------------------------------------------------------------------------
-- 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 (ℕ; 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) → ℕ → A
fold z s zero    = z
fold z s (suc n) = s (fold z s n)
 
iterate : (A → A) → A → ℕ → A
iterate f x zero    = x
iterate f x (suc n) = iterate f (f x) n
 
add : (0# : A) (1+ : A → A) → ℕ → A → A
add 0# 1+ n z = fold z 1+ n
 
mul : (0# : A) (1+ : A → A) → (+ : A → A → A) → (ℕ → A → A)
mul 0# 1+ _+_ n x = fold 0# (λ 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 (λ 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  m) ⟩
  fold z s (m + )  ≡⟨ 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  (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
                         (λ l → sym (+-assoc n l p)) m ⟩
  fold  (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)
                         (λ l → sym (*-assoc m l p)) n ⟩
  fold  (m *_) n * p ≡⟨ cong (_* p) (^-is-fold n) ⟩
  m ^ n * p           ∎