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/Context/Base.agda       | 36 +++++++++++++++----------
 src/Cfe/Context/Properties.agda | 59 ++++++++++++++++++++++++++++++++++++++++-
 2 files changed, 80 insertions(+), 15 deletions(-)

(limited to 'src/Cfe/Context')

diff --git a/src/Cfe/Context/Base.agda b/src/Cfe/Context/Base.agda
index dcd8056..6b7a9dc 100644
--- a/src/Cfe/Context/Base.agda
+++ b/src/Cfe/Context/Base.agda
@@ -1,6 +1,6 @@
 {-# OPTIONS --without-K --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary using (Setoid; Rel)
 
 module Cfe.Context.Base
   {c ℓ} (over : Setoid c ℓ)
@@ -8,18 +8,23 @@ module Cfe.Context.Base
 
 open import Cfe.Type over
 open import Data.Empty
-open import Data.Fin as F
+open import Data.Fin as F hiding (cast)
 open import Data.Fin.Properties hiding (≤-trans)
 open import Data.Nat as ℕ hiding (_⊔_)
 open import Data.Nat.Properties
+open import Data.Product
 open import Data.Vec
 open import Level renaming (suc to lsuc)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary
 
-reduce≥′ : ∀ {m n} → .(m ℕ.≤ n) → (i : Fin n) → .(toℕ i ≥ m) → Fin (n ∸ m)
+cast : ∀ {a A m n} → .(m ≡ n) → Vec {a} A m → Vec {a} A n
+cast {m = 0} {0} eq [] = []
+cast {m = suc _} {suc n} eq (x ∷ xs) = x ∷ cast (cong ℕ.pred eq) xs
+
+reduce≥′ : ∀ {m n} → .(m ℕ.≤ n) → (i : Fin n) → toℕ i ≥ m → Fin (n ∸ m)
 reduce≥′ {ℕ.zero} {n} m≤n i i≥m = i
-reduce≥′ {suc m} {suc n} m≤n (suc i) i≥m = reduce≥′ (pred-mono m≤n) i (pred-mono i≥m)
+reduce≥′ {suc m} {suc n} m≤n (suc i) (s≤s i≥m) = reduce≥′ (pred-mono m≤n) i i≥m
 
 private
   insert′ : ∀ {a A m n} → Vec {a} A (n ∸ m) → m ℕ.≤ n → m ≢ 0 → (i : Fin (n ∸ ℕ.pred m)) → A → Vec A (n ∸ ℕ.pred m)
@@ -29,9 +34,9 @@ private
   insert′ {a} {A} {suc (suc m)} {suc ℕ.zero} xs m≤n _ i x = ⊥-elim (<⇒≱ (s≤s (s≤s z≤n)) m≤n)
   insert′ {a} {A} {suc (suc m)} {suc (suc n)} xs m≤n _ i x = insert′ {m = suc m} xs (pred-mono m≤n) (λ ()) i x
 
-  reduce≥′-mono : ∀ {m n} → .(m≤n : m ℕ.≤ n) → (i j : Fin n) → .(i≥m : toℕ i ≥ m) → (i≤j : i F.≤ j) → reduce≥′ m≤n i i≥m F.≤ reduce≥′ m≤n j (≤-trans i≥m i≤j)
+  reduce≥′-mono : ∀ {m n} → .(m≤n : m ℕ.≤ n) → (i j : Fin n) → (i≥m : toℕ i ≥ m) → (i≤j : i F.≤ j) → reduce≥′ m≤n i i≥m F.≤ reduce≥′ m≤n j (≤-trans i≥m i≤j)
   reduce≥′-mono {ℕ.zero} {n} m≤n i j i≥m i≤j = i≤j
-  reduce≥′-mono {suc m} {suc n} m≤n (suc i) (suc j) i≥m i≤j = reduce≥′-mono (pred-mono m≤n) i j (pred-mono i≥m) (pred-mono i≤j)
+  reduce≥′-mono {suc m} {suc n} m≤n (suc i) (suc j) (s≤s i≥m) (s≤s i≤j) = reduce≥′-mono (pred-mono m≤n) i j i≥m i≤j
 
   remove′ : ∀ {a A m} → Vec {a} A m → .(m ≢ 0) → Fin m → Vec A (ℕ.pred m)
   remove′ (x ∷ xs) m≢0 F.zero = xs
@@ -48,25 +53,25 @@ record Context n : Set (c ⊔ lsuc ℓ) where
     Γ : Vec (Type ℓ ℓ) (n ∸ m)
     Δ : Vec (Type ℓ ℓ) m
 
-wkn₁ : ∀ {n} → (Γ,Δ : Context n) → (i : Fin (suc n)) → .(toℕ i ≥ Context.m Γ,Δ) → Type ℓ ℓ → Context (suc n)
+wkn₁ : ∀ {n} → (Γ,Δ : Context n) → (i : Fin (suc n)) → (toℕ i ≥ Context.m Γ,Δ) → Type ℓ ℓ → Context (suc n)
 wkn₁ Γ,Δ i i≥m τ = record
   { m≤n = ≤-step m≤n
-  ; Γ = subst (Vec (Type ℓ ℓ)) (sym (+-∸-assoc 1 m≤n)) (insert Γ (F.cast (+-∸-assoc 1 m≤n) (reduce≥′ (≤-step m≤n) i i≥m)) τ)
+  ; Γ = cast (sym (+-∸-assoc 1 m≤n)) (insert Γ (F.cast (+-∸-assoc 1 m≤n) (reduce≥′ (≤-step m≤n) i i≥m)) τ)
   ; Δ = Δ
   }
   where
   open Context Γ,Δ
 
 wkn₂ : ∀ {n} → (Γ,Δ : Context n) → (i : Fin (suc n)) → toℕ i ℕ.≤ Context.m Γ,Δ → Type ℓ ℓ → Context (suc n)
-wkn₂ Γ,Δ i i<m τ = record
+wkn₂ Γ,Δ i i≤m τ = record
   { m≤n = s≤s m≤n
   ; Γ = Γ
-  ; Δ = insert Δ (fromℕ< (s≤s i<m)) τ
+  ; Δ = insert Δ (fromℕ< (s≤s i≤m)) τ
   }
   where
   open Context Γ,Δ
 
-rotate₁ : ∀ {n} → (Γ,Δ : Context n) → (i j : Fin n) → toℕ i ≥ Context.m Γ,Δ → .(i F.≤ j) → Context n
+rotate₁ : ∀ {n} → (Γ,Δ : Context n) → (i j : Fin n) → toℕ i ≥ Context.m Γ,Δ → (i F.≤ j) → Context n
 rotate₁ {n} Γ,Δ i j i≥m i≤j = record
   { m≤n = m≤n
   ; Γ = 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) Γ
@@ -104,8 +109,8 @@ transfer {n} Γ,Δ i j i<m 1+j≥m with Context.m Γ,Δ ℕ.≟ 0
   where
   open Context Γ,Δ
 
-cons : ∀ {n} → Type ℓ ℓ → Context n → Context (suc n)
-cons {n} τ Γ,Δ = record
+cons : ∀ {n} → Context n → Type ℓ ℓ → Context (suc n)
+cons {n} Γ,Δ τ = record
   { m≤n = s≤s m≤n
   ; Γ = Γ
   ; Δ = τ ∷ Δ
@@ -116,8 +121,11 @@ cons {n} τ Γ,Δ = record
 shift : ∀ {n} → Context n → Context n
 shift {n} Γ,Δ = record
   { m≤n = z≤n
-  ; Γ = subst (Vec (Type ℓ ℓ)) (trans (sym (+-∸-assoc m m≤n)) (m+n∸m≡n m n)) (Δ ++ Γ)
+  ; Γ = cast (trans (sym (+-∸-assoc m m≤n)) (m+n∸m≡n m n)) (Δ ++ Γ)
   ; Δ = []
   }
   where
   open Context Γ,Δ
+
+_≋_ : ∀ {n} → Rel (Context n) (c ⊔ lsuc ℓ)
+Γ,Δ ≋ Γ,Δ′ = Σ (Context.m Γ,Δ ≡ Context.m Γ,Δ′) λ {refl → Context.Γ Γ,Δ ≡ Context.Γ Γ,Δ′ × Context.Δ Γ,Δ ≡ Context.Δ Γ,Δ′}
diff --git a/src/Cfe/Context/Properties.agda b/src/Cfe/Context/Properties.agda
index 2acaf72..2761fae 100644
--- a/src/Cfe/Context/Properties.agda
+++ b/src/Cfe/Context/Properties.agda
@@ -1,7 +1,64 @@
 {-# OPTIONS --without-K --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary using (Setoid; Symmetric)
 
 module Cfe.Context.Properties
   {c ℓ} (over : Setoid c ℓ)
   where
+
+open import Cfe.Context.Base over as C
+open import Cfe.Type over
+open import Data.Fin as F
+open import Data.Nat as ℕ
+open import Data.Nat.Properties
+open import Data.Product
+open import Data.Vec
+open import Function
+open import Relation.Binary.PropositionalEquality
+
+≋-sym : ∀ {n} → Symmetric (_≋_ {n})
+≋-sym (refl , refl , refl) = refl , refl , refl
+
+cast-involutive : ∀ {a A k m n} .(k≡m : k ≡ m) .(m≡n : m ≡ n) .(k≡n : _) xs → C.cast m≡n (C.cast {a} {A} k≡m xs) ≡ C.cast k≡n xs
+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)
+
+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 ,
+  eq Δ Γ m≤n i i≥m τ ,
+  refl
+  where
+  eq : ∀ {a A m n} xs ys .(m≤n : m ℕ.≤ n) i (i≥m : toℕ i ≥ m) y →
+       C.cast {a} {A}
+         (trans (sym (+-∸-assoc m (≤-step m≤n))) (m+n∸m≡n m (suc n)))
+         (xs ++ C.cast (sym (+-∸-assoc 1 m≤n)) (insert ys (F.cast (+-∸-assoc 1 m≤n) (reduce≥′ (≤-step m≤n) i i≥m)) y)) ≡
+       C.cast refl (insert (C.cast (trans (sym (+-∸-assoc m m≤n)) (m+n∸m≡n m n)) (xs ++ ys)) (F.cast refl i) y)
+  eq [] [] m≤n zero i≥m y = refl
+  eq [] (x ∷ ys) m≤n zero i≥m y = refl
+  eq [] (x ∷ ys) m≤n (suc i) i≥m y = cong (x ∷_) (eq [] ys z≤n i z≤n y)
+  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)
+
+wkn₂-shift : ∀ {n} (Γ,Δ : Context n) i i≤m τ → shift (wkn₂ Γ,Δ i i≤m τ) ≋ wkn₁ (shift Γ,Δ) i z≤n τ
+wkn₂-shift record { m = m ; m≤n = m≤n ; Γ = Γ ; Δ = Δ } i i≤m τ =
+  refl ,
+  eq Δ Γ m≤n i i≤m τ ,
+  refl
+  where
+  eq : ∀ {a A m n} xs ys .(m≤n : m ℕ.≤ n) i (i≤m : toℕ i ℕ.≤ m) y →
+       C.cast {a} {A}
+         (trans (sym (+-∸-assoc (suc m) (s≤s m≤n))) (m+n∸m≡n (suc m) (suc n)))
+         (insert xs (fromℕ< (s≤s i≤m)) y ++ ys) ≡
+       C.cast
+         (sym (+-∸-assoc 1 z≤n))
+         (insert (C.cast (trans (sym (+-∸-assoc m m≤n)) (m+n∸m≡n m n)) (xs ++ ys))
+                 (F.cast (+-∸-assoc 1 z≤n) (reduce≥′ (≤-step z≤n) i z≤n)) y)
+  eq [] [] m≤n zero i≤m y = refl
+  eq [] (x ∷ ys) m≤n zero i≤m y = cong (λ z → y ∷ x ∷ z) (sym (cast-involutive refl refl refl ys))
+  eq {m = suc m} {suc n} (x ∷ xs) ys m≤n zero i≤m y =
+    cong (λ z → y ∷ x ∷ z)
+         (sym (cast-involutive (trans (sym (+-∸-assoc m (pred-mono m≤n))) (m+n∸m≡n m n))
+                               refl
+                               (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)
-- 
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