chapter More, save for primitive numbers, in the untyped lambda calculus.
 Your code goes here
Multistep reduction is transitive
In our formulation of the reflexive transitive closure of reduction,
i.e., the —↠
relation, there is not an explicit rule for
transitivity. Instead the relation mimics the structure of lists by
providing a case for an empty reduction sequence and a case for adding
one reduction to the front of a reduction sequence. The following is
the proof of transitivity, which has the same structure as the append
function _++_
on lists.
—↠trans : ∀{Γ}{A}{L M N : Γ ⊢ A} → L —↠ M → M —↠ N → L —↠ N —↠trans (M ∎) mn = mn —↠trans (L —→⟨ r ⟩ lm) mn = L —→⟨ r ⟩ (—↠trans lm mn)
The following notation makes it convenient to employ
transitivity of —↠
.
infixr 2 _—↠⟨_⟩_ _—↠⟨_⟩_ : ∀ {Γ A} (L : Γ ⊢ A) {M N : Γ ⊢ A} → L —↠ M → M —↠ N  → L —↠ N L —↠⟨ L—↠M ⟩ M—↠N = —↠trans L—↠M M—↠N
Multistep reduction is a congruence
Recall from Chapter Induction that a
relation R
is a congruence for a given function f
if it is
preserved by that function, i.e., if R x y
then R (f x) (f y)
.
The term constructors ƛ_
and _·_
are functions, and so
the notion of congruence applies to them as well. Furthermore, when a
relation is a congruence for all of the term constructors, we
say that the relation is a congruence for the language in question, in
this case the untyped lambda calculus.
The rules ξ₁
, ξ₂
, and ζ
ensure that the reduction relation is a
congruence for the untyped lambda calculus. The multistep reduction
relation —↠
is also a congruence, which we prove in the following
three lemmas.
appLcong : ∀ {Γ} {L L' M : Γ ⊢ ★} → L —↠ L'  → L · M —↠ L' · M appLcong {Γ}{L}{L'}{M} (L ∎) = L · M ∎ appLcong {Γ}{L}{L'}{M} (L —→⟨ r ⟩ rs) = L · M —→⟨ ξ₁ r ⟩ appLcong rs
The proof of appLcong
is by induction on the reduction sequence L —↠ L'
.
 Suppose
L —↠ L
byL ∎
. Then we haveL · M —↠ L · M
byL · M ∎
.  Suppose
L —↠ L''
byL —→⟨ r ⟩ rs
, soL —→ L'
byr
andL' —↠ L''
byrs
. We haveL · M —→ L' · M
byξ₁ r
andL' · M —↠ L'' · M
by the induction hypothesis applied tors
. We conclude thatL · M —↠ L'' · M
by putting these two facts together using_—→⟨_⟩_
.
The proofs of appRcong
and abscong
follow the same pattern as
the proof for appLcong
.
appRcong : ∀ {Γ} {L M M' : Γ ⊢ ★} → M —↠ M'  → L · M —↠ L · M' appRcong {Γ}{L}{M}{M'} (M ∎) = L · M ∎ appRcong {Γ}{L}{M}{M'} (M —→⟨ r ⟩ rs) = L · M —→⟨ ξ₂ r ⟩ appRcong rs
abscong : ∀ {Γ} {N N' : Γ , ★ ⊢ ★} → N —↠ N'  → ƛ N —↠ ƛ N' abscong (M ∎) = ƛ M ∎ abscong (L —→⟨ r ⟩ rs) = ƛ L —→⟨ ζ r ⟩ abscong rs
Unicode
This chapter uses the following unicode:
★ U+2605 BLACK STAR (\st)
The \st
command permits navigation among many different stars;
the one we use is number 7.