From c75cd9dc061083260a26a1e668d981815ca1ac1a Mon Sep 17 00:00:00 2001
From: wadler <wadler@inf.ed.ac.uk>
Date: Fri, 23 Feb 2018 13:45:53 +0100
Subject: [PATCH] further work on DeBruijn

---
 src/extra/DeBruijn.lagda | 90 +++++++++++++++++++++++++++++++++++-----
 1 file changed, 80 insertions(+), 10 deletions(-)

diff --git a/src/extra/DeBruijn.lagda b/src/extra/DeBruijn.lagda
index 60a4ad76..66b71256 100644
--- a/src/extra/DeBruijn.lagda
+++ b/src/extra/DeBruijn.lagda
@@ -10,7 +10,7 @@ module DeBruijn where
 import Relation.Binary.PropositionalEquality as Eq
 open Eq using (_≡_; refl; sym; trans; cong)
 open Eq.≡-Reasoning
-open import Data.Nat using (ℕ; zero; suc)
+open import Data.Nat using (ℕ; zero; suc; _+_)
 open import Data.Product using (_×_; proj₁; proj₂) renaming (_,_ to ⟨_,_⟩)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
 open import Relation.Nullary using (¬_)
@@ -20,12 +20,14 @@ open import Data.Empty using (⊥; ⊥-elim)
 open import Function using (_∘_)
 \end{code}
 
-## Representation
+## Syntax
 
 \begin{code}
+infixr 4 _⇒_
+
 data Type : Set where
-  `ℕ : Type
-  _`→_ : Type → Type → Type
+  o : Type
+  _⇒_ : Type → Type → Type
 
 data Env : Set where
   ε : Env
@@ -37,23 +39,91 @@ data Var : Env → Type → Set where
 
 data Exp : Env → Type → Set where
   var : ∀ {Γ : Env} {A : Type} → Var Γ A → Exp Γ A
-  abs : ∀ {Γ : Env} {A B : Type} → Exp (Γ , A) B → Exp Γ (A `→ B)
-  app : ∀ {Γ : Env} {A B : Type} → Exp Γ (A `→ B) → Exp Γ A → Exp Γ B
+  abs : ∀ {Γ : Env} {A B : Type} → Exp (Γ , A) B → Exp Γ (A ⇒ B)
+  app : ∀ {Γ : Env} {A B : Type} → Exp Γ (A ⇒ B) → Exp Γ A → Exp Γ B
 
 type : Type → Set
-type `ℕ = ℕ
-type (A `→ B) = type A → type B
+type o = ℕ
+type (A ⇒ B) = type A → type B
 
 env : Env → Set
 env ε = ⊤
 env (Γ , A) = env Γ × type A
+\end{code}
 
+# Test code
+
+\begin{code}
+Church : Type
+Church = (o ⇒ o) ⇒ o ⇒ o
+
+plus : Exp ε (Church ⇒ Church ⇒ Church)
+plus =  (abs (abs (abs (abs (app (app (var (suc (suc (suc zero)))) (var (suc zero))) (app (app (var (suc (suc zero))) (var (suc zero))) (var zero)))))))
+
+one : Exp ε Church
+one =  (abs (abs (app (var (suc zero)) (var zero))))
+
+two : Exp ε Church
+two = (app (app plus one) one)
+
+four : Exp ε Church
+four = (app (app plus two) two)
+\end{code}
+
+
+# Denotational semantics
+
+\begin{code}
 lookup : ∀ {Γ : Env} {A : Type} → Var Γ A → env Γ → type A
 lookup zero ⟨ ρ , v ⟩ = v
 lookup (suc n) ⟨ ρ , v ⟩ = lookup n ρ
 
 eval : ∀ {Γ : Env} {A : Type} → Exp Γ A → env Γ → type A
-eval (var n) ρ  = lookup n ρ
-eval (abs N) ρ  = λ{ v → eval N ⟨ ρ , v ⟩ }
+eval (var n) ρ  =  lookup n ρ
+eval (abs N) ρ  =  λ{ v → eval N ⟨ ρ , v ⟩ }
 eval (app L M) ρ  =  eval L ρ (eval M ρ)
+
+ex : eval four tt suc 0 ≡ 4
+ex = refl
+\end{code}
+
+# Operational semantics
+
+## Substitution
+
+\begin{code}
+extr : ∀ {Γ Δ : Env} {B : Type} → (∀ {A : Type} → Var Γ A → Var Δ A) → Var Δ B → (∀ {A : Type} → Var (Γ , B) A → Var Δ A)
+extr ρ v zero     =  v
+extr ρ v (suc k)  =  ρ k
+
+ren : ∀ {Γ Δ : Env} → (∀ {A : Type} → Var Γ A → Var Δ A) → (∀ {A : Type} → Exp Γ A → Exp Δ A)
+ren ρ (var n)    =  var (ρ n)
+ren ρ (app L M)  =  app (ren ρ L) (ren ρ M)
+ren ρ (abs N)    =  abs (ren (extr (suc ∘ ρ) zero) N)
+\end{code}
+
+\begin{code}
+ext :  ∀ {Γ Δ : Env} {B : Type} → (∀ {A : Type} → Var Γ A → Exp Δ A) → Exp Δ B → (∀ {A : Type} → Var (Γ , B) A → Exp Δ A)
+ext ρ v zero     =  v
+ext ρ v (suc k)  =  ρ k
+
+emp : ∀ {Γ : Env} → (∀ {A : Type} → Var Γ A → Exp Γ A)
+emp k = var k
+
+sub : ∀ {Γ Δ : Env} → (∀ {A : Type} → Var Γ A → Exp Δ A) → (∀ {A : Type} → Exp Γ A → Exp Δ A)
+sub ρ (var n)    =  ρ n
+sub ρ (app L M)  =  app (sub ρ L) (sub ρ M)
+sub ρ (abs N)    =  abs (sub (ext (ren suc ∘ ρ) (var zero)) N)
+
+subst : ∀ {Γ : Env} {A B : Type} → Exp (Γ , A) B → Exp Γ A → Exp Γ B
+subst N M =  sub (ext emp M) N 
+\end{code}
+
+## Reduction step
+
+\begin{code}
+data Val : {Γ : Env} {A : Type} → Exp Γ A → Set where
+
+data Step : {Γ : Env} {A : Type} → Exp Γ A → Exp Γ A → Set where
+  beta : ∀ {Γ : Env} {A B : Type} {N : Exp (Γ , A) B} {M : Exp Γ A} → Step (app (abs N) M) (subst N M) 
 \end{code}