From 9e89f36e3fc6210b270d673c30691530015278fb Mon Sep 17 00:00:00 2001
From: Chloe Brown <chloe.brown.00@outlook.com>
Date: Mon, 22 Mar 2021 16:43:49 +0000
Subject: Prove transfer.

---
 src/Cfe/Context/Properties.agda | 57 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 57 insertions(+)

(limited to 'src/Cfe/Context/Properties.agda')

diff --git a/src/Cfe/Context/Properties.agda b/src/Cfe/Context/Properties.agda
index 2761fae..230c18b 100644
--- a/src/Cfe/Context/Properties.agda
+++ b/src/Cfe/Context/Properties.agda
@@ -8,6 +8,7 @@ module Cfe.Context.Properties
 
 open import Cfe.Context.Base over as C
 open import Cfe.Type over
+open import Data.Empty
 open import Data.Fin as F
 open import Data.Nat as ℕ
 open import Data.Nat.Properties
@@ -23,6 +24,11 @@ cast-involutive : ∀ {a A k m n} .(k≡m : k ≡ m) .(m≡n : m ≡ n) .(k≡n
 cast-involutive {k = zero} {zero} {zero} k≡m m≡n k≡n [] = refl
 cast-involutive {k = suc _} {suc _} {suc _} k≡m m≡n k≡n (x ∷ xs) = cong (x ∷_) (cast-involutive (cong ℕ.pred k≡m) (cong ℕ.pred m≡n) (cong ℕ.pred k≡n) xs)
 
+cast-insert : ∀ {a A m n} xs .(m≡n : _) i j .(_ : toℕ i ≡ toℕ j) y → C.cast {a} {A} {suc m} {suc n} (cong suc m≡n) (insert xs i y) ≡ insert (C.cast m≡n xs) j y
+cast-insert [] m≡n zero zero _ y = refl
+cast-insert (x ∷ xs) m≡n zero zero _ y = refl
+cast-insert {m = suc _} {n = suc _} (x ∷ xs) m≡n (suc i) (suc j) i≡j y = cong (x ∷_) (cast-insert xs (cong ℕ.pred m≡n) i j (cong ℕ.pred i≡j) y)
+
 wkn₁-shift : ∀ {n} (Γ,Δ : Context n) i i≥m τ → shift (wkn₁ Γ,Δ i i≥m τ) ≋ wkn₁ (shift Γ,Δ) i z≤n τ
 wkn₁-shift record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ } i i≥m τ =
   refl ,
@@ -62,3 +68,54 @@ wkn₂-shift record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ } i i≤m τ =
                                (trans (sym (+-∸-assoc m (pred-mono m≤n))) (m+n∸m≡n m n))
                                (xs ++ ys)))
   eq {m = suc m} {suc n} (x ∷ xs) ys m≤n (suc i) (s≤s i≤m) y = cong (x ∷_) (eq xs ys (pred-mono m≤n) i i≤m y)
+
+rotate₁-shift : ∀ {n} (Γ,Δ : Context n) i j i≥m i≤j → rotate₁ (shift Γ,Δ) i j z≤n i≤j ≋ shift (rotate₁ Γ,Δ i j i≥m i≤j)
+rotate₁-shift record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ } i j i≥m i≤j =
+  refl ,
+  eq Γ Δ m≤n i j i≥m i≤j ,
+  refl
+  where
+  eq : ∀ {a A m n} xs ys .(m≤n : m ℕ.≤ n) i j i≥m i≤j →
+    rotate {a} {A} i j i≤j (C.cast (trans (sym (+-∸-assoc m m≤n)) (m+n∸m≡n m n)) (ys ++ xs)) ≡
+    C.cast (trans (sym (+-∸-assoc m m≤n)) (m+n∸m≡n m n)) (ys ++ rotate (reduce≥′ 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)
+  eq {m = zero} {suc _} (x ∷ xs) [] _ zero j _ _ = sym (cast-insert xs refl j j refl x)
+  eq {m = zero} (x ∷ xs) [] _ (suc i) (suc j) _ i≤j = cong (x ∷_) (eq xs [] z≤n i j z≤n (pred-mono i≤j))
+  eq {m = suc _} {suc _} xs (y ∷ ys) m≤n (suc i) (suc j) (s≤s i≥m) (s≤s i≤j) = cong (y ∷_) (eq xs ys (pred-mono m≤n) i j i≥m i≤j)
+
+transfer-cons : ∀ {n} (Γ,Δ : Context n) i j i<m 1+j≥m τ → transfer (cons Γ,Δ τ) (suc i) (suc j) (s≤s i<m) (s≤s 1+j≥m) ≋ cons (transfer Γ,Δ i j i<m 1+j≥m) τ
+transfer-cons record { m = suc m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ } i j i<m 1+j≥m τ =
+  refl , eq₁ Γ Δ m≤n (fromℕ< i<m) j 1+j≥m τ , eq₂ Δ (fromℕ< i<m) τ
+  where
+  eq₁ : ∀ {a A m n} xs ys .(m≤n : suc m ℕ.≤ n) (i : Fin (suc m)) j .(1+j≥m : _) y →
+    insert′ {a} {A} xs (s≤s m≤n) (λ ()) (reduce≥′ (≤-step m≤n) (suc j) 1+j≥m) (lookup (y ∷ ys) (suc i)) ≡
+    insert′ xs m≤n (λ ()) (reduce≥′ (pred-mono (≤-step m≤n)) j (pred-mono 1+j≥m)) (lookup ys i)
+  eq₁ {m = zero} {suc _} xs ys m≤n i j 1+j≥m y = refl
+  eq₁ {m = suc m} xs ys m≤n i zero 1+j≥m x = ⊥-elim (<⇒≱ (s≤s (s≤s z≤n)) (≤-recomputable 1+j≥m))
+  eq₁ {m = suc m} {suc _} xs (x ∷ ys) m≤n i (suc j) 1+j≥m y = refl
+
+  eq₂ : ∀ {a A m} ys (i : Fin (suc m)) y →
+    remove′ {a} {A} (y ∷ ys) (λ ()) (suc i) ≡ y ∷ remove′ ys (λ ()) i
+  eq₂ (x ∷ ys) i y = refl
+
+transfer-shift : ∀ {n} (Γ,Δ : Context n) i j i<m 1+j≥m → rotate₁ (shift Γ,Δ) i j z≤n (pred-mono (≤-trans i<m 1+j≥m)) ≋ shift (transfer Γ,Δ i j i<m 1+j≥m)
+transfer-shift record { m = suc m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ } i j i<m 1+j≥m =
+  refl ,
+  eq Γ Δ m≤n i j i<m 1+j≥m ,
+  refl
+  where
+  eq : ∀ {a A m n} xs ys .(m≤n : suc m ℕ.≤ n) i j i<m .(1+j≥m : _) →
+    rotate {a} {A} i j
+      (pred-mono {_} {suc (toℕ j)} (≤-trans i<m 1+j≥m))
+      (C.cast (trans (sym (+-∸-assoc (suc m) m≤n)) (m+n∸m≡n (suc m) n)) (ys ++ xs)) ≡
+    C.cast
+      (trans (sym (+-∸-assoc m (pred-mono (≤-step m≤n)))) (m+n∸m≡n (suc m) n))
+      ( remove′ ys (λ ()) (fromℕ< i<m) ++
+        insert′ xs m≤n (λ ())
+          (reduce≥′ (pred-mono (≤-step m≤n)) j (pred-mono 1+j≥m))
+          (lookup ys (fromℕ< i<m)))
+  eq {m = zero} {suc _} xs (y ∷ []) m≤n zero zero i<m 1+j≥m = refl
+  eq {m = zero} {suc (suc _)} (x ∷ xs) (y ∷ []) _ zero (suc j) _ _ = cong (x ∷_) (eq xs (y ∷ []) (s≤s z≤n) zero j (s≤s z≤n) (s≤s z≤n))
+  eq {m = zero} {suc _} _ (_ ∷ []) _ (suc _) _ (s≤s ()) _
+  eq {m = suc _} {suc _} _ (_ ∷ _) _ _ zero _ 1+j≥m = ⊥-elim (<⇒≱ (s≤s (s≤s z≤n)) (≤-recomputable 1+j≥m))
+  eq {m = suc _} {suc (suc _)} xs (x ∷ y ∷ ys) m≤n zero (suc j) i<m 1+j≥m = cong (y ∷_) (eq xs (x ∷ ys) (pred-mono m≤n) zero j (s≤s z≤n) (pred-mono 1+j≥m))
+  eq {m = suc _} {suc (suc _)} xs (x ∷ y ∷ ys) m≤n (suc i) (suc j) (s≤s i<m) 1+j≥m = cong (x ∷_) (eq xs (y ∷ ys) (pred-mono m≤n) i j i<m (pred-mono 1+j≥m))
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