module NatMap where

open import Agda.Primitive using (Level)
open import Data.Nat using (ℕ; _<?_; _≟_)
open import Data.Nat.Properties using (<-cmp)
open import Data.String using (String; _++_)
open import Data.List using (List; []; _∷_)
open import Data.Product using (_×_; _,_)
open import Relation.Nullary using (yes; no)
open import Relation.Binary using (Tri)
open import Agda.Builtin.Equality using (_≡_; refl)

variable
    a : Level
    A : Set a

-- It's easiest to reason about a linear-insertion map.

NatMap : (A : Set a) → Set a
NatMap A = List (ℕ × A)

insert : ℕ → A → NatMap A -> NatMap A
insert n a [] = (n , a) ∷ []
insert n a l@(x@(n' , a') ∷ xs) with n <? n'
...                             | yes n≡n' = (n , a) ∷ l
...                             | no n≢n' = x ∷ insert n a xs


testInsert₁ : insert 3 "third" [] ≡ (3 , "third") ∷ []
testInsert₁ = refl

testInsert₂ : insert 4 "fourth" ((3 , "third") ∷ []) ≡ (3 , "third") ∷ (4 , "fourth") ∷ []
testInsert₂ = refl

testInsert₃ : insert 2 "second" ((3 , "third") ∷ (4 , "fourth") ∷ []) ≡ (2 , "second") ∷ (3 , "third") ∷ (4 , "fourth") ∷ []
testInsert₃ = refl

{-# TERMINATING #-}
merge : (A -> A -> A) -> NatMap A -> NatMap A -> NatMap A
merge _ [] m₂ = m₂
merge _ m₁ [] = m₁
merge f m₁@(x₁@(n₁ , a₁) ∷ xs₁) m₂@(x₂@(n₂ , a₂) ∷ xs₂) with <-cmp n₁ n₂
...                                                     | Tri.tri< _ _ _ = x₁ ∷ merge f xs₁ m₂
...                                                     | Tri.tri> _ _ _ = x₂ ∷ merge f m₁ xs₂
...                                                     | Tri.tri≈ _ _ _ = (n₁ , f a₁ a₂) ∷ merge f xs₁ xs₂

testMerge : merge (_++_) ((1 , "one") ∷ (2 , "two") ∷ []) ((2 , "two") ∷ (3 , "three") ∷ []) ≡ (1 , "one") ∷ (2 , "twotwo") ∷ (3 , "three") ∷ []
testMerge = refl

-- Note to self: will it be tricky to define a Semilattice instance for this?
--               I'm worried because proofs will likely require knowing the ordering -- merge certainly relies on it, but
--               we can't prove commutativity if the NatMap type includes a proof of ordering, because then we'll need
--               to compare `<=` for equality, which are going to be defined in terms of `=`...
--
--               We might need to generalize Lattice and friends to use ≈. But then,
--               pain in the ass for products because they'll need to lift ≈ ...