Change judgement to use variable contexts.

Add some useful context transformations.

2 files changed, 95 insertions, 23 deletions

diff --git a/src/Cfe/Expression/Base.agda b/src/Cfe/Expression/Base.agda
index 2023a71..c3367c2 100644
--- a/src/Cfe/Expression/Base.agda
+++ b/src/Cfe/Expression/Base.agda
@@ -66,16 +66,16 @@ Var j [ e′ / i ] with i F.≟ j
 ... | no i≢j = Var (punchOut i≢j)
 μ e [ e′ / i ] = μ (e [ wkn e′ F.zero / suc i ])
 
-shift : ∀ {n} → Expression n → (i j : Fin n) → .(_ : i F.≤ j) → Expression n
-shift ⊥ _ _ _ = ⊥
-shift ε _ _ _ = ε
-shift (Char x) _ _ _ = Char x
-shift (e₁ ∨ e₂) i j i≤j = shift e₁ i j i≤j ∨ shift e₂ i j i≤j
-shift (e₁ ∙ e₂) i j i≤j = shift e₁ i j i≤j ∙ shift e₂ i j i≤j
-shift {suc n} (Var k) i j _ with i F.≟ k
+rotate : ∀ {n} → Expression n → (i j : Fin n) → .(_ : i F.≤ j) → Expression n
+rotate ⊥ _ _ _ = ⊥
+rotate ε _ _ _ = ε
+rotate (Char x) _ _ _ = Char x
+rotate (e₁ ∨ e₂) i j i≤j = rotate e₁ i j i≤j ∨ rotate e₂ i j i≤j
+rotate (e₁ ∙ e₂) i j i≤j = rotate e₁ i j i≤j ∙ rotate e₂ i j i≤j
+rotate {suc n} (Var k) i j _ with i F.≟ k
 ... | yes i≡k = Var j
 ... | no i≢k = Var (punchIn j (punchOut i≢k))
-shift (μ e) i j i≤j = μ (shift e (suc i) (suc j) (s≤s i≤j))
+rotate (μ e) i j i≤j = μ (rotate e (suc i) (suc j) (s≤s i≤j))
 
 ⟦_⟧ : ∀ {n : ℕ} → Expression n → Vec (Language (c ⊔ ℓ)) n → Language (c ⊔ ℓ)
 ⟦ ⊥ ⟧ _ = Lift (c ⊔ ℓ) ∅
diff --git a/src/Cfe/Judgement/Base.agda b/src/Cfe/Judgement/Base.agda
index 475968c..4bb7b67 100644
--- a/src/Cfe/Judgement/Base.agda
+++ b/src/Cfe/Judgement/Base.agda
@@ -6,16 +6,44 @@ module Cfe.Judgement.Base
   {c ℓ} (over : Setoid c ℓ)
   where
 
-open import Cfe.Expression over renaming (shift to shiftₑ)
+open import Cfe.Expression over hiding (rotate)
 open import Cfe.Type over renaming (_∙_ to _∙ₜ_; _∨_ to _∨ₜ_)
 open import Cfe.Type.Construct.Lift over
+open import Data.Empty using (⊥-elim)
 open import Data.Fin as F
-open import Data.Fin.Properties
+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 hiding (_⊛_) renaming (lookup to lookup′)
+open import Function
 open import Level hiding (Lift) renaming (suc to lsuc)
 open import Relation.Binary.PropositionalEquality
+open import Relation.Nullary
+
+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)
+  insert′ {a} {A} {ℕ.zero} {n} xs m≤n m≢0 i x = ⊥-elim (m≢0 refl)
+  insert′ {a} {A} {suc ℕ.zero} {suc _} xs _ _ F.zero x = x ∷ xs
+  insert′ {a} {A} {suc ℕ.zero} {suc (suc n)} (y ∷ xs) _ _ (suc i) x = y ∷ insert′ {m = suc ℕ.zero} {suc n} xs (s≤s z≤n) (λ ()) i x
+  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≥′ : ∀ {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≥′-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)
+
+  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
+  remove′ (x ∷ y ∷ xs) m≢0 (suc i) = x ∷ remove′ (y ∷ xs) (λ ()) i
+
+  rotate : ∀ {a A n} → (i j : Fin n) → .(i F.≤ j) → Vec {a} A n → Vec A n
+  rotate F.zero j i≤j (x ∷ xs) = insert xs j x
+  rotate (suc i) (suc j) i≤j (x ∷ xs) = x ∷ (rotate i j (pred-mono i≤j) xs)
 
 record Context n : Set (c ⊔ lsuc ℓ) where
   field
@@ -24,20 +52,64 @@ record Context n : Set (c ⊔ lsuc ℓ) where
     Γ : Vec (Type ℓ ℓ) (n ∸ m)
     Δ : Vec (Type ℓ ℓ) m
 
--- Fin n → Fin n∸m
-
   lookup : (i : Fin n) → toℕ i ≥ m → Type ℓ ℓ
-  lookup i i≥m = lookup′ Γ (reduce≥
-      (F.cast (begin-equality
-        n ≡˘⟨ m+n∸m≡n m n ⟩
-        m ℕ.+ n ∸ m ≡⟨ +-∸-assoc m m≤n ⟩
-        m ℕ.+ (n ∸ m) ∎) i)
-      (begin
-        m ≤⟨ i≥m ⟩
-        toℕ i ≡˘⟨ toℕ-cast _ i ⟩
-        toℕ (F.cast _ i) ∎))
-    where
-    open ≤-Reasoning
+  lookup i i≥m = lookup′ Γ (reduce≥′ m≤n i i≥m)
+
+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)
+  ; 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
+  { m≤n = s≤s m≤n
+  ; Γ = Γ
+  ; 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 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
+  ; Δ = Δ
+  }
+  where
+  open Context Γ,Δ
+
+rotate₂ : ∀ {n} → (Γ,Δ : Context n) → (i j : Fin n) → (toℕ j ℕ.< Context.m Γ,Δ) → (i F.≤ j) → Context n
+rotate₂ {n} Γ,Δ i j j<m i≤j = record
+  { m≤n = m≤n
+  ; Γ = Γ
+  ; Δ = rotate
+    (fromℕ< (≤-trans (s≤s i≤j) j<m))
+    (fromℕ< j<m)
+    (begin
+      toℕ (fromℕ< (≤-trans (s≤s i≤j) j<m)) ≡⟨ toℕ-fromℕ< (≤-trans (s≤s i≤j) j<m) ⟩
+      toℕ i ≤⟨ i≤j ⟩
+      toℕ j ≡˘⟨ toℕ-fromℕ< j<m ⟩
+      toℕ (fromℕ< j<m) ∎)
+    Δ
+  }
+  where
+  open Context Γ,Δ
+  open ≤-Reasoning
+
+transfer : ∀ {n} → (Γ,Δ : Context n) → (i j : Fin n) → (toℕ i ℕ.< Context.m Γ,Δ) → (suc (toℕ j) ≥ Context.m Γ,Δ) → Context n
+transfer {n} Γ,Δ i j i<m 1+j≥m with Context.m Γ,Δ ℕ.≟ 0
+... | yes m≡0 = ⊥-elim (m<n⇒n≢0 i<m m≡0)
+... | no m≢0 = record
+  { m≤n = pred-mono (≤-step m≤n)
+  ; Γ = insert′ Γ m≤n m≢0 (reduce≥′ (pred-mono (≤-step m≤n)) j (pred-mono 1+j≥m)) (lookup′ Δ (fromℕ< i<m))
+  ; Δ = remove′ Δ m≢0 (fromℕ< i<m)
+  }
+  where
+  open Context Γ,Δ
 
 cons : ∀ {n} → Type ℓ ℓ → Context n → Context (suc n)
 cons {n} τ Γ,Δ = record