---
title : "Typed: Typed Lambda term representation"
permalink : /Typed
---
## Imports
\begin{code}
module Typed where
\end{code}
\begin{code}
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong; cong₂; _≢_)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.List using (List; []; _∷_; _++_; map; foldr; filter)
open import Data.Nat using (ℕ; zero; suc; _+_)
open import Data.String using (String; _≟_)
open import Data.Product using (_×_; proj₁; proj₂) renaming (_,_ to ⟨_,_⟩)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Function using (_∘_)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Relation.Nullary.Negation using (¬?)
open import Collections
pattern [_] x = x ∷ []
pattern [_,_] x y = x ∷ y ∷ []
pattern [_,_,_] x y z = x ∷ y ∷ z ∷ []
\end{code}
## Syntax
\begin{code}
infix 4 _wf
infix 4 _∉_
infix 4 _∋_`:_
infix 4 _⊢_`:_
infixl 5 _,_`:_
infixr 6 _`→_
infix 6 `λ_`→_
infixl 7 `if0_then_else_
infix 8 `suc_ `pred_ `Y_
infixl 9 _·_
infix 10 S_
Id : Set
Id = String
data Type : Set where
`ℕ : Type
_`→_ : Type → Type → Type
data Env : Set where
ε : Env
_,_`:_ : Env → Id → Type → Env
data Term : Set where
`_ : Id → Term
`λ_`→_ : Id → Term → Term
_·_ : Term → Term → Term
`zero : Term
`suc_ : Term → Term
`pred_ : Term → Term
`if0_then_else_ : Term → Term → Term → Term
`Y_ : Term → Term
data _∋_`:_ : Env → Id → Type → Set where
Z : ∀ {Γ A x}
--------------------
→ Γ , x `: A ∋ x `: A
S_ : ∀ {Γ A B x w}
→ Γ ∋ w `: B
--------------------
→ Γ , x `: A ∋ w `: B
_∉_ : Id → Env → Set
x ∉ Γ = ∀ {A} → ¬ (Γ ∋ x `: A)
data _⊢_`:_ : Env → Term → Type → Set where
Ax : ∀ {Γ A x}
→ Γ ∋ x `: A
--------------
→ Γ ⊢ ` x `: A
⊢λ : ∀ {Γ x N A B}
→ x ∉ Γ
→ Γ , x `: A ⊢ N `: B
--------------------------
→ Γ ⊢ (`λ x `→ N) `: A `→ B
_·_ : ∀ {Γ L M A B}
→ Γ ⊢ L `: A `→ B
→ Γ ⊢ M `: A
----------------
→ Γ ⊢ L · M `: B
⊢zero : ∀ {Γ}
----------------
→ Γ ⊢ `zero `: `ℕ
⊢suc : ∀ {Γ M}
→ Γ ⊢ M `: `ℕ
-----------------
→ Γ ⊢ `suc M `: `ℕ
⊢pred : ∀ {Γ M}
→ Γ ⊢ M `: `ℕ
------------------
→ Γ ⊢ `pred M `: `ℕ
⊢if0 : ∀ {Γ L M N A}
→ Γ ⊢ L `: `ℕ
→ Γ ⊢ M `: A
→ Γ ⊢ N `: A
------------------------------
→ Γ ⊢ `if0 L then M else N `: A
⊢Y : ∀ {Γ M A}
→ Γ ⊢ M `: A `→ A
----------------
→ Γ ⊢ `Y M `: A
data _wf : Env → Set where
empty :
-----
ε wf
extend : ∀ {Γ x A}
→ Γ wf
→ x ∉ Γ
-------------------------
→ (Γ , x `: A) wf
\end{code}
## Test examples
\begin{code}
two : Term
two = `suc `suc `zero
⊢two : ε ⊢ two `: `ℕ
⊢two = (⊢suc (⊢suc ⊢zero))
plus : Term
plus = `Y (`λ "p" `→ `λ "m" `→ `λ "n" `→ `if0 ` "m" then ` "n" else ` "p" · (`pred ` "m") · ` "n")
⊢plus : ε ⊢ plus `: `ℕ `→ `ℕ `→ `ℕ
⊢plus = (⊢Y (⊢λ p∉ (⊢λ m∉ (⊢λ n∉
(⊢if0 (Ax ⊢m) (Ax ⊢n) (Ax ⊢p · (⊢pred (Ax ⊢m)) · Ax ⊢n))))))
where
⊢p = S S Z
⊢m = S Z
⊢n = Z
Γ₀ = ε
Γ₁ = Γ₀ , "p" `: `ℕ `→ `ℕ `→ `ℕ
Γ₂ = Γ₁ , "m" `: `ℕ
p∉ : "p" ∉ Γ₀
p∉ ()
m∉ : "m" ∉ Γ₁
m∉ (S ())
n∉ : "n" ∉ Γ₂
n∉ (S S ())
four : Term
four = plus · two · two
⊢four : ε ⊢ four `: `ℕ
⊢four = ⊢plus · ⊢two · ⊢two
Ch : Type
Ch = (`ℕ `→ `ℕ) `→ `ℕ `→ `ℕ
twoCh : Term
twoCh = `λ "s" `→ `λ "z" `→ (` "s" · (` "s" · ` "z"))
⊢twoCh : ε ⊢ twoCh `: Ch
⊢twoCh = (⊢λ s∉ (⊢λ z∉ (Ax ⊢s · (Ax ⊢s · Ax ⊢z))))
where
⊢s = S Z
⊢z = Z
Γ₀ = ε
Γ₁ = Γ₀ , "s" `: `ℕ `→ `ℕ
s∉ : "s" ∉ ε
s∉ ()
z∉ : "z" ∉ Γ₁
z∉ (S ())
plusCh : Term
plusCh = `λ "m" `→ `λ "n" `→ `λ "s" `→ `λ "z" `→
` "m" · ` "s" · (` "n" · ` "s" · ` "z")
⊢plusCh : ε ⊢ plusCh `: Ch `→ Ch `→ Ch
⊢plusCh = (⊢λ m∉ (⊢λ n∉ (⊢λ s∉ (⊢λ z∉ (Ax ⊢m · Ax ⊢s · (Ax ⊢n · Ax ⊢s · Ax ⊢z))))))
where
⊢m = S S S Z
⊢n = S S Z
⊢s = S Z
⊢z = Z
Γ₀ = ε
Γ₁ = Γ₀ , "m" `: Ch
Γ₂ = Γ₁ , "n" `: Ch
Γ₃ = Γ₂ , "s" `: `ℕ `→ `ℕ
m∉ : "m" ∉ Γ₀
m∉ ()
n∉ : "n" ∉ Γ₁
n∉ (S ())
s∉ : "s" ∉ Γ₂
s∉ (S S ())
z∉ : "z" ∉ Γ₃
z∉ (S S S ())
fromCh : Term
fromCh = `λ "m" `→ ` "m" · (`λ "s" `→ `suc ` "s") · `zero
⊢fromCh : ε ⊢ fromCh `: Ch `→ `ℕ
⊢fromCh = (⊢λ m∉ (Ax ⊢m · (⊢λ s∉ (⊢suc (Ax ⊢s))) · ⊢zero))
where
⊢m = Z
⊢s = Z
Γ₀ = ε
Γ₁ = Γ₀ , "m" `: Ch
m∉ : "m" ∉ Γ₀
m∉ ()
s∉ : "s" ∉ Γ₁
s∉ (S ())
fourCh : Term
fourCh = fromCh · (plusCh · twoCh · twoCh)
⊢fourCh : ε ⊢ fourCh `: `ℕ
⊢fourCh = ⊢fromCh · (⊢plusCh · ⊢twoCh · ⊢twoCh)
\end{code}
## Erasure
\begin{code}
lookup : ∀ {Γ x A} → Γ ∋ x `: A → Id
lookup {Γ , x `: A} Z = x
lookup {Γ , x `: A} (S ⊢w) = lookup {Γ} ⊢w
erase : ∀ {Γ M A} → Γ ⊢ M `: A → Term
erase (Ax ⊢w) = ` lookup ⊢w
erase (⊢λ {x = x} x∉ ⊢N) = `λ x `→ erase ⊢N
erase (⊢L · ⊢M) = erase ⊢L · erase ⊢M
erase (⊢zero) = `zero
erase (⊢suc ⊢M) = `suc (erase ⊢M)
erase (⊢pred ⊢M) = `pred (erase ⊢M)
erase (⊢if0 ⊢L ⊢M ⊢N) = `if0 (erase ⊢L) then (erase ⊢M) else (erase ⊢N)
erase (⊢Y ⊢M) = `Y (erase ⊢M)
\end{code}
### Properties of erasure
\begin{code}
cong₃ : ∀ {A B C D : Set} (f : A → B → C → D) {s t u v x y} →
s ≡ t → u ≡ v → x ≡ y → f s u x ≡ f t v y
cong₃ f refl refl refl = refl
lookup-lemma : ∀ {Γ x A} → (⊢x : Γ ∋ x `: A) → lookup ⊢x ≡ x
lookup-lemma Z = refl
lookup-lemma (S ⊢w) = lookup-lemma ⊢w
erase-lemma : ∀ {Γ M A} → (⊢M : Γ ⊢ M `: A) → erase ⊢M ≡ M
erase-lemma (Ax ⊢x) = cong `_ (lookup-lemma ⊢x)
erase-lemma (⊢λ {x = x} x∉ ⊢N) = cong (`λ x `→_) (erase-lemma ⊢N)
erase-lemma (⊢L · ⊢M) = cong₂ _·_ (erase-lemma ⊢L) (erase-lemma ⊢M)
erase-lemma (⊢zero) = refl
erase-lemma (⊢suc ⊢M) = cong `suc_ (erase-lemma ⊢M)
erase-lemma (⊢pred ⊢M) = cong `pred_ (erase-lemma ⊢M)
erase-lemma (⊢if0 ⊢L ⊢M ⊢N) = cong₃ `if0_then_else_
(erase-lemma ⊢L) (erase-lemma ⊢M) (erase-lemma ⊢N)
erase-lemma (⊢Y ⊢M) = cong `Y_ (erase-lemma ⊢M)
\end{code}
## Substitution
### Lists as sets
\begin{code}
open Collections.CollectionDec (Id) (_≟_)
\end{code}
### Free variables
\begin{code}
free : Term → List Id
free (` x) = [ x ]
free (`λ x `→ N) = free N \\ x
free (L · M) = free L ++ free M
free (`zero) = []
free (`suc M) = free M
free (`pred M) = free M
free (`if0 L then M else N) = free L ++ free M ++ free N
free (`Y M) = free M
\end{code}
### Identifier maps
\begin{code}
∅ : Id → Term
∅ x = ` x
infixl 5 _,_↦_
_,_↦_ : (Id → Term) → Id → Term → (Id → Term)
(ρ , x ↦ M) w with w ≟ x
... | yes _ = M
... | no _ = ρ w
\end{code}
### Substitution
\begin{code}
subst : (Id → Term) → Term → Term
subst ρ (` x) = ρ x
subst ρ (`λ x `→ N) = `λ x `→ subst (ρ , x ↦ ` x) N
subst ρ (L · M) = subst ρ L · subst ρ M
subst ρ (`zero) = `zero
subst ρ (`suc M) = `suc (subst ρ M)
subst ρ (`pred M) = `pred (subst ρ M)
subst ρ (`if0 L then M else N)
= `if0 (subst ρ L) then (subst ρ M) else (subst ρ N)
subst ρ (`Y M) = `Y (subst ρ M)
_[_:=_] : Term → Id → Term → Term
N [ x := M ] = subst (∅ , x ↦ M) N
\end{code}
### Testing substitution
\begin{code}
_ : (` "s" · ` "s" · ` "z") [ "z" := `zero ] ≡ (` "s" · ` "s" · `zero)
_ = refl
_ : (` "s" · ` "s" · ` "z") [ "s" := (`λ "m" `→ `suc ` "m") ] [ "z" := `zero ]
≡ (`λ "m" `→ `suc ` "m") · (`λ "m" `→ `suc ` "m") · `zero
_ = refl
_ : (`λ "m" `→ ` "m" · ` "n") [ "n" := ` "p" · ` "q" ]
≡ `λ "m" `→ ` "m" · (` "p" · ` "q")
_ = refl
_ : subst (∅ , "m" ↦ ` "p" , "n" ↦ ` "q") (` "m" · ` "n") ≡ (` "p" · ` "q")
_ = refl
\end{code}
## Values
\begin{code}
data Value : Term → Set where
Zero :
----------
Value `zero
Suc : ∀ {V}
→ Value V
--------------
→ Value (`suc V)
Fun : ∀ {x N}
---------------
→ Value (`λ x `→ N)
\end{code}
## Reduction
\begin{code}
infix 4 _⟶_
data _⟶_ : Term → Term → Set where
ξ-·₁ : ∀ {L L′ M}
→ L ⟶ L′
-----------------
→ L · M ⟶ L′ · M
ξ-·₂ : ∀ {V M M′}
→ Value V
→ M ⟶ M′
-----------------
→ V · M ⟶ V · M′
β-`→ : ∀ {x N V}
→ Value V
---------------------------------
→ (`λ x `→ N) · V ⟶ N [ x := V ]
ξ-suc : ∀ {M M′}
→ M ⟶ M′
-------------------
→ `suc M ⟶ `suc M′
ξ-pred : ∀ {M M′}
→ M ⟶ M′
---------------------
→ `pred M ⟶ `pred M′
β-pred-zero :
----------------------
`pred `zero ⟶ `zero
β-pred-suc : ∀ {V}
→ Value V
---------------------
→ `pred (`suc V) ⟶ V
ξ-if0 : ∀ {L L′ M N}
→ L ⟶ L′
-----------------------------------------------
→ `if0 L then M else N ⟶ `if0 L′ then M else N
β-if0-zero : ∀ {M N}
-------------------------------
→ `if0 `zero then M else N ⟶ M
β-if0-suc : ∀ {V M N}
→ Value V
----------------------------------
→ `if0 (`suc V) then M else N ⟶ N
ξ-Y : ∀ {M M′}
→ M ⟶ M′
---------------
→ `Y M ⟶ `Y M′
β-Y : ∀ {V x N}
→ Value V
→ V ≡ `λ x `→ N
-------------------------
→ `Y V ⟶ N [ x := `Y V ]
\end{code}
## Reflexive and transitive closure
\begin{code}
infix 2 _⟶*_
infix 1 begin_
infixr 2 _⟶⟨_⟩_
infix 3 _∎
data _⟶*_ : Term → Term → Set where
_∎ : ∀ (M : Term)
-------------
→ M ⟶* M
_⟶⟨_⟩_ : ∀ (L : Term) {M N}
→ L ⟶ M
→ M ⟶* N
---------
→ L ⟶* N
begin_ : ∀ {M N} → (M ⟶* N) → (M ⟶* N)
begin M⟶*N = M⟶*N
\end{code}
## Canonical forms
\begin{code}
data Canonical : Term → Type → Set where
Zero :
-------------------
Canonical `zero `ℕ
Suc : ∀ {V}
→ Canonical V `ℕ
----------------------
→ Canonical (`suc V) `ℕ
Fun : ∀ {x N A B}
→ ε , x `: A ⊢ N `: B
-------------------------------
→ Canonical (`λ x `→ N) (A `→ B)
\end{code}
## Canonical forms lemma
Every typed value is canonical.
\begin{code}
canonical : ∀ {V A}
→ ε ⊢ V `: A
→ Value V
-------------
→ Canonical V A
canonical ⊢zero Zero = Zero
canonical (⊢suc ⊢V) (Suc VV) = Suc (canonical ⊢V VV)
canonical (⊢λ x∉ ⊢N) Fun = Fun ⊢N
\end{code}
Every canonical form has a type and a value.
\begin{code}
type : ∀ {V A}
→ Canonical V A
--------------
→ ε ⊢ V `: A
type Zero = ⊢zero
type (Suc CV) = ⊢suc (type CV)
type (Fun {x = x} ⊢N) = ⊢λ x∉ ⊢N
where
x∉ : x ∉ ε
x∉ ()
value : ∀ {V A}
→ Canonical V A
-------------
→ Value V
value Zero = Zero
value (Suc CV) = Suc (value CV)
value (Fun ⊢N) = Fun
\end{code}
## Progress
\begin{code}
data Progress (M : Term) (A : Type) : Set where
step : ∀ {N}
→ M ⟶ N
----------
→ Progress M A
done :
Canonical M A
-------------
→ Progress M A
progress : ∀ {M A} → ε ⊢ M `: A → Progress M A
progress (Ax ())
progress (⊢λ x∉ ⊢N) = done (Fun ⊢N)
progress (⊢L · ⊢M) with progress ⊢L
... | step L⟶L′ = step (ξ-·₁ L⟶L′)
... | done (Fun _) with progress ⊢M
... | step M⟶M′ = step (ξ-·₂ Fun M⟶M′)
... | done CM = step (β-`→ (value CM))
progress ⊢zero = done Zero
progress (⊢suc ⊢M) with progress ⊢M
... | step M⟶M′ = step (ξ-suc M⟶M′)
... | done CM = done (Suc CM)
progress (⊢pred ⊢M) with progress ⊢M
... | step M⟶M′ = step (ξ-pred M⟶M′)
... | done Zero = step β-pred-zero
... | done (Suc CM) = step (β-pred-suc (value CM))
progress (⊢if0 ⊢L ⊢M ⊢N) with progress ⊢L
... | step L⟶L′ = step (ξ-if0 L⟶L′)
... | done Zero = step β-if0-zero
... | done (Suc CM) = step (β-if0-suc (value CM))
progress (⊢Y ⊢M) with progress ⊢M
... | step M⟶M′ = step (ξ-Y M⟶M′)
... | done (Fun _) = step (β-Y Fun refl)
\end{code}
## Preservation
### Domain of an environment
\begin{code}
{-
dom : Env → List Id
dom ε = []
dom (Γ , x `: A) = x ∷ dom Γ
dom-lemma : ∀ {Γ y B} → Γ ∋ y `: B → y ∈ dom Γ
dom-lemma Z = here
dom-lemma (S x≢y ⊢y) = there (dom-lemma ⊢y)
free-lemma : ∀ {Γ M A} → Γ ⊢ M `: A → free M ⊆ dom Γ
free-lemma (Ax ⊢x) w∈ with w∈
... | here = dom-lemma ⊢x
... | there ()
free-lemma {Γ} (⊢λ {N = N} ⊢N) = ∷-to-\\ (free-lemma ⊢N)
free-lemma (⊢L · ⊢M) w∈ with ++-to-⊎ w∈
... | inj₁ ∈L = free-lemma ⊢L ∈L
... | inj₂ ∈M = free-lemma ⊢M ∈M
free-lemma ⊢zero ()
free-lemma (⊢suc ⊢M) w∈ = free-lemma ⊢M w∈
free-lemma (⊢pred ⊢M) w∈ = free-lemma ⊢M w∈
free-lemma (⊢if0 ⊢L ⊢M ⊢N) w∈
with ++-to-⊎ w∈
... | inj₁ ∈L = free-lemma ⊢L ∈L
... | inj₂ ∈MN with ++-to-⊎ ∈MN
... | inj₁ ∈M = free-lemma ⊢M ∈M
... | inj₂ ∈N = free-lemma ⊢N ∈N
free-lemma (⊢Y ⊢M) w∈ = free-lemma ⊢M w∈
-}
\end{code}
### Renaming
Let's try an example. The result I want to prove is:
⊢subst : ∀ {Γ Δ ρ}
→ (∀ {x A} → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
-----------------------------------------------
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ subst ρ M `: A)
For this to work, I need to know that neither `Δ` or any of the
bound variables in `ρ x` will collide with any bound variable in `M`.
How can I establish this?
In particular, I need to check that the conditions for ordinary
substitution are sufficient to establish the required invariants.
In that case we have:
⊢substitution : ∀ {Γ x A N B M} →
Γ , x `: A ⊢ N `: B →
Γ ⊢ M `: A →
--------------------
Γ ⊢ N [ x := M ] `: B
Here, since `N` is well-typed, none of it's bound variables collide
with `Γ`, and hence cannot collide with any free variable of `M`.
*But* we can't make a similar guarantee for the *bound* variables
of `M`, so substitution may break the invariants. Here are examples:
(`λ "x" `→ `λ "y" `→ ` "x") (`λ "y" `→ ` "y")
⟶
(`λ "y" → (`λ "y" `→ ` "y"))
ε , "z" `: `ℕ ⊢ (`λ "x" `→ `λ "y" → ` "x" · ` "y" · ` "z") (`λ "y" `→ ` "y" · ` "z")
⟶
ε , "z" `: `ℕ ⊢ (`λ "y" → (`λ "y" `→ ` "y" · ` "z") · ` "y" · ` "z")
This doesn't maintain the invariant, but doesn't break either.
But I don't know how to prove it never breaks. Maybe I can come
up with an example that does break after a few steps. Or, maybe
I don't need the nested variables to be unique. Maybe all I need
is for the free variables in each `ρ x` to be distinct from any
of the bound variables in `N`. But this requires every bound
variable in `N` to not appear in `Γ`. Not clear how to maintain
such a condition without the invariant, so I don't know how
the proof works. Bugger!
Consider a term with free variables, where every bound
variable of the term is distinct from any free variable.
(This is trivially true for a closed term.) Question: if
I never reduce under lambda, do I ever need
to perform renaming?
It's easy to come up with a counter-example if I allow
reduction under lambda.
(λ y → (λ x → λ y → x y) y) ⟶ (λ y → (λ y′ → y y′))
The above requires renaming. But if I remove the outer lambda
(λ x → λ y → x y) y ⟶ (λ y → (λ y′ → y y′))
then the term on the left violates the condition on free
variables, and any term I can think of that causes problems
also violates the condition. So I may be able to do something
here.
\begin{code}
{-
⊢rename : ∀ {Γ Δ xs}
→ (∀ {x A} → Γ ∋ x `: A → Δ ∋ x `: A)
--------------------------------------------------
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ M `: A)
⊢rename ⊢σ (Ax ⊢x) = Ax (⊢σ ⊢x)
⊢rename {Γ} {Δ} ⊢σ (⊢λ {x = x} {N = N} {A = A} x∉Γ ⊢N)
= ⊢λ x∉Δ (⊢rename {Γ′} {Δ′} ⊢σ′ ⊢N)
where
Γ′ = Γ , x `: A
Δ′ = Δ , x `: A
xs′ = x ∷ xs
⊢σ′ : ∀ {w B} → w ∈ xs′ → Γ′ ∋ w `: B → Δ′ ∋ w `: B
⊢σ′ w∈′ Z = Z
⊢σ′ w∈′ (S w≢ ⊢w) = S w≢ (⊢σ ∈w ⊢w)
where
∈w = there⁻¹ w∈′ w≢
⊆xs′ : free N ⊆ xs′
⊆xs′ = \\-to-∷ ⊆xs
⊢rename ⊢σ ⊆xs (⊢L · ⊢M) = ⊢rename ⊢σ L⊆ ⊢L · ⊢rename ⊢σ M⊆ ⊢M
where
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
M⊆ = trans-⊆ ⊆-++₂ ⊆xs
⊢rename ⊢σ ⊆xs (⊢zero) = ⊢zero
⊢rename ⊢σ ⊆xs (⊢suc ⊢M) = ⊢suc (⊢rename ⊢σ ⊆xs ⊢M)
⊢rename ⊢σ ⊆xs (⊢pred ⊢M) = ⊢pred (⊢rename ⊢σ ⊆xs ⊢M)
⊢rename ⊢σ ⊆xs (⊢if0 {L = L} ⊢L ⊢M ⊢N)
= ⊢if0 (⊢rename ⊢σ L⊆ ⊢L) (⊢rename ⊢σ M⊆ ⊢M) (⊢rename ⊢σ N⊆ ⊢N)
where
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
M⊆ = trans-⊆ ⊆-++₁ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
N⊆ = trans-⊆ ⊆-++₂ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
⊢rename ⊢σ ⊆xs (⊢Y ⊢M) = ⊢Y (⊢rename ⊢σ ⊆xs ⊢M)
-}
\end{code}
### Substitution preserves types
\begin{code}
{-
⊢subst : ∀ {Γ Δ xs ys ρ}
→ (∀ {x} → x ∈ xs → free (ρ x) ⊆ ys)
→ (∀ {x A} → x ∈ xs → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
-------------------------------------------------------------
→ (∀ {M A} → free M ⊆ xs → Γ ⊢ M `: A → Δ ⊢ subst ys ρ M `: A)
⊢subst Σ ⊢ρ ⊆xs (Ax ⊢x)
= ⊢ρ (⊆xs here) ⊢x
⊢subst {Γ} {Δ} {xs} {ys} {ρ} Σ ⊢ρ ⊆xs (⊢λ {x = x} {N = N} {A = A} ⊢N)
= ⊢λ {x = y} {A = A} (⊢subst {Γ′} {Δ′} {xs′} {ys′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N)
where
y = fresh ys
Γ′ = Γ , x `: A
Δ′ = Δ , y `: A
xs′ = x ∷ xs
ys′ = y ∷ ys
ρ′ = ρ , x ↦ ` y
Σ′ : ∀ {w} → w ∈ xs′ → free (ρ′ w) ⊆ ys′
Σ′ {w} w∈′ with w ≟ x
... | yes refl = ⊆-++₁
... | no w≢ = ⊆-++₂ ∘ Σ (there⁻¹ w∈′ w≢)
⊆xs′ : free N ⊆ xs′
⊆xs′ = \\-to-∷ ⊆xs
⊢σ : ∀ {w C} → w ∈ ys → Δ ∋ w `: C → Δ′ ∋ w `: C
⊢σ w∈ ⊢w = S (fresh-lemma w∈) ⊢w
⊢ρ′ : ∀ {w C} → w ∈ xs′ → Γ′ ∋ w `: C → Δ′ ⊢ ρ′ w `: C
⊢ρ′ {w} _ Z with w ≟ x
... | yes _ = Ax Z
... | no w≢ = ⊥-elim (w≢ refl)
⊢ρ′ {w} w∈′ (S w≢ ⊢w) with w ≟ x
... | yes refl = ⊥-elim (w≢ refl)
... | no _ = ⊢rename {Δ} {Δ′} {ys} ⊢σ (Σ w∈) (⊢ρ w∈ ⊢w)
where
w∈ = there⁻¹ w∈′ w≢
⊢subst Σ ⊢ρ ⊆xs (⊢L · ⊢M)
= ⊢subst Σ ⊢ρ L⊆ ⊢L · ⊢subst Σ ⊢ρ M⊆ ⊢M
where
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
M⊆ = trans-⊆ ⊆-++₂ ⊆xs
⊢subst Σ ⊢ρ ⊆xs ⊢zero = ⊢zero
⊢subst Σ ⊢ρ ⊆xs (⊢suc ⊢M) = ⊢suc (⊢subst Σ ⊢ρ ⊆xs ⊢M)
⊢subst Σ ⊢ρ ⊆xs (⊢pred ⊢M) = ⊢pred (⊢subst Σ ⊢ρ ⊆xs ⊢M)
⊢subst Σ ⊢ρ ⊆xs (⊢if0 {L = L} ⊢L ⊢M ⊢N)
= ⊢if0 (⊢subst Σ ⊢ρ L⊆ ⊢L) (⊢subst Σ ⊢ρ M⊆ ⊢M) (⊢subst Σ ⊢ρ N⊆ ⊢N)
where
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
M⊆ = trans-⊆ ⊆-++₁ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
N⊆ = trans-⊆ ⊆-++₂ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
⊢subst Σ ⊢ρ ⊆xs (⊢Y ⊢M) = ⊢Y (⊢subst Σ ⊢ρ ⊆xs ⊢M)
⊢substitution : ∀ {Γ x A N B M} →
Γ , x `: A ⊢ N `: B →
Γ ⊢ M `: A →
--------------------
Γ ⊢ N [ x := M ] `: B
⊢substitution {Γ} {x} {A} {N} {B} {M} ⊢N ⊢M =
⊢subst {Γ′} {Γ} {xs} {ys} {ρ} Σ ⊢ρ {N} {B} ⊆xs ⊢N
where
Γ′ = Γ , x `: A
xs = free N
ys = free M ++ (free N \\ x)
ρ = ∅ , x ↦ M
Σ : ∀ {w} → w ∈ xs → free (ρ w) ⊆ ys
Σ {w} w∈ y∈ with w ≟ x
... | yes _ = ⊆-++₁ y∈
... | no w≢ rewrite ∈-[_] y∈ = ⊆-++₂ (∈-≢-to-\\ w∈ w≢)
⊢ρ : ∀ {w B} → w ∈ xs → Γ′ ∋ w `: B → Γ ⊢ ρ w `: B
⊢ρ {w} w∈ Z with w ≟ x
... | yes _ = ⊢M
... | no w≢ = ⊥-elim (w≢ refl)
⊢ρ {w} w∈ (S w≢ ⊢w) with w ≟ x
... | yes refl = ⊥-elim (w≢ refl)
... | no _ = Ax ⊢w
⊆xs : free N ⊆ xs
⊆xs x∈ = x∈
-}
\end{code}
### Preservation
\begin{code}
{-
preservation : ∀ {Γ M N A}
→ Γ ⊢ M `: A
→ M ⟶ N
---------
→ Γ ⊢ N `: A
preservation (Ax ⊢x) ()
preservation (⊢λ ⊢N) ()
preservation (⊢L · ⊢M) (ξ-·₁ L⟶) = preservation ⊢L L⟶ · ⊢M
preservation (⊢V · ⊢M) (ξ-·₂ _ M⟶) = ⊢V · preservation ⊢M M⟶
preservation ((⊢λ ⊢N) · ⊢W) (β-`→ _) = ⊢substitution ⊢N ⊢W
preservation (⊢zero) ()
preservation (⊢suc ⊢M) (ξ-suc M⟶) = ⊢suc (preservation ⊢M M⟶)
preservation (⊢pred ⊢M) (ξ-pred M⟶) = ⊢pred (preservation ⊢M M⟶)
preservation (⊢pred ⊢zero) (β-pred-zero) = ⊢zero
preservation (⊢pred (⊢suc ⊢M)) (β-pred-suc _) = ⊢M
preservation (⊢if0 ⊢L ⊢M ⊢N) (ξ-if0 L⟶) = ⊢if0 (preservation ⊢L L⟶) ⊢M ⊢N
preservation (⊢if0 ⊢zero ⊢M ⊢N) β-if0-zero = ⊢M
preservation (⊢if0 (⊢suc ⊢V) ⊢M ⊢N) (β-if0-suc _) = ⊢N
preservation (⊢Y ⊢M) (ξ-Y M⟶) = ⊢Y (preservation ⊢M M⟶)
preservation (⊢Y (⊢λ ⊢N)) (β-Y _ refl) = ⊢substitution ⊢N (⊢Y (⊢λ ⊢N))
-}
\end{code}
## Normalise
\begin{code}
{-
data Normalise {M A} (⊢M : ε ⊢ M `: A) : Set where
out-of-gas : ∀ {N} → M ⟶* N → ε ⊢ N `: A → Normalise ⊢M
normal : ∀ {V} → ℕ → Canonical V A → M ⟶* V → Normalise ⊢M
normalise : ∀ {L A} → ℕ → (⊢L : ε ⊢ L `: A) → Normalise ⊢L
normalise {L} zero ⊢L = out-of-gas (L ∎) ⊢L
normalise {L} (suc m) ⊢L with progress ⊢L
... | done CL = normal (suc m) CL (L ∎)
... | step L⟶M with preservation ⊢L L⟶M
... | ⊢M with normalise m ⊢M
... | out-of-gas M⟶*N ⊢N = out-of-gas (L ⟶⟨ L⟶M ⟩ M⟶*N) ⊢N
... | normal n CV M⟶*V = normal n CV (L ⟶⟨ L⟶M ⟩ M⟶*V)
-}
\end{code}