From 9c72c8ed0c3e10b5dafb41e438285b08f244ba68 Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Sun, 21 Mar 2021 13:14:22 +0000
Subject: Prove judgement weakening.

---
 src/Cfe/Judgement/Base.agda       |  2 +-
 src/Cfe/Judgement/Properties.agda | 60 +++++++++++++++++++++++++++++++++++++++
 2 files changed, 61 insertions(+), 1 deletion(-)

(limited to 'src/Cfe/Judgement')

diff --git a/src/Cfe/Judgement/Base.agda b/src/Cfe/Judgement/Base.agda
index 6b42598..0e5417a 100644
--- a/src/Cfe/Judgement/Base.agda
+++ b/src/Cfe/Judgement/Base.agda
@@ -22,6 +22,6 @@ data _⊢_∶_ : {n : ℕ} → Context n → Expression n → Type ℓ ℓ → S
   Char : ∀ {n} {Γ,Δ : Context n} c → Γ,Δ ⊢ Char c ∶ Lift ℓ ℓ τ[ c ]
   Bot : ∀ {n} {Γ,Δ : Context n} → Γ,Δ ⊢ ⊥ ∶ Lift ℓ ℓ τ⊥
   Var : ∀ {n} {Γ,Δ : Context n} {i} (i≥m : toℕ i ≥ _) → Γ,Δ ⊢ Var i ∶ lookup (Context.Γ Γ,Δ) (reduce≥′ (Context.m≤n Γ,Δ) i i≥m)
-  Fix : ∀ {n} {Γ,Δ : Context n} {e τ} → cons τ Γ,Δ ⊢ e ∶ τ → Γ,Δ ⊢ μ e ∶ τ
+  Fix : ∀ {n} {Γ,Δ : Context n} {e τ} → cons Γ,Δ τ ⊢ e ∶ τ → Γ,Δ ⊢ μ e ∶ τ
   Cat : ∀ {n} {Γ,Δ : Context n} {e₁ e₂ τ₁ τ₂} → Γ,Δ ⊢ e₁ ∶ τ₁ → shift Γ,Δ ⊢ e₂ ∶ τ₂ → (τ₁⊛τ₂ : τ₁ ⊛ τ₂) → Γ,Δ ⊢ e₁ ∙ e₂ ∶ τ₁ ∙ₜ τ₂
   Vee : ∀ {n} {Γ,Δ : Context n} {e₁ e₂ τ₁ τ₂} → Γ,Δ ⊢ e₁ ∶ τ₁ → Γ,Δ ⊢ e₂ ∶ τ₂ → (τ₁#τ₂ : τ₁ # τ₂) → Γ,Δ ⊢ e₁ ∨ e₂ ∶ τ₁ ∨ₜ τ₂
diff --git a/src/Cfe/Judgement/Properties.agda b/src/Cfe/Judgement/Properties.agda
index b901ced..7f357f0 100644
--- a/src/Cfe/Judgement/Properties.agda
+++ b/src/Cfe/Judgement/Properties.agda
@@ -5,3 +5,63 @@ open import Relation.Binary using (Setoid)
 module Cfe.Judgement.Properties
   {c ℓ} (over : Setoid c ℓ)
   where
+
+open import Cfe.Context over renaming (wkn₁ to cwkn₁; wkn₂ to cwkn₂; _≋_ to _≋ᶜ_)
+open import Cfe.Expression over
+open import Cfe.Judgement.Base over
+open import Data.Fin
+open import Data.Nat as ℕ
+open import Data.Nat.Properties
+open import Data.Product
+open import Data.Vec
+open import Relation.Binary.PropositionalEquality
+
+toℕ-punchIn : ∀ {n} i j → toℕ j ℕ.≤ toℕ (punchIn {n} i j)
+toℕ-punchIn zero j = n≤1+n (toℕ j)
+toℕ-punchIn (suc i) zero = ≤-refl
+toℕ-punchIn (suc i) (suc j) = s≤s (toℕ-punchIn i j)
+
+congᶜ : ∀ {n} {Γ,Δ Γ,Δ′ : Context n} {e τ} → Γ,Δ ≋ᶜ Γ,Δ′ → Γ,Δ ⊢ e ∶ τ → Γ,Δ′ ⊢ e ∶ τ
+congᶜ {Γ,Δ = Γ,Δ} {Γ,Δ′} (refl , refl , refl) Γ,Δ⊢e∶τ with ≤-irrelevant (Context.m≤n Γ,Δ) (Context.m≤n Γ,Δ′)
+... | refl = Γ,Δ⊢e∶τ
+
+congᵗ : ∀ {n} {Γ,Δ : Context n} {e τ τ′} → τ ≡ τ′ → Γ,Δ ⊢ e ∶ τ → Γ,Δ ⊢ e ∶ τ′
+congᵗ refl Γ,Δ⊢e∶τ = Γ,Δ⊢e∶τ
+
+wkn₁ : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ i τ′ i≥m → cwkn₁ Γ,Δ i i≥m τ′ ⊢ wkn e i ∶ τ
+wkn₁ Eps i τ′ i≥m = Eps
+wkn₁ (Char c) i τ′ i≥m = Char c
+wkn₁ Bot i τ′ i≥m = Bot
+wkn₁ {Γ,Δ = Γ,Δ} (Var {i = j} j≥m) i τ′ i≥m = congᵗ (τ≡τ′ Γ,Δ i j i≥m j≥m τ′) (Var (≤-trans j≥m (toℕ-punchIn i j)))
+  where
+  open Context Γ,Δ
+  τ≡τ′ : ∀ {n} (Γ,Δ : Context n) i j i≥m j≥m τ → lookup (Context.Γ (cwkn₁ Γ,Δ i i≥m τ)) (reduce≥′ (≤-step (Context.m≤n Γ,Δ)) (punchIn i j) (≤-trans j≥m (toℕ-punchIn i j))) ≡ lookup (Context.Γ Γ,Δ) (reduce≥′ (Context.m≤n Γ,Δ) j j≥m)
+  τ≡τ′ {suc _} record { m = zero ; m≤n = _ ; Γ = (_ ∷ _) ; Δ = _ } zero zero _ _ _ = refl
+  τ≡τ′ {suc n} record { m = zero ; m≤n = _ ; Γ = (_ ∷ Γ) ; Δ = Δ } zero (suc j) _ _ τ = τ≡τ′ (record { m≤n = z≤n ; Γ = Γ ; Δ = Δ }) zero j z≤n z≤n τ
+  τ≡τ′ {suc n} record { m = zero ; m≤n = _ ; Γ = (_ ∷ _) ; Δ = _ } (suc _) zero _ _ τ = refl
+  τ≡τ′ {suc n} record { m = zero ; m≤n = _ ; Γ = (_ ∷ Γ) ; Δ = Δ } (suc i) (suc j) _ _ τ = τ≡τ′ (record { m≤n = z≤n ; Γ = Γ ; Δ = Δ}) i j z≤n z≤n τ
+  τ≡τ′ {suc n} record { m = (suc m) ; m≤n = (s≤s m≤n) ; Γ = Γ ; Δ = (_ ∷ Δ) } (suc i) (suc j) (s≤s i≥m) (s≤s j≥m) τ = τ≡τ′ (record { m≤n = m≤n ; Γ = Γ ; Δ = Δ}) i j i≥m j≥m τ
+wkn₁ (Fix Γ,Δ⊢e∶τ) i τ′ i≥m = Fix (wkn₁ Γ,Δ⊢e∶τ (suc i) τ′ (s≤s i≥m))
+wkn₁ {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i τ′ i≥m = Cat (wkn₁ Γ,Δ⊢e₁∶τ₁ i τ′ i≥m) (congᶜ (≋-sym (wkn₁-shift Γ,Δ i i≥m τ′)) (wkn₁ Δ++Γ,∙⊢e₂∶τ₂ i τ′ z≤n)) τ₁⊛τ₂
+wkn₁ (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i τ′ i≥m = Vee (wkn₁ Γ,Δ⊢e₁∶τ₁ i τ′ i≥m) (wkn₁ Γ,Δ⊢e₂∶τ₂ i τ′ i≥m) τ₁#τ₂
+
+wkn₂ : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ i τ′ i≤m → cwkn₂ Γ,Δ i i≤m τ′ ⊢ wkn e i ∶ τ
+wkn₂ Eps i τ′ i≤m = Eps
+wkn₂ (Char c) i τ′ i≤m = Char c
+wkn₂ Bot i τ′ i≤m = Bot
+wkn₂ {Γ,Δ = Γ,Δ} (Var {i = j} j≥m) i τ′ i≤m =
+  congᵗ
+    (τ≡τ′ (Context.Γ Γ,Δ) (Context.m≤n Γ,Δ) i j i≤m j≥m)
+    (Var (punchIn[i,j]≥m i j i≤m j≥m))
+  where
+  punchIn[i,j]≥m : ∀ {m n} i j → toℕ i ℕ.≤ m → toℕ j ≥ m → toℕ (punchIn {n} i j) ≥ suc m
+  punchIn[i,j]≥m {m} zero j i≤m j≥m = s≤s j≥m
+  punchIn[i,j]≥m {suc m} (suc i) (suc j) (s≤s i≤m) (s≤s j≥m) = s≤s (punchIn[i,j]≥m i j i≤m j≥m)
+
+  τ≡τ′ : ∀ {a A m n} xs m≤n i j i≤m j≥m → lookup {a} {A} xs (reduce≥′ {suc m} (s≤s m≤n) (punchIn {n} i j) (punchIn[i,j]≥m i j i≤m j≥m)) ≡ lookup xs (reduce≥′ m≤n j j≥m)
+  τ≡τ′ {m = zero} xs m≤n zero j i≤m j≥m = refl
+  τ≡τ′ {m = suc _} xs m≤n zero (suc j) i≤m (s≤s j≥m) = τ≡τ′ xs (pred-mono m≤n) zero j z≤n j≥m
+  τ≡τ′ {m = suc _} xs m≤n (suc i) (suc j) (s≤s i≤m) (s≤s j≥m) = τ≡τ′ xs (pred-mono m≤n) i j i≤m j≥m
+wkn₂ (Fix Γ,Δ⊢e∶τ) i τ′ i≤m = Fix (wkn₂ Γ,Δ⊢e∶τ (suc i) τ′ (s≤s i≤m))
+wkn₂ {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i τ′ i≤m = Cat (wkn₂ Γ,Δ⊢e₁∶τ₁ i τ′ i≤m) (congᶜ (≋-sym (wkn₂-shift Γ,Δ i i≤m τ′)) (wkn₁ Δ++Γ,∙⊢e₂∶τ₂ i τ′ z≤n)) τ₁⊛τ₂
+wkn₂ (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i τ′ i≤m = Vee (wkn₂ Γ,Δ⊢e₁∶τ₁ i τ′ i≤m) (wkn₂ Γ,Δ⊢e₂∶τ₂ i τ′ i≤m) τ₁#τ₂
-- 
cgit v1.2.3