From f5f3a900d097b26363627d7e1a6a673667fec807 Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Mon, 19 Feb 2024 12:52:36 -0600
Subject: Add glue function for finite sets

---
 FinMerge.agda | 43 +++++++++++++++++++++++++++++++++++++------
 1 file changed, 37 insertions(+), 6 deletions(-)

diff --git a/FinMerge.agda b/FinMerge.agda
index a327602..d8b38fd 100644
--- a/FinMerge.agda
+++ b/FinMerge.agda
@@ -1,13 +1,15 @@
 module FinMerge where
 
-open import Data.Fin using (Fin; splitAt; join; fromℕ<; cast)
-open import Data.Nat using (ℕ; _+_; _≤_; _<_)
-open import Data.Nat.Properties using (+-comm)
+open import Data.Empty using (⊥-elim)
+open import Data.Fin using (Fin; splitAt; join; fromℕ<; cast; toℕ; #_) renaming (_<_ to _<′_)
+open import Data.Fin.Properties using (¬Fin0)
+open import Data.Nat using (ℕ; _+_; _≤_; _<_ ; z<s)
+open import Data.Nat.Properties using (+-comm; <⇒≤)
 open import Data.Sum.Base using (_⊎_)
 open import Data.Product using (_×_; _,_; Σ-syntax; map₂)
 open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym)
 open import Relation.Binary.PropositionalEquality.Properties using (module ≡-Reasoning)
-open import Function using (_$_)
+open import Function using (id ; _∘_ ; _$_)
 
 
 _<_≤_ : ℕ → ℕ → ℕ → Set
@@ -16,6 +18,24 @@ _<_≤_ i j k = (i < j) × (j ≤ k)
 _<_<_ : ℕ → ℕ → ℕ → Set
 _<_<_ i j k = (i < j) × (j < k)
 
+toℕ< : {n : ℕ} → (i : Fin n) → toℕ i < n
+toℕ< Fin.zero = z<s
+toℕ< (Fin.suc i) = _≤_.s≤s (toℕ< i)
+
+data Ordering {n : ℕ} : Fin n → Fin n → Set where
+  less : ∀ {i j} → toℕ i < toℕ j < n → Ordering i j
+  equal : ∀ {i j} → toℕ i ≡ toℕ j → Ordering i j
+  greater : ∀ {i j} → toℕ j < toℕ i < n → Ordering i j
+
+compare : ∀ {n : ℕ} (i j : Fin n) → Ordering i j
+compare Fin.zero Fin.zero = equal refl
+compare Fin.zero j@(Fin.suc _) = less (z<s , toℕ< j)
+compare i@(Fin.suc _) Fin.zero = greater (z<s , toℕ< i)
+compare (Fin.suc i) (Fin.suc j) with compare i j
+... | less (i<j , j<n) = less (_≤_.s≤s i<j , _≤_.s≤s j<n)
+... | equal i≡j = equal (cong ℕ.suc i≡j)
+... | greater (j<i , i<n) = greater (_≤_.s≤s j<i , _≤_.s≤s i<n)
+
 private
   variable
     m n : ℕ
@@ -32,8 +52,8 @@ splitℕ  _≤_.z≤n = _ , refl
 splitℕ (_≤_.s≤s le) = map₂ (cong ℕ.suc) (splitℕ le)
 
 -- Merge two elements of a finite set
-merge : (i j : ℕ) → i < j ≤ n → Fin (ℕ.suc n) → Fin n
-merge {n} i j (lt , le) x with splitℕ le
+merge : {i j : ℕ} → i < j ≤ n → Fin (ℕ.suc n) → Fin n
+merge {n} {i} {j} (lt , le) x with splitℕ le
 ... | j′ , n≡j+j′ =
     cast (sym n≡j+j′) $
     join j j′ $
@@ -49,3 +69,14 @@ merge {n} i j (lt , le) x with splitℕ le
         ℕ.suc (j′ + j)  ≡⟨⟩
         ℕ.suc j′ + j    ≡⟨ +-comm (ℕ.suc j′) j ⟩
         j + ℕ.suc j′ ∎
+
+-- 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)
+glue {ℕ.zero} {n} _ _ = n , id
+glue {ℕ.suc _} {ℕ.zero} f _ = ⊥-elim (¬Fin0 (f (# 0)))
+glue {ℕ.suc _} {ℕ.suc n} f g with glue (f ∘ Fin.suc) (g ∘ Fin.suc)
+... | ℕ.zero , h = ⊥-elim (¬Fin0 (h (# 0)))
+... | ℕ.suc x , h with compare (h (f (# 0))) (h (g (# 0)))
+...     | less (f0<g0 , _≤_.s≤s g0<n) = x , merge (f0<g0 , g0<n) ∘ h
+...     | equal f0≡g0 = ℕ.suc x , h
+...     | greater (g0<f0 , _≤_.s≤s f0<n) = x , merge (g0<f0 , f0<n) ∘ h
-- 
cgit v1.2.3