From a92f724a46a78af74121c44bbb06c4ec51f9555e Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Tue, 23 Mar 2021 12:19:30 +0000
Subject: Replace transfer with shift.

Prove substitution in the unguarded context.
---
 src/Cfe/Judgement/Base.agda       |   2 +-
 src/Cfe/Judgement/Properties.agda | 223 +++++++++++++++++---------------------
 2 files changed, 98 insertions(+), 127 deletions(-)

(limited to 'src/Cfe/Judgement')

diff --git a/src/Cfe/Judgement/Base.agda b/src/Cfe/Judgement/Base.agda
index 0e5417a..4be0256 100644
--- a/src/Cfe/Judgement/Base.agda
+++ b/src/Cfe/Judgement/Base.agda
@@ -21,7 +21,7 @@ data _⊢_∶_ : {n : ℕ} → Context n → Expression n → Type ℓ ℓ → S
   Eps : ∀ {n} {Γ,Δ : Context n} → Γ,Δ ⊢ ε ∶ Lift ℓ ℓ τε
   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)
+  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≥m)
   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 053ab73..1e79a81 100644
--- a/src/Cfe/Judgement/Properties.agda
+++ b/src/Cfe/Judgement/Properties.agda
@@ -6,19 +6,14 @@ module Cfe.Judgement.Properties
   {c ℓ} (over : Setoid c ℓ)
   where
 
-open import Cfe.Context over
-  renaming
-  ( wkn₁ to cwkn₁
-  ; wkn₂ to cwkn₂
-  ; rotate to crotate
-  ; rotate₁ to crotate₁
-  ; transfer to ctransfer
-  ; _≋_ to _≋ᶜ_
-  )
-open import Cfe.Expression over
+open import Cfe.Context over as C
+  hiding
+  ( shift≤ ; wkn₁ ; wkn₂ )
+  renaming (_≋_ to _≋ᶜ_; ≋-sym to ≋ᶜ-sym; ≋-trans to ≋ᶜ-trans )
+open import Cfe.Expression over as E
 open import Cfe.Judgement.Base over
 open import Data.Empty
-open import Data.Fin as F
+open import Data.Fin as F hiding (splitAt)
 open import Data.Fin.Properties hiding (≤-refl; ≤-trans; ≤-irrelevant)
 open import Data.Nat as ℕ
 open import Data.Nat.Properties
@@ -29,10 +24,19 @@ open import Function
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary
 
-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)
+private
+  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)
+
+  punchIn[i,j]≥m : ∀ {n m i j} → toℕ i ℕ.≤ m → toℕ j ≥ m → toℕ (punchIn {n} i j) ≥ suc m
+  punchIn[i,j]≥m {i = zero} i≤m j≥m = s≤s j≥m
+  punchIn[i,j]≥m {i = suc i} {suc j} (s≤s i≤m) (s≤s j≥m) = s≤s (punchIn[i,j]≥m i≤m j≥m)
+
+  punchOut≥m : ∀ {n m i j} → (i≢j : i ≢ j) → toℕ {suc n} i ≥ m → toℕ j ≥ m → toℕ (punchOut i≢j) ≥ m
+  punchOut≥m {m = zero} _ z≤n _ = z≤n
+  punchOut≥m {n = suc _} {.(suc _)} {suc _} {suc _} i≢j (s≤s i≥m) (s≤s j≥m) = s≤s (punchOut≥m (i≢j ∘ cong suc) i≥m j≥m)
 
 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′)
@@ -41,121 +45,88 @@ congᶜ {Γ,Δ = Γ,Δ} {Γ,Δ′} (refl , refl , refl) Γ,Δ⊢e∶τ with ≤-
 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)))
+wkn₁ : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ {i} i≥m τ′ → C.wkn₁ {i = i} Γ,Δ i≥m τ′ ⊢ wkn e i ∶ τ
+wkn₁ Eps i≥m τ′ = Eps
+wkn₁ (Char c) i≥m τ′ = Char c
+wkn₁ Bot i≥m τ′ = Bot
+wkn₁ {Γ,Δ = record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ }} (Var {i = j} j≥m) {i = i} i≥m τ′ =
+  congᵗ (τ≡τ′ Γ m≤n 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
+  τ≡τ′ : ∀ {a A n m i j} xs (m≤n : m ℕ.≤ n) (i≥m : toℕ i ≥ _) j≥m x →
+    lookup {a} {A}
+      (insert′ xs (s≤s m≤n) (reduce≥′ (≤-step m≤n) i≥m) x)
+      (reduce≥′ (≤-step m≤n) (≤-trans j≥m (toℕ-punchIn i j))) ≡
+    lookup xs (reduce≥′ m≤n j≥m)
+  τ≡τ′ {i = zero} {j} (y ∷ xs) z≤n z≤n z≤n x = refl
+  τ≡τ′ {i = suc i} {zero} (y ∷ xs) z≤n z≤n z≤n x = refl
+  τ≡τ′ {i = suc i} {suc j} (y ∷ xs) z≤n z≤n z≤n x = τ≡τ′ {i = i} xs z≤n z≤n z≤n x
+  τ≡τ′ {i = suc i} {suc j} xs (s≤s m≤n) (s≤s i≥m) (s≤s j≥m) x = τ≡τ′ xs m≤n i≥m j≥m x
+wkn₁ (Fix Γ,Δ⊢e∶τ) i≥m τ′ = Fix (wkn₁ Γ,Δ⊢e∶τ (s≤s i≥m) τ′)
+wkn₁ {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i≥m τ′ = Cat (wkn₁ Γ,Δ⊢e₁∶τ₁ i≥m τ′) (congᶜ (≋ᶜ-sym (shift≤-wkn₁-comm Γ,Δ z≤n i≥m τ′)) (wkn₁ Δ++Γ,∙⊢e₂∶τ₂ z≤n τ′)) τ₁⊛τ₂
+wkn₁ (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i≥m τ′ = Vee (wkn₁ Γ,Δ⊢e₁∶τ₁ i≥m τ′) (wkn₁ Γ,Δ⊢e₂∶τ₂ i≥m τ′) 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))
+wkn₂ : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ {i} i≤m τ′ → C.wkn₂ {i = i} Γ,Δ i≤m τ′ ⊢ wkn e i ∶ τ
+wkn₂ Eps i≤m τ′ = Eps
+wkn₂ (Char c) i≤m τ′ = Char c
+wkn₂ Bot i≤m τ′ = Bot
+wkn₂ {Γ,Δ = record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ }}(Var {i = j} j≥m) i≤m τ′ = congᵗ (τ≡τ′ Γ m≤n i≤m j≥m) (Var (punchIn[i,j]≥m 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 n m i j} xs (m≤n : m ℕ.≤ n) (i≤m : toℕ i ℕ.≤ _) (j≥m : toℕ j ≥ _) →
+         lookup {a} {A} xs (reduce≥′ (s≤s m≤n) (punchIn[i,j]≥m i≤m j≥m)) ≡
+         lookup xs (reduce≥′ m≤n j≥m)
+  τ≡τ′ {i = zero} _ z≤n _ _ = refl
+  τ≡τ′ {i = zero} _ (s≤s _) _ _ = refl
+  τ≡τ′ {i = suc i} {suc j} xs (s≤s m≤n) (s≤s i≤m) (s≤s j≥m) = τ≡τ′ xs m≤n i≤m j≥m
+wkn₂ (Fix Γ,Δ⊢e∶τ) i≤m τ′ = Fix (wkn₂ Γ,Δ⊢e∶τ (s≤s i≤m) τ′)
+wkn₂ {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i≤m τ′ = Cat (wkn₂ Γ,Δ⊢e₁∶τ₁ i≤m τ′) (congᶜ (≋ᶜ-sym (shift≤-wkn₂-comm-≤ Γ,Δ z≤n i≤m τ′)) (wkn₁ Δ++Γ,∙⊢e₂∶τ₂ z≤n τ′)) τ₁⊛τ₂
+wkn₂ (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i≤m τ′ = Vee (wkn₂ Γ,Δ⊢e₁∶τ₁ i≤m τ′) (wkn₂ Γ,Δ⊢e₂∶τ₂ i≤m τ′) τ₁#τ₂
 
-  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
-
-rotate₁ : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ i j i≥m i≤j → crotate₁ Γ,Δ i j i≥m i≤j ⊢ rotate e i j i≤j ∶ τ
-rotate₁ Eps i j i≥m i≤j = Eps
-rotate₁ (Char c) i j i≥m i≤j = Char c
-rotate₁ Bot i j i≥m i≤j = Bot
-rotate₁ {suc n} {Γ,Δ = record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ }} (Var {i = k} k≥m) i j i≥m i≤j with i F.≟ k
-... | yes refl = congᵗ (τ≡τ′ Γ m≤n i j i≥m i≤j) (Var (≤-trans i≥m i≤j))
-  where
-    τ≡τ′ : ∀ {a A m n} xs m≤n i j i≥m i≤j → lookup {a} {A} (crotate (reduce≥′ {m} {n} m≤n i i≥m) (reduce≥′ m≤n j (≤-trans i≥m i≤j)) (reduce≥′-mono m≤n i j i≥m i≤j) xs) (reduce≥′ m≤n j (≤-trans i≥m i≤j)) ≡ lookup xs (reduce≥′ m≤n i i≥m)
-    τ≡τ′ {m = zero} (x ∷ xs) m≤n zero j i≥m i≤j = insert-lookup xs j x
-    τ≡τ′ {m = zero} (x ∷ xs) m≤n (suc i) (suc j) i≥m i≤j = τ≡τ′ xs z≤n i j z≤n (pred-mono i≤j)
-    τ≡τ′ {m = suc m} {suc n} xs m≤n (suc i) (suc j) (s≤s i≥m) (s≤s i≤j) = τ≡τ′ xs (pred-mono m≤n) i j i≥m i≤j
-... | no i≢k = congᵗ (τ≡τ′ Γ m≤n i j k i≢k i≥m i≤j k≥m) (Var (punchIn-punchOut≥m i j k i≢k i≥m i≤j k≥m))
+shift≤ : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ {i} (i≤m : i ℕ.≤ _) → C.shift≤ Γ,Δ i≤m ⊢ e ∶ τ
+shift≤ Eps i≤m = Eps
+shift≤ (Char c) i≤m = Char c
+shift≤ Bot i≤m = Bot
+shift≤ {Γ,Δ = record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ }} (Var {i = j} j≥m) i≤m =
+  congᵗ (τ≡τ′ Γ Δ m≤n i≤m j≥m) (Var (≤-trans i≤m j≥m))
   where
-  punchIn-punchOut≥m : ∀ {m n} (i j k : Fin (suc n)) (i≢k : i ≢ k) → toℕ i ≥ m → i F.≤ j → toℕ k ≥ m → toℕ (punchIn j (punchOut i≢k)) ≥ m
-  punchIn-punchOut≥m {zero} _ _ _ _ _ _ _ = z≤n
-  punchIn-punchOut≥m {suc _} zero _ zero i≢k _ _ _ = ⊥-elim (i≢k refl)
-  punchIn-punchOut≥m {suc _} zero zero (suc _) _ _ _ k≥m = k≥m
-  punchIn-punchOut≥m {suc _} {suc _} (suc i) (suc j) (suc k) i≢k (s≤s i≥m) (s≤s i≤j) (s≤s k≥m) = s≤s (punchIn-punchOut≥m i j k (i≢k ∘ cong suc) i≥m i≤j k≥m)
+  τ≡τ′ : ∀ {a A n m i j} xs ys (m≤n : m ℕ.≤ n) (i≤m : i ℕ.≤ m) (j≥m : toℕ j ≥ m) →
+         lookup {a} {A} (drop′ m≤n i≤m (ys ++ xs)) (reduce≥′ (≤-trans i≤m m≤n) (≤-trans i≤m j≥m)) ≡
+         lookup xs (reduce≥′ m≤n j≥m)
+  τ≡τ′ xs [] z≤n z≤n z≤n = refl
+  τ≡τ′ {j = suc j} xs (x ∷ ys) (s≤s m≤n) z≤n (s≤s j≥m) = τ≡τ′ xs ys m≤n z≤n j≥m
+  τ≡τ′ {j = suc j} xs (x ∷ ys) (s≤s m≤n) (s≤s i≤m) (s≤s j≥m) = τ≡τ′ xs ys m≤n i≤m j≥m
+shift≤ {Γ,Δ = Γ,Δ} {τ = τ} (Fix Γ,Δ⊢e∶τ) {i} i≤m = Fix (congᶜ (shift≤-wkn₂-comm-> Γ,Δ z≤n i≤m τ) (shift≤ Γ,Δ⊢e∶τ (s≤s i≤m)))
+shift≤ {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i≤m =
+  Cat (shift≤ Γ,Δ⊢e₁∶τ₁ i≤m)
+      (congᶜ (≋ᶜ-trans (shift≤-identity (shift Γ,Δ))
+                       (≋ᶜ-sym (shift≤-idem Γ,Δ z≤n i≤m)))
+             (shift≤ Δ++Γ,∙⊢e₂∶τ₂ z≤n))
+      τ₁⊛τ₂
+shift≤ (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i≤m = Vee (shift≤ Γ,Δ⊢e₁∶τ₁ i≤m) (shift≤ Γ,Δ⊢e₂∶τ₂ i≤m) τ₁#τ₂
 
-  τ≡τ′ : ∀ {a A m n} xs m≤n i j k i≢k i≥m i≤j k≥m →
-    lookup {a} {A}
-      (crotate
-        (reduce≥′ {m} {suc n} m≤n i i≥m)
-        (reduce≥′ m≤n j (≤-trans i≥m i≤j))
-        (reduce≥′-mono m≤n i j i≥m i≤j) xs)
-        (reduce≥′
-          m≤n
-          (punchIn j (punchOut i≢k))
-          (punchIn-punchOut≥m i j k i≢k i≥m i≤j k≥m)) ≡
-    lookup xs (reduce≥′ m≤n k k≥m)
-  τ≡τ′ {m = zero} _ _ zero _ zero i≢k _ _ _ = ⊥-elim (i≢k refl)
-  τ≡τ′ {m = zero} (_ ∷ _) _ zero zero (suc _) _ _ _ _ = refl
-  τ≡τ′ {m = zero} (_ ∷ _ ∷ _) _ zero (suc _) (suc zero) _ _ _ _ = refl
-  τ≡τ′ {m = zero} (x ∷ _ ∷ xs) _ zero (suc j) (suc (suc k)) _ _ _ _ = τ≡τ′ (x ∷ xs) z≤n zero j (suc k) (λ ()) z≤n z≤n z≤n
-  τ≡τ′ {m = zero} (_ ∷ _ ∷ _) _ (suc _) (suc _) zero _ _ _ _ = refl
-  τ≡τ′ {m = zero} (_ ∷ x ∷ xs) _ (suc i) (suc j) (suc k) i≢k _ i≤j _ = τ≡τ′ (x ∷ xs) z≤n i j k (i≢k ∘ cong suc) z≤n (pred-mono i≤j) z≤n
-  τ≡τ′ {m = suc m} {suc _} xs m≤n (suc i) (suc j) (suc k) i≢k i≥m i≤j k≥m = τ≡τ′ xs (pred-mono m≤n) i j k (i≢k ∘ cong suc) (pred-mono i≥m) (pred-mono i≤j) (pred-mono k≥m)
-rotate₁ (Fix Γ,Δ⊢e∶τ) i j i≥m i≤j = Fix (rotate₁ Γ,Δ⊢e∶τ (suc i) (suc j) (s≤s i≥m) (s≤s i≤j))
-rotate₁ {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i j i≥m i≤j = Cat (rotate₁ Γ,Δ⊢e₁∶τ₁ i j i≥m i≤j) (congᶜ (rotate₁-shift Γ,Δ i j i≥m i≤j) (rotate₁ Δ++Γ,∙⊢e₂∶τ₂ i j z≤n i≤j)) τ₁⊛τ₂
-rotate₁ (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i j i≥m i≤j = Vee (rotate₁ Γ,Δ⊢e₁∶τ₁ i j i≥m i≤j) (rotate₁ Γ,Δ⊢e₂∶τ₂ i j i≥m i≤j) τ₁#τ₂
-
-transfer : ∀ {n} {Γ,Δ : Context n} {e τ} → Γ,Δ ⊢ e ∶ τ → ∀ i j i<m 1+j≥m → ctransfer Γ,Δ i j i<m 1+j≥m ⊢ rotate e i j (pred-mono (≤-trans i<m 1+j≥m)) ∶ τ
-transfer Eps i j i<m 1+j≥m = Eps
-transfer (Char c) i j i<m 1+j≥m = Char c
-transfer Bot i j i<m 1+j≥m = Bot
-transfer {suc n} {Γ,Δ = record { m = suc m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ }} (Var {i = k} k≥m) i j i<m 1+j≥m with suc m ℕ.≟ 0 | i F.≟ k
-... | no m≢0 | yes refl = ⊥-elim (<⇒≱ i<m k≥m)
-... | no m≢0 | no i≢k = congᵗ (τ≡τ′ Γ (lookup Δ (fromℕ< i<m)) m≢0 m≤n i j k i<m 1+j≥m k≥m i≢k) (Var (punchIn≥m i j k i≢k i<m 1+j≥m k≥m))
+subst₁ : ∀ {n} {Γ,Δ : Context n} {e τ i τ′} (i≥m : toℕ i ≥ _) → C.wkn₁ Γ,Δ i≥m τ′ ⊢ e ∶ τ → ∀ {e′} → Γ,Δ ⊢ e′ ∶ τ′ → Γ,Δ ⊢ e [ e′ / i ] ∶ τ
+subst₁ _ Eps Γ,Δ⊢e′∶τ′ = Eps
+subst₁ _ (Char c) Γ,Δ⊢e′∶τ′ = Char c
+subst₁ _ Bot Γ,Δ⊢e′∶τ′ = Bot
+subst₁ {Γ,Δ = record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ }} {i = i} {τ′} i≥m (Var {i = j} j≥m) Γ,Δ⊢e′∶τ′ with  i F.≟ j
+... | no i≢j =
+  congᵗ (sym (τ≡τ′ Γ m≤n i≢j i≥m j≥m τ′)) (Var (punchOut≥m i≢j i≥m j≥m))
   where
-  punchIn≥m : ∀ {m n} (i j k : Fin (suc n)) (i≢k : i ≢ k) → toℕ i ℕ.< m → .(suc (toℕ j) ≥ m) → toℕ k ≥ m → toℕ (punchIn j (punchOut i≢k)) ≥ ℕ.pred m
-  punchIn≥m {suc zero} _ _ _ _ _ _ _ = z≤n
-  punchIn≥m {suc (suc _)} zero _ zero i≢k _ _ _ = ⊥-elim (i≢k refl)
-  punchIn≥m {suc (suc _)} zero zero (suc _) _ _ _ (s≤s k≥m) = ≤-step k≥m
-  punchIn≥m {suc (suc _)} {suc _} (suc i) zero (suc k) i≢k (s≤s i<m) 1+j≥m (s≤s k≥m) = ⊥-elim (<⇒≱ (s≤s (s≤s z≤n)) (≤-recomputable 1+j≥m))
-  punchIn≥m {suc (suc _)} zero (suc _) (suc zero) _ _ _ (s≤s k≥m) = k≥m
-  punchIn≥m {suc (suc _)} zero (suc j) (suc (suc k)) _ _ 1+j≥m (s≤s k≥m) = s≤s (punchIn≥m zero j (suc k) (λ ()) (s≤s z≤n) (pred-mono 1+j≥m) k≥m)
-  punchIn≥m {suc (suc _)} {suc _} (suc i) (suc j) (suc k) i≢k (s≤s i<m) 1+j≥m (s≤s k≥m) = s≤s (punchIn≥m i j k (i≢k ∘ cong suc) i<m (pred-mono 1+j≥m) k≥m)
-
-  τ≡τ′ : ∀ {a A m n} xs y m≢0 .(m≤n : _) i j k i<m .(1+j≥m : _) k≥m i≢k →
-    lookup
-      (insert′ {a} {A} {m} {suc n} xs m≤n m≢0
-        (reduce≥′ (pred-mono (≤-step m≤n)) j (pred-mono 1+j≥m))
-        y)
-      (reduce≥′
-        (pred-mono (≤-step m≤n))
-        (punchIn j (punchOut i≢k))
-        (punchIn≥m i j k i≢k i<m 1+j≥m k≥m)) ≡
-    lookup xs (reduce≥′ m≤n k k≥m)
-  τ≡τ′ {m = suc zero} _ _ _ _ zero zero (suc _) _ _ (s≤s z≤n) _ = refl
-  τ≡τ′ {m = suc zero} _ _ _ _ (suc _) zero (suc _) (s≤s ()) _ _ _
-  τ≡τ′ {m = suc zero} (_ ∷ _) _ _ _ zero (suc _) (suc zero) _ _ (s≤s z≤n) _ = refl
-  τ≡τ′ {m = suc zero} (_ ∷ xs) y _ _ zero (suc j) (suc (suc k)) _ _ (s≤s z≤n) _ = τ≡τ′ xs y (λ ()) (s≤s (z≤n)) zero j (suc k) (s≤s z≤n) (s≤s z≤n) (s≤s z≤n) (λ ())
-  τ≡τ′ {m = suc (suc _)} _ _ _ _ _ zero _ _ 1+j≥m _ _ = ⊥-elim (<⇒≱ (s≤s (s≤s z≤n)) (≤-recomputable 1+j≥m))
-  τ≡τ′ {m = suc (suc _)} _ _ _ _ zero _ (suc zero) _ _ (s≤s ()) _
-  τ≡τ′ {m = suc (suc _)} {suc (suc _)} xs y _ m≤n zero (suc j) (suc (suc k)) _ 1+j≥m (s≤s k≥m) _ = τ≡τ′ xs y (λ ()) (pred-mono m≤n) zero j (suc k) (s≤s z≤n) (pred-mono 1+j≥m) k≥m (λ ())
-  τ≡τ′ {m = suc (suc m)} xs y m≢0 m≤n (suc i) (suc j) (suc zero) (s≤s i<m) 1+j≥m (s≤s k≥m) i≢k = τ≡τ′ xs y (λ ()) (pred-mono m≤n) i j zero i<m (pred-mono 1+j≥m) k≥m (i≢k ∘ cong suc)
-  τ≡τ′ {m = suc (suc m)} xs y m≢0 m≤n (suc i) (suc j) (suc (suc k)) (s≤s i<m) 1+j≥m (s≤s k≥m) i≢k = τ≡τ′ xs y (λ ()) (pred-mono m≤n) i j (suc k) i<m (pred-mono 1+j≥m) k≥m (i≢k ∘ cong suc)
-transfer {Γ,Δ = Γ,Δ} {τ = τ} (Fix Γ,Δ⊢e∶τ) i j i<m 1+j≥m = Fix (congᶜ (transfer-cons Γ,Δ i j i<m 1+j≥m τ) (transfer Γ,Δ⊢e∶τ (suc i) (suc j) (s≤s i<m) (s≤s 1+j≥m)))
-transfer {Γ,Δ = Γ,Δ} (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) i j i<m 1+j≥m = Cat (transfer Γ,Δ⊢e₁∶τ₁ i j i<m 1+j≥m) (congᶜ (transfer-shift Γ,Δ i j i<m 1+j≥m) (rotate₁ Δ++Γ,∙⊢e₂∶τ₂ i j z≤n (pred-mono (≤-trans i<m 1+j≥m)))) τ₁⊛τ₂
-transfer (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) i j i<m 1+j≥m = Vee (transfer Γ,Δ⊢e₁∶τ₁ i j i<m 1+j≥m) (transfer Γ,Δ⊢e₂∶τ₂ i j i<m 1+j≥m) τ₁#τ₂
+  τ≡τ′ : ∀ {a A n m i j} xs (m≤n : m ℕ.≤ n) (i≢j : i ≢ j) (i≥m : toℕ i ≥ m) (j≥m : toℕ j ≥ m) x →
+        lookup {a} {A} (insert′ xs (s≤s m≤n) (reduce≥′ (≤-step m≤n) i≥m) x) (reduce≥′ (≤-step m≤n) j≥m) ≡
+        lookup xs (reduce≥′ m≤n (punchOut≥m i≢j i≥m j≥m))
+  τ≡τ′ {i = zero} {zero} xs m≤n i≢j i≥m j≥m x = ⊥-elim (i≢j refl)
+  τ≡τ′ {n = suc n} {i = zero} {suc j} xs z≤n i≢j z≤n z≤n x = refl
+  τ≡τ′ {n = suc n} {i = suc i} {zero} (y ∷ xs) z≤n i≢j z≤n z≤n x = refl
+  τ≡τ′ {n = suc n} {i = suc i} {suc j} (y ∷ xs) z≤n i≢j z≤n z≤n x = τ≡τ′ xs z≤n (i≢j ∘ cong suc) z≤n z≤n x
+  τ≡τ′ {n = suc n} {i = suc i} {suc j} xs (s≤s m≤n) i≢j (s≤s i≥m) (s≤s j≥m) x = τ≡τ′ xs m≤n (i≢j ∘ cong suc) i≥m j≥m x
+... | yes refl with ≤-irrelevant i≥m j≥m
+... | refl = congᵗ (sym (τ≡τ′ Γ m≤n i≥m τ′)) Γ,Δ⊢e′∶τ′
+  where
+  τ≡τ′ : ∀ {a A n m i} xs (m≤n : m ℕ.≤ n) (i≥m : toℕ i ≥ m) x →
+         lookup {a} {A} (insert′ xs (s≤s m≤n) (reduce≥′ (≤-step m≤n) i≥m) x) (reduce≥′ (≤-step m≤n) i≥m) ≡ x
+  τ≡τ′ {i = zero} xs z≤n z≤n x = refl
+  τ≡τ′ {i = suc i} (y ∷ xs) z≤n z≤n x = insert-lookup xs i x
+  τ≡τ′ {i = suc i} xs (s≤s m≤n) (s≤s i≥m) = τ≡τ′ xs m≤n i≥m
+subst₁ {τ = τ} i≥m (Fix Γ,Δ⊢e∶τ) Γ,Δ⊢e′∶τ′ = Fix (subst₁ (s≤s i≥m) Γ,Δ⊢e∶τ (wkn₂ Γ,Δ⊢e′∶τ′ z≤n τ))
+subst₁ {Γ,Δ = Γ,Δ} {i = i} {τ′} i≥m (Cat Γ,Δ⊢e₁∶τ₁ Δ++Γ,∙⊢e₂∶τ₂ τ₁⊛τ₂) Γ,Δ⊢e′∶τ′ = Cat (subst₁ i≥m Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e′∶τ′) (subst₁ z≤n (congᶜ (shift≤-wkn₁-comm Γ,Δ z≤n i≥m τ′) Δ++Γ,∙⊢e₂∶τ₂) (shift≤ Γ,Δ⊢e′∶τ′ z≤n)) τ₁⊛τ₂
+subst₁ i≥m (Vee Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e₂∶τ₂ τ₁#τ₂) Γ,Δ⊢e′∶τ′ = Vee (subst₁ i≥m Γ,Δ⊢e₁∶τ₁ Γ,Δ⊢e′∶τ′) (subst₁ i≥m Γ,Δ⊢e₂∶τ₂ Γ,Δ⊢e′∶τ′) τ₁#τ₂
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