Remove with-abstractions

1 files changed, 32 insertions, 13 deletions

diff --git a/FinMerge.agda b/FinMerge.agda
index c62fe14..c75e982 100644
--- a/FinMerge.agda
+++ b/FinMerge.agda
@@ -4,10 +4,10 @@ module FinMerge where
 open import Data.Empty using (⊥-elim)
 open import Data.Fin using (Fin; fromℕ<; toℕ; #_)
 open import Data.Fin.Properties using (¬Fin0)
-open import Data.Nat using (ℕ; _+_; _≤_; _<_ ; z<s; s≤s)
+open import Data.Nat using (ℕ; _+_; _≤_; z≤n; s≤s; _<_ ; z<s)
 open import Data.Nat.Properties using (≤-trans)
 open import Data.Sum.Base using (_⊎_)
-open import Data.Product using (_×_; _,_; Σ-syntax; map₂; proj₂)
+open import Data.Product using (_×_; _,_; Σ-syntax; map₂; proj₁; proj₂)
 open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym)
 open import Relation.Binary.PropositionalEquality.Properties using (module ≡-Reasoning)
 open import Function using (id ; _∘_ ; _$_)
@@ -22,20 +22,39 @@ private
 
 -- Send the specified m to nothing
 pluck : m ≤ n → Fin (ℕ.suc n) → Maybe (Fin n)
-pluck _≤_.z≤n Fin.zero = nothing
-pluck _≤_.z≤n (Fin.suc x) = just x
-pluck (_≤_.s≤s m) Fin.zero = just Fin.zero
-pluck (_≤_.s≤s m) (Fin.suc x) = Data.Maybe.Base.map Fin.suc (pluck m x)
+pluck z≤n Fin.zero = nothing
+pluck z≤n (Fin.suc x) = just x
+pluck (s≤s m) Fin.zero = just Fin.zero
+pluck (s≤s m) (Fin.suc x) = Data.Maybe.Base.map Fin.suc (pluck m x)
+
+-- Send nothing to the specified m
+unpluck : m ≤ n → Maybe (Fin n) → Fin (ℕ.suc n)
+unpluck z≤n nothing = Fin.zero
+unpluck z≤n (just x) = Fin.suc x
+unpluck (s≤s m) nothing = Fin.suc (unpluck m nothing)
+unpluck (s≤s m) (just Fin.zero) = Fin.zero
+unpluck (s≤s m) (just (Fin.suc x)) = Fin.suc (unpluck m (just x))
 
 -- Merge two elements of a finite set
 merge : {i j : ℕ} → i < j ≤ n → Fin (ℕ.suc n) → Fin n
 merge (lt , le) x = fromMaybe (fromℕ< (≤-trans lt le)) (pluck le x)
 
-glue-once : Fin (ℕ.suc n) → Fin (ℕ.suc n) → Σ[ x ∈ ℕ ] (Fin (ℕ.suc n) → Fin x)
-glue-once {n} f0 g0 with compare f0 g0
-... | less (f0<g0 , s≤s g0≤n) = n , merge (f0<g0 , g0≤n)
-... | equal f0≡g0 = ℕ.suc n , id
-... | greater (g0<f0 , s≤s f0≤n) = n , merge (g0<f0 , f0≤n)
+-- Merge two elements of a finite set
+unmerge : {i j : ℕ} → i < j ≤ n → Fin n → Fin (ℕ.suc n)
+unmerge (lt , le) x = unpluck le (just x)
+
+glue-once : {i j : Fin (ℕ.suc n)} → Ordering i j → Σ[ x ∈ ℕ ] (Fin (ℕ.suc n) → Fin x)
+glue-once {n} (less (i<j , s≤s j≤n)) = n , merge (i<j , j≤n)
+glue-once {n} (equal i≡j) = ℕ.suc n , id
+glue-once {n} (greater (j<i , s≤s i≤n)) = n , merge (j<i , i≤n)
+
+glue-unglue-once
+    : {i j : Fin (ℕ.suc n)}
+    → Ordering i j
+    → Σ[ x ∈ ℕ ] ((Fin (ℕ.suc n) → Fin x) × (Fin x → Fin (ℕ.suc n)))
+glue-unglue-once {n} (less (i<j , s≤s j≤n)) = n , merge (i<j , j≤n) , unmerge (i<j , j≤n)
+glue-unglue-once {n} (equal i≡j) = ℕ.suc n , id , id
+glue-unglue-once {n} (greater (j<i , s≤s i≤n)) = n , merge (j<i , i≤n) , unmerge (j<i , i≤n)
 
 -- Glue together the image of two finite-set functions
 glue : (Fin m → Fin n) → (Fin m → Fin n) → Σ[ x ∈ ℕ ] (Fin n → Fin x)
@@ -43,7 +62,7 @@ glue {ℕ.zero} {n} _ _ = n , id
 glue {ℕ.suc _} {ℕ.zero} f _ = ⊥-elim (¬Fin0 (f (# 0)))
 glue {ℕ.suc _} {ℕ.suc _} f g with glue (f ∘ Fin.suc) (g ∘ Fin.suc)
 ... | ℕ.zero , h = ⊥-elim (¬Fin0 (h (# 0)))
-... | ℕ.suc _ , h = map₂ (_∘ h) (glue-once (h (f (# 0))) (h (g (# 0))))
+... | ℕ.suc _ , h = map₂ (_∘ h) (glue-once (compare (h (f (# 0))) (h (g (# 0)))))
 
 -- Glue together the image of two finite-set functions, iterative
 glue-iter
@@ -55,5 +74,5 @@ glue-iter
 glue-iter {ℕ.zero} {n} {y} f g h = y , h
 glue-iter {ℕ.suc m} {n} {ℕ.zero} f g h = ⊥-elim (¬Fin0 (f (# 0)))
 glue-iter {ℕ.suc m} {n} {ℕ.suc y} f g h =
-    let p = proj₂ (glue-once (f (# 0)) (g (# 0))) in
+    let p = proj₁ (proj₂ (glue-unglue-once (compare (f (# 0)) (g (# 0))))) in
     glue-iter (p ∘ f ∘ Fin.suc) (p ∘ g ∘ Fin.suc) (p ∘ h)