Height

------------------------------------------------------------------------
-- The Agda standard library
--
-- Types and functions which are used to keep track of height
-- invariants in AVL Trees
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Data.Tree.AVL.Height where

open import Data.Nat.Base
open import Data.Fin.Base using (Fin; zero; suc)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)

private
  variable
    i j m n : ℕ


ℕ₂ = Fin 
pattern 0# = zero
pattern 1# = suc zero
pattern ## = suc (suc ())

-- Addition.

infixl  _⊕_

_⊕_ : ℕ₂ → ℕ → ℕ
0# ⊕ n = n
1# ⊕ n =  + n

-- pred[ i ⊕ n ] = pred (i ⊕ n).

pred[_⊕_] : ℕ₂ → ℕ → ℕ
pred[ i ⊕ zero  ] = 
pred[ i ⊕ suc n ] = i ⊕ n

infix  _∼_⊔_

-- If i ∼ j ⊔ m, then the difference between i and j is at most 1,
-- and the maximum of i and j is m. _∼_⊔_ is used to record the
-- balance factor of the AVL trees, and also to ensure that the
-- absolute value of the balance factor is never more than 1.

data _∼_⊔_ : ℕ → ℕ → ℕ → Set where
  ∼+ :     n ∼  + n ⊔  + n
  ∼0 :     n ∼ n     ⊔ n
  ∼- :  + n ∼ n     ⊔  + n

-- Some lemmas.

max∼ : i ∼ j ⊔ m → m ∼ i ⊔ m
max∼ ∼+ = ∼-
max∼ ∼0 = ∼0
max∼ ∼- = ∼0

∼max : i ∼ j ⊔ m → j ∼ m ⊔ m
∼max ∼+ = ∼0
∼max ∼0 = ∼0
∼max ∼- = ∼+

0∼⊔ :  ∼ j ⊔ m → j ≡ m
0∼⊔ ∼+ = refl
0∼⊔ ∼0 = refl

∼0⊔ : i ∼  ⊔ m → i ≡ m
∼0⊔ ∼- = refl
∼0⊔ ∼0 = refl