------------------------------------------------------------------------
-- The Agda standard library
--
-- A generalisation of the arithmetic operations
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Nat.GeneralisedArithmetic where

open import Data.Nat.Base using (; zero; suc; _+_; _*_; _^_)
open import Data.Nat.Properties using (+-comm; +-assoc; *-identityˡ;
  *-assoc)
open import Function.Base using (_∘′_; _∘_; id)
open import Level using (Level)
open import Relation.Binary.PropositionalEquality.Core
  using (_≡_; refl; cong; sym)
open import Relation.Binary.PropositionalEquality.Properties
  using (module ≡-Reasoning)
open ≡-Reasoning

private
  variable
    a : Level
    A : Set a

fold : A  (A  A)    A
fold z s zero    = z
fold z s (suc n) = s (fold z s n)

iterate : (A  A)  A    A
iterate f x zero    = x
iterate f x (suc n) = iterate f (f x) n

add : (0# : A) (1+ : A  A)    A  A
add 0# 1+ n z = fold z 1+ n

mul : (0# : A) (1+ : A  A)  (+ : A  A  A)  (  A  A)
mul 0# 1+ _+_ n x = fold 0#  s  x + s) n

-- Properties

fold-+ :  (z : A) (s : A  A) m {n} 
         fold z s (m + n)  fold (fold z s n) s m
fold-+ z s zero    = refl
fold-+ z s (suc m) = cong s (fold-+ z s m)

fold-k :  (z : A) (s : A  A) {k} m 
         fold k (s ∘′_) m z  fold (k z) s m
fold-k z s zero    = refl
fold-k z s (suc m) = cong s (fold-k z s m)

fold-* :  (z : A) (s : A  A) m {n} 
         fold z s (m * n)  fold z (fold id (s ∘_) n) m
fold-* z s zero        = refl
fold-* z s (suc m) {n} = let +n = fold id (s ∘′_) n in begin
  fold z s (n + m * n)        ≡⟨ fold-+ z s n 
  fold (fold z s (m * n)) s n ≡⟨ cong  z  fold z s n) (fold-* z s m) 
  fold (fold z +n m) s n      ≡⟨ sym (fold-k _ s n) 
  fold z +n (suc m)           

fold-pull :  (z : A) (s : A  A) (g : A  A  A) (p : A)
            (eqz : g z p  p)
            (eqs :  l  s (g l p)  g (s l) p) 
             m  fold p s m  g (fold z s m) p
fold-pull z s _ _ eqz _ zero    = sym eqz
fold-pull z s g p eqz eqs (suc m) = begin
  s (fold p s m)       ≡⟨ cong s (fold-pull z s g p eqz eqs m) 
  s (g (fold z s m) p) ≡⟨ eqs (fold z s m) 
  g (s (fold z s m)) p 

iterate-is-fold :  (z : A) s m  fold z s m  iterate s z m
iterate-is-fold z s zero    = refl
iterate-is-fold z s (suc m) = begin
  fold z s (suc m)  ≡⟨ cong (fold z s) (+-comm 1 m) 
  fold z s (m + 1)  ≡⟨ fold-+ z s m 
  fold (s z) s m    ≡⟨ iterate-is-fold (s z) s m 
  iterate s (s z) m 

id-is-fold :  m  fold zero suc m  m
id-is-fold zero    = refl
id-is-fold (suc m) = cong suc (id-is-fold m)

+-is-fold :  m {n}  fold n suc m  m + n
+-is-fold zero    = refl
+-is-fold (suc m) = cong suc (+-is-fold m)

*-is-fold :  m {n}  fold zero (n +_) m  m * n
*-is-fold zero        = refl
*-is-fold (suc m) {n} = cong (n +_) (*-is-fold m)

^-is-fold :  {m} n  fold 1 (m *_) n  m ^ n
^-is-fold     zero    = refl
^-is-fold {m} (suc n) = cong (m *_) (^-is-fold n)

*+-is-fold :  m n {p}  fold p (n +_) m  m * n + p
*+-is-fold m n {p} = begin
  fold p (n +_) m     ≡⟨ fold-pull _ _ _+_ p refl
                          l  sym (+-assoc n l p)) m 
  fold 0 (n +_) m + p ≡⟨ cong (_+ p) (*-is-fold m) 
  m * n + p           

^*-is-fold :  m n {p}  fold p (m *_) n  m ^ n * p
^*-is-fold m n {p} = begin
  fold p (m *_) n     ≡⟨ fold-pull _ _ _*_ p (*-identityˡ p)
                          l  sym (*-assoc m l p)) n 
  fold 1 (m *_) n * p ≡⟨ cong (_* p) (^-is-fold n) 
  m ^ n * p