---
title : "Typed: Raw terms with types (broken)"
permalink : /Typed
---
This version uses raw terms. Substitution presumes that no
generation of fresh names is required.
The substitution algorithm is based on one by McBride.
It is given a map from names to terms. Say the mapping of a
name is trivial if it takes a name to a term consisting of
just the variable with that name. No fresh names are required
if the mapping on each variable is either trivial or to a
closed term. However, the proof of correctness currently
contains a hole, and may be difficult to repair.
## 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 (¬?)
import Collections
pattern [_] x = x ∷ []
pattern [_,_] x y = x ∷ y ∷ []
pattern [_,_,_] x y z = x ∷ y ∷ z ∷ []
\end{code}
## Syntax
\begin{code}
infix 4 _∋_`:_
infix 4 _⊢_`:_
infixl 5 _,_`:_
infixr 6 _`→_
infix 6 `λ_`→_
infixl 7 `if0_then_else_
infix 8 `suc_ `pred_ `Y_
infixl 9 _·_
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}
→ w ≢ x
→ Γ ∋ w `: B
--------------------
→ Γ , x `: A ∋ w `: B
data _⊢_`:_ : Env → Term → Type → Set where
Ax : ∀ {Γ A x}
→ Γ ∋ x `: A
--------------
→ Γ ⊢ ` x `: A
⊢λ : ∀ {Γ x N A B}
→ 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
\end{code}
## Test examples
\begin{code}
s≢z : "s" ≢ "z"
s≢z ()
n≢z : "n" ≢ "z"
n≢z ()
n≢s : "n" ≢ "s"
n≢s ()
m≢z : "m" ≢ "z"
m≢z ()
m≢s : "m" ≢ "s"
m≢s ()
m≢n : "m" ≢ "n"
m≢n ()
p≢n : "p" ≢ "n"
p≢n ()
p≢m : "p" ≢ "m"
p≢m ()
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 `suc (` "p" · (`pred (` "m")) · ` "n"))
⊢plus : ε ⊢ plus `: `ℕ `→ `ℕ `→ `ℕ
⊢plus = (⊢Y (⊢λ (⊢λ (⊢λ (⊢if0 (Ax ⊢m) (Ax ⊢n) (⊢suc (Ax ⊢p · (⊢pred (Ax ⊢m)) · Ax ⊢n)))))))
where
⊢p = S p≢n (S p≢m Z)
⊢m = S m≢n Z
⊢n = Z
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 (⊢λ (Ax ⊢s · (Ax ⊢s · Ax ⊢z))))
where
⊢s = S s≢z Z
⊢z = Z
plusCh : Term
plusCh = `λ "m" `→ `λ "n" `→ `λ "s" `→ `λ "z" `→
` "m" · ` "s" · (` "n" · ` "s" · ` "z")
⊢plusCh : ε ⊢ plusCh `: Ch `→ Ch `→ Ch
⊢plusCh = (⊢λ (⊢λ (⊢λ (⊢λ (Ax ⊢m · Ax ⊢s · (Ax ⊢n · Ax ⊢s · Ax ⊢z))))))
where
⊢m = S m≢z (S m≢s (S m≢n Z))
⊢n = S n≢z (S n≢s Z)
⊢s = S s≢z Z
⊢z = Z
fromCh : Term
fromCh = `λ "m" `→ ` "m" · (`λ "s" `→ `suc ` "s") · `zero
⊢fromCh : ε ⊢ fromCh `: Ch `→ `ℕ
⊢fromCh = (⊢λ (Ax ⊢m · (⊢λ (⊢suc (Ax ⊢s))) · ⊢zero))
where
⊢m = Z
⊢s = Z
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≢ ⊢w) = lookup {Γ} ⊢w
erase : ∀ {Γ M A} → Γ ⊢ M `: A → Term
erase (Ax ⊢w) = ` lookup ⊢w
erase (⊢λ {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≢ ⊢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} ⊢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 (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" ↦ two , "n" ↦ four) (` "m" · ` "n") ≡ (two · four)
_ = 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 : ∀ {F x N}
→ F ≡ `λ x `→ N
-------------------------
→ `Y F ⟶ N [ x := `Y F ]
\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}
## Sample execution
\begin{code}
_ : plus · two · two ⟶* (`suc (`suc (`suc (`suc `zero))))
_ =
begin
plus · two · two
⟶⟨ ξ-·₁ (ξ-·₁ (β-Y refl)) ⟩
(`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
`suc (plus · (`pred (` "m")) · (` "n")))) · two · two
⟶⟨ ξ-·₁ (β-→ (Suc (Suc Zero))) ⟩
(`λ "n" `→ `if0 two then ` "n" else
`suc (plus · (`pred two) · (` "n"))) · two
⟶⟨ β-→ (Suc (Suc Zero)) ⟩
`if0 two then two else
`suc (plus · (`pred two) · two)
⟶⟨ β-if0-suc (Suc Zero) ⟩
`suc (plus · (`pred two) · two)
⟶⟨ ξ-suc (ξ-·₁ (ξ-·₁ (β-Y refl))) ⟩
`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
`suc (plus · (`pred (` "m")) · (` "n")))) · (`pred two) · two)
⟶⟨ ξ-suc (ξ-·₁ (ξ-·₂ Fun (β-pred-suc (Suc Zero)))) ⟩
`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
`suc (plus · (`pred (` "m")) · (` "n")))) · (`suc `zero) · two)
⟶⟨ ξ-suc (ξ-·₁ (β-→ (Suc Zero))) ⟩
`suc ((`λ "n" `→ `if0 `suc `zero then ` "n" else
`suc (plus · (`pred (`suc `zero)) · (` "n")))) · two
⟶⟨ ξ-suc (β-→ (Suc (Suc Zero))) ⟩
`suc (`if0 `suc `zero then two else
`suc (plus · (`pred (`suc `zero)) · two))
⟶⟨ ξ-suc (β-if0-suc Zero) ⟩
`suc (`suc (plus · (`pred (`suc `zero)) · two))
⟶⟨ ξ-suc (ξ-suc (ξ-·₁ (ξ-·₁ (β-Y refl)))) ⟩
`suc (`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
`suc (plus · (`pred (` "m")) · (` "n")))) · (`pred (`suc `zero)) · two))
⟶⟨ ξ-suc (ξ-suc (ξ-·₁ (ξ-·₂ Fun (β-pred-suc Zero)))) ⟩
`suc (`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
`suc (plus · (`pred (` "m")) · (` "n")))) · `zero · two))
⟶⟨ ξ-suc (ξ-suc (ξ-·₁ (β-→ Zero))) ⟩
`suc (`suc ((`λ "n" `→ `if0 `zero then ` "n" else
`suc (plus · (`pred `zero) · (` "n"))) · two))
⟶⟨ ξ-suc (ξ-suc (β-→ (Suc (Suc Zero)))) ⟩
`suc (`suc (`if0 `zero then two else
`suc (plus · (`pred `zero) · two)))
⟶⟨ ξ-suc (ξ-suc β-if0-zero) ⟩
`suc (`suc (`suc (`suc `zero)))
∎
\end{code}
## Values do not reduce
Values do not reduce.
\begin{code}
Val-⟶ : ∀ {M N} → Value M → ¬ (M ⟶ N)
Val-⟶ Fun ()
Val-⟶ Zero ()
Val-⟶ (Suc VM) (ξ-suc M⟶N) = Val-⟶ VM M⟶N
\end{code}
As a corollary, terms that reduce are not values.
\begin{code}
⟶-Val : ∀ {M N} → (M ⟶ N) → ¬ Value M
⟶-Val M⟶N VM = Val-⟶ VM M⟶N
\end{code}
## Reduction is deterministic
\begin{code}
det : ∀ {M M′ M″}
→ (M ⟶ M′)
→ (M ⟶ M″)
----------
→ M′ ≡ M″
det (ξ-·₁ L⟶L′) (ξ-·₁ L⟶L″) = cong₂ _·_ (det L⟶L′ L⟶L″) refl
det (ξ-·₁ L⟶L′) (ξ-·₂ VL _) = ⊥-elim (Val-⟶ VL L⟶L′)
det (ξ-·₁ L⟶L′) (β-→ _) = ⊥-elim (Val-⟶ Fun L⟶L′)
det (ξ-·₂ VL _) (ξ-·₁ L⟶L″) = ⊥-elim (Val-⟶ VL L⟶L″)
det (ξ-·₂ _ M⟶M′) (ξ-·₂ _ M⟶M″) = cong₂ _·_ refl (det M⟶M′ M⟶M″)
det (ξ-·₂ _ M⟶M′) (β-→ VM) = ⊥-elim (Val-⟶ VM M⟶M′)
det (β-→ VM) (ξ-·₁ L⟶L″) = ⊥-elim (Val-⟶ Fun L⟶L″)
det (β-→ VM) (ξ-·₂ _ M⟶M″) = ⊥-elim (Val-⟶ VM M⟶M″)
det (β-→ _) (β-→ _) = refl
det (ξ-suc M⟶M′) (ξ-suc M⟶M″) = cong `suc_ (det M⟶M′ M⟶M″)
det (ξ-pred M⟶M′) (ξ-pred M⟶M″) = cong `pred_ (det M⟶M′ M⟶M″)
det (ξ-pred M⟶M′) β-pred-zero = ⊥-elim (Val-⟶ Zero M⟶M′)
det (ξ-pred M⟶M′) (β-pred-suc VM) = ⊥-elim (Val-⟶ (Suc VM) M⟶M′)
det β-pred-zero (ξ-pred M⟶M′) = ⊥-elim (Val-⟶ Zero M⟶M′)
det β-pred-zero β-pred-zero = refl
det (β-pred-suc VM) (ξ-pred M⟶M′) = ⊥-elim (Val-⟶ (Suc VM) M⟶M′)
det (β-pred-suc _) (β-pred-suc _) = refl
det (ξ-if0 L⟶L′) (ξ-if0 L⟶L″) = cong₃ `if0_then_else_ (det L⟶L′ L⟶L″) refl refl
det (ξ-if0 L⟶L′) β-if0-zero = ⊥-elim (Val-⟶ Zero L⟶L′)
det (ξ-if0 L⟶L′) (β-if0-suc VL) = ⊥-elim (Val-⟶ (Suc VL) L⟶L′)
det β-if0-zero (ξ-if0 L⟶L″) = ⊥-elim (Val-⟶ Zero L⟶L″)
det β-if0-zero β-if0-zero = refl
det (β-if0-suc VL) (ξ-if0 L⟶L″) = ⊥-elim (Val-⟶ (Suc VL) L⟶L″)
det (β-if0-suc _) (β-if0-suc _) = refl
det (ξ-Y M⟶M′) (ξ-Y M⟶M″) = cong `Y_ (det M⟶M′ M⟶M″)
det (ξ-Y M⟶M′) (β-Y refl) = ⊥-elim (Val-⟶ Fun M⟶M′)
det (β-Y refl) (ξ-Y M⟶M″) = ⊥-elim (Val-⟶ Fun M⟶M″)
det (β-Y refl) (β-Y refl) = refl
\end{code}
Almost half the lines in the above proof are redundant, for example
det (ξ-·₁ L⟶L′) (ξ-·₂ VL _) = ⊥-elim (Val-⟶ VL L⟶L′)
det (ξ-·₂ VL _) (ξ-·₁ L⟶L″) = ⊥-elim (Val-⟶ VL L⟶L″)
are essentially identical. What we might like to do is delete the
redundant lines and add
det M⟶M′ M⟶M″ = sym (det M⟶M″ M⟶M′)
to the bottom of the proof. But this does not work. The termination
checker complains, because the arguments have merely switched order
and neither is smaller.
## 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 ⊢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 ⊢N
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 ⊢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 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
\begin{code}
⊢rename : ∀ {Γ Δ}
→ (∀ {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} ⊢N)
= ⊢λ (⊢rename {Γ′} {Δ′} ⊢σ′ ⊢N)
where
Γ′ = Γ , x `: A
Δ′ = Δ , x `: A
⊢σ′ : ∀ {w B} → Γ′ ∋ w `: B → Δ′ ∋ w `: B
⊢σ′ Z = Z
⊢σ′ (S w≢ ⊢w) = S w≢ (⊢σ ⊢w)
⊢rename ⊢σ (⊢L · ⊢M) = ⊢rename ⊢σ ⊢L · ⊢rename ⊢σ ⊢M
⊢rename ⊢σ (⊢zero) = ⊢zero
⊢rename ⊢σ (⊢suc ⊢M) = ⊢suc (⊢rename ⊢σ ⊢M)
⊢rename ⊢σ (⊢pred ⊢M) = ⊢pred (⊢rename ⊢σ ⊢M)
⊢rename ⊢σ (⊢if0 ⊢L ⊢M ⊢N)
= ⊢if0 (⊢rename ⊢σ ⊢L) (⊢rename ⊢σ ⊢M) (⊢rename ⊢σ ⊢N)
⊢rename ⊢σ (⊢Y ⊢M) = ⊢Y (⊢rename ⊢σ ⊢M)
\end{code}
### Substitution preserves types
\begin{code}
-- trivial : Set
-- trivial = ∀ ρ x → ρ x ≡ ` x ⊎ closed (ρ x)
⊢subst : ∀ {Γ Δ ρ}
-- → (∀ {x A} → Γ ∋ x `: A → trivial ρ x)
→ (∀ {x A} → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
-------------------------------------------------
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ subst ρ M `: A)
⊢subst ⊢ρ (Ax ⊢x) = ⊢ρ ⊢x
⊢subst {Γ} {Δ} {ρ} ⊢ρ (⊢λ {x = x} {N = N} {A = A} ⊢N)
= ⊢λ {x = x} {A = A} (⊢subst {Γ′} {Δ′} {ρ′} ⊢ρ′ ⊢N)
where
Γ′ = Γ , x `: A
Δ′ = Δ , x `: A
ρ′ = ρ , x ↦ ` x
⊢σ : ∀ {w C} → Δ ∋ w `: C → Δ′ ∋ w `: C
⊢σ ⊢w = S {!!} ⊢w
⊢ρ′ : ∀ {w C} → Γ′ ∋ w `: C → Δ′ ⊢ ρ′ w `: C
⊢ρ′ {w} Z with w ≟ x
... | yes _ = Ax Z
... | no w≢ = ⊥-elim (w≢ refl)
⊢ρ′ {w} (S w≢ ⊢w) with w ≟ x
... | yes refl = ⊥-elim (w≢ refl)
... | no _ = ⊢rename {Δ} {Δ′} ⊢σ (⊢ρ ⊢w)
⊢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 ⊢M ⊢N)
= ⊢if0 (⊢subst ⊢ρ ⊢L) (⊢subst ⊢ρ ⊢M) (⊢subst ⊢ρ ⊢N)
⊢subst ⊢ρ (⊢Y ⊢M) = ⊢Y (⊢subst ⊢ρ ⊢M)
\end{code}
Let's look at examples. Assume `M` is closed. Example 1.
subst (∅ , "x" ↦ M) (`λ "y" `→ ` "x") ≡ `λ "y" `→ M
Example 2.
subst (∅ , "y" ↦ N , "x" ↦ M) (`λ "y" `→ ` "x" · ` "y")
≡
`λ "y" `→ subst (∅ , "y" ↦ ` N , "x" ↦ M , "y" ↦ ` "y") (` "x" · ` "y")
≡
`λ "y" `→ (M · ` "y")
Before I wrote: "The hypotheses of the theorem appear to be violated. Drat!"
But let's assume that ``M `: A``, ``N `: B``, and the lambda bound `y` has type `C`.
Then ``Γ ∋ y `: B`` will not hold for the extended `ρ` because of interference
by the earlier `y`. So I'm not sure the hypothesis is violated.
\begin{code}
⊢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} {ρ} ⊢ρ {N} {B} ⊢N
where
Γ′ = Γ , x `: A
ρ = ∅ , x ↦ M
⊢ρ : ∀ {w B} → Γ′ ∋ w `: B → Γ ⊢ ρ w `: B
⊢ρ {w} Z with w ≟ x
... | yes _ = ⊢M
... | no w≢ = ⊥-elim (w≢ refl)
⊢ρ {w} (S w≢ ⊢w) with w ≟ x
... | yes refl = ⊥-elim (w≢ refl)
... | no _ = Ax ⊢w
\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}