```------------------------------------------------------------------------
-- The Agda standard library
--
-- Homomorphism proofs for addition over polynomials
------------------------------------------------------------------------

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

open import Tactic.RingSolver.Core.Polynomial.Parameters

{r₁ r₂ r₃ r₄}
(homo : Homomorphism r₁ r₂ r₃ r₄)
where

open import Data.Nat            as ℕ  using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
open import Data.Nat.Properties as ℕₚ using (≤′-trans)
open import Data.Product.Base         using (_,_; _×_; proj₂)
open import Data.List.Base            using (_∷_; [])
open import Data.List.Kleene
open import Data.Vec.Base             using (Vec)
open import Function.Base using (_⟨_⟩_; flip)
open import Relation.Unary

import Relation.Binary.PropositionalEquality.Core as ≡

open Homomorphism homo hiding (_^_)
open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo
open import Tactic.RingSolver.Core.Polynomial.Base from
open import Tactic.RingSolver.Core.Polynomial.Reasoning to
open import Tactic.RingSolver.Core.Polynomial.Semantics homo
open import Algebra.Properties.Semiring.Exp.TCOptimised semiring

mutual
⊞-hom : ∀ {n} (xs ys : Poly n) →
∀ ρ → ⟦ xs ⊞ ys ⟧ ρ ≈ ⟦ xs ⟧ ρ + ⟦ ys ⟧ ρ
⊞-hom (xs ⊐ i≤n) (ys ⊐ j≤n) = ⊞-match-hom (inj-compare i≤n j≤n) xs ys

⊞-match-hom : ∀ {i j n} {i≤n : i ≤′ n} {j≤n : j ≤′ n}
(i-cmp-j : InjectionOrdering i≤n j≤n)
(xs : FlatPoly i) (ys : FlatPoly j) →
∀ ρ → ⟦ ⊞-match i-cmp-j xs ys ⟧ ρ ≈ ⟦ xs ⊐ i≤n ⟧ ρ + ⟦ ys ⊐ j≤n ⟧ ρ
⊞-match-hom (inj-eq ij≤n) (Κ x) (Κ y) Ρ = +-homo x y
⊞-match-hom (inj-eq ij≤n) (⅀ (x Δ i & xs)) (⅀ (y Δ j & ys)) Ρ =
begin
⟦ ⊞-zip (compare i j) x xs y ys ⊐↓ ij≤n ⟧ Ρ
≈⟨ ⊐↓-hom (⊞-zip (compare i j) x xs y ys) ij≤n Ρ ⟩
⅀?⟦ ⊞-zip (compare i j) x xs y ys ⟧ (drop-1 ij≤n Ρ)
≈⟨ ⊞-zip-hom (compare i j) x xs y ys (drop-1 ij≤n Ρ) ⟩
⅀⟦ x Δ i & xs ⟧ (drop-1 ij≤n Ρ) + ⅀⟦ y Δ j & ys ⟧ (drop-1 ij≤n Ρ)
∎
⊞-match-hom (inj-gt i≤n j≤i-1) (⅀ xs) ys Ρ =
let (ρ , Ρ′) = drop-1 i≤n Ρ
in
begin
⟦ ⊞-inj j≤i-1 ys xs ⊐↓ i≤n ⟧ Ρ
≈⟨ ⊐↓-hom (⊞-inj j≤i-1 ys xs) i≤n Ρ ⟩
⅀?⟦ ⊞-inj j≤i-1 ys xs ⟧ (drop-1 i≤n Ρ)
≈⟨ ⊞-inj-hom j≤i-1 ys xs ρ Ρ′ ⟩
⟦ ys ⊐ j≤i-1 ⟧ (proj₂ (drop-1 i≤n Ρ)) + ⅀⟦ xs ⟧ (drop-1 i≤n Ρ)
≈⟨ ≪+ trans-join-hom j≤i-1 i≤n ys Ρ ⟩
⟦ ys ⊐ (≤′-step j≤i-1 ⟨ ≤′-trans ⟩ i≤n) ⟧ Ρ + ⅀⟦ xs ⟧ (drop-1 i≤n Ρ)
≈⟨ +-comm _ _ ⟩
⅀⟦ xs ⟧ (drop-1 i≤n Ρ) + ⟦ ys ⊐ (≤′-step j≤i-1 ⟨ ≤′-trans ⟩ i≤n) ⟧ Ρ
∎
⊞-match-hom (inj-lt i≤j-1 j≤n) xs (⅀ ys) Ρ =
let (ρ , Ρ′) = drop-1 j≤n Ρ
in
begin
⟦ ⊞-inj i≤j-1 xs ys ⊐↓ j≤n ⟧ Ρ
≈⟨ ⊐↓-hom (⊞-inj i≤j-1 xs ys) j≤n Ρ ⟩
⅀?⟦ ⊞-inj i≤j-1 xs ys ⟧ (drop-1 j≤n Ρ)
≈⟨ ⊞-inj-hom i≤j-1 xs ys ρ Ρ′ ⟩
⟦ xs ⊐ i≤j-1 ⟧ (proj₂ (drop-1 j≤n Ρ)) + ⅀⟦ ys ⟧ (drop-1 j≤n Ρ)
≈⟨ ≪+ trans-join-hom i≤j-1 j≤n xs Ρ ⟩
⟦ xs ⊐ (≤′-step i≤j-1 ⟨ ≤′-trans ⟩ j≤n) ⟧ Ρ + ⅀⟦ ys ⟧ (drop-1 j≤n Ρ)
∎

⊞-inj-hom : ∀ {i k}
→ (i≤k : i ≤′ k)
→ (x : FlatPoly i)
→ (ys : Coeff k +)
→ (ρ : Carrier)
→ (Ρ : Vec Carrier k)
→ ⅀?⟦ ⊞-inj i≤k x ys ⟧ (ρ , Ρ) ≈ ⟦ x ⊐ i≤k ⟧ Ρ + ⅀⟦ ys ⟧ (ρ , Ρ)
⊞-inj-hom i≤k xs (y ⊐ j≤k ≠0 Δ 0 & []) ρ Ρ =
begin
⅀?⟦ ⊞-match (inj-compare j≤k i≤k) y xs Δ 0 ∷↓ [] ⟧ (ρ , Ρ)
≈⟨ ∷↓-hom-0 ((⊞-match (inj-compare j≤k i≤k) y xs)) [] ρ Ρ ⟩
⟦ ⊞-match (inj-compare j≤k i≤k) y xs ⟧ Ρ
≈⟨ ⊞-match-hom (inj-compare j≤k i≤k) y xs Ρ ⟩
(⟦ y ⊐ j≤k ⟧ Ρ + ⟦ xs ⊐ i≤k ⟧ Ρ)
≈⟨ +-comm _ _ ⟩
⟦ xs ⊐ i≤k ⟧ Ρ + ( ⟦ y ⊐ j≤k ⟧ Ρ)
∎
⊞-inj-hom i≤k xs (y ⊐ j≤k ≠0 Δ 0 & (∹ ys )) ρ Ρ =
begin
⅀?⟦ ⊞-match (inj-compare j≤k i≤k) y xs Δ 0 ∷↓ (∹ ys ) ⟧ (ρ , Ρ)
≈⟨ ∷↓-hom-0 ((⊞-match (inj-compare j≤k i≤k) y xs)) (∹ ys ) ρ Ρ ⟩
ρ * ⅀⟦ ys ⟧ (ρ , Ρ) + ⟦ ⊞-match (inj-compare j≤k i≤k) y xs ⟧ Ρ
≈⟨ +≫ ⊞-match-hom (inj-compare j≤k i≤k) y xs Ρ ⟩
ρ * ⅀⟦ ys ⟧ (ρ , Ρ) + (⟦ y ⊐ j≤k ⟧ Ρ + ⟦ xs ⊐ i≤k ⟧ Ρ)
≈⟨ sym (+-assoc _ _ _) ⟨ trans ⟩ +-comm _ _ ⟩
⟦ xs ⊐ i≤k ⟧ Ρ + (ρ * ⅀⟦ ys ⟧ (ρ , Ρ) + ⟦ y ⊐ j≤k ⟧ Ρ)
∎
⊞-inj-hom i≤k xs (y Δ suc j & ys) ρ Ρ =
begin
⅀?⟦ ⊞-inj i≤k xs (y Δ suc j & ys) ⟧ (ρ , Ρ)
≡⟨⟩
⅀?⟦ xs ⊐ i≤k Δ 0 ∷↓ (∹ y Δ j & ys ) ⟧ (ρ , Ρ)
≈⟨ ∷↓-hom-0 (xs ⊐ i≤k) (∹ y Δ j & ys ) ρ Ρ ⟩
ρ * ⅀⟦ y Δ j & ys ⟧ (ρ , Ρ) + ⟦ xs ⊐ i≤k ⟧ Ρ
≈⟨ +-comm _ _ ⟩
⟦ xs ⊐ i≤k ⟧ Ρ + ρ * ⅀⟦ y Δ j & ys ⟧ (ρ , Ρ)
≈⟨ +≫ (
begin
ρ * ⅀⟦ y Δ j & ys ⟧ (ρ , Ρ)
≡⟨⟩
ρ * (((poly y , ys) ⟦∷⟧ (ρ , Ρ)) *⟨ ρ ⟩^ j)
≈⟨ *≫ pow-opt _ ρ j ⟩
ρ * (ρ ^ j * ((poly y , ys) ⟦∷⟧ (ρ , Ρ)))
≈⟨ sym (*-assoc ρ _ _) ⟩
ρ * ρ ^ j * ((poly y , ys) ⟦∷⟧ (ρ , Ρ))
≈⟨ ≪* sym (pow-suc ρ j) ⟩
⅀⟦ y Δ suc j & ys ⟧ (ρ , Ρ)
∎) ⟩
⟦ xs ⊐ i≤k ⟧ Ρ + ⅀⟦ y Δ suc j & ys ⟧ (ρ , Ρ)
∎

⊞-coeffs-hom : ∀ {n} (xs : Coeff n *) (ys : Coeff n *) →
∀ ρ → ⅀?⟦ ⊞-coeffs xs ys ⟧ ρ ≈ ⅀?⟦ xs ⟧ ρ + ⅀?⟦ ys ⟧ ρ
⊞-coeffs-hom [] ys Ρ         = sym (+-identityˡ (⅀?⟦ ys ⟧ Ρ))
⊞-coeffs-hom (∹ x Δ i & xs ) = ⊞-zip-r-hom i x xs

⊞-zip-hom : ∀ {n i j}
→ (c : ℕ.Ordering i j)
→ (x : NonZero n)
→ (xs : Coeff n *)
→ (y : NonZero n)
→ (ys : Coeff n *)
→ ∀ ρ → ⅀?⟦ ⊞-zip c x xs y ys ⟧ ρ ≈ ⅀⟦ x Δ i & xs ⟧ ρ + ⅀⟦ y Δ j & ys ⟧ ρ
⊞-zip-hom (ℕ.equal i) (x ≠0) xs (y ≠0) ys (ρ , Ρ) =
let x′  = ⟦ x ⟧ Ρ
y′  = ⟦ y ⟧ Ρ
xs′ = ⅀?⟦ xs ⟧ (ρ , Ρ)
ys′ = ⅀?⟦ ys ⟧ (ρ , Ρ)
in
begin
⅀?⟦ x ⊞ y Δ i ∷↓ ⊞-coeffs xs ys ⟧ (ρ , Ρ)
≈⟨ ∷↓-hom (x ⊞ y) i (⊞-coeffs xs ys) ρ Ρ ⟩
ρ ^ i * (((x ⊞ y) , (⊞-coeffs xs ys)) ⟦∷⟧ (ρ , Ρ))
≈⟨ *≫ ⟦∷⟧-hom (x ⊞ y) (⊞-coeffs xs ys) ρ Ρ ⟩
ρ ^ i * (ρ * ⅀?⟦ ⊞-coeffs xs ys ⟧ (ρ , Ρ) + ⟦ x ⊞ y ⟧ Ρ)
≈⟨ *≫ begin
ρ * ⅀?⟦ ⊞-coeffs xs ys ⟧ (ρ , Ρ) + ⟦ x ⊞ y ⟧ Ρ
≈⟨ (*≫ ⊞-coeffs-hom xs ys (ρ , Ρ)) ⟨ +-cong ⟩ ⊞-hom x y Ρ  ⟩
ρ * (xs′ + ys′) + (x′ + y′)
≈⟨ ≪+ distribˡ ρ xs′ ys′ ⟩
ρ * xs′ + ρ * ys′ + (x′ + y′)
≈⟨ +-assoc _ _ _ ⟨ trans ⟩ (+≫ (sym (+-assoc _ _ _) ⟨ trans ⟩ (≪+ +-comm _ _))) ⟩
ρ * xs′ + (x′ + ρ * ys′ + y′)
≈⟨ (+≫ +-assoc _ _ _) ⟨ trans ⟩ sym (+-assoc _ _ _) ⟩
(ρ * xs′ + x′) + (ρ * ys′ + y′)
∎ ⟩
ρ ^ i * ((ρ * xs′ + x′) + (ρ * ys′ + y′))
≈⟨ distribˡ (ρ ^ i) _ _ ⟩
ρ ^ i * (ρ * xs′ + x′) + ρ ^ i * (ρ * ys′ + y′)
≈⟨ sym (pow-opt _ ρ i ⟨ +-cong ⟩ pow-opt _ ρ i) ⟩
(ρ * xs′ + x′) *⟨ ρ ⟩^ i + (ρ * ys′ + y′) *⟨ ρ ⟩^ i
≈⟨ pow-mul-cong (sym (⟦∷⟧-hom x xs ρ Ρ)) ρ i ⟨ +-cong ⟩ pow-mul-cong (sym (⟦∷⟧-hom y ys ρ Ρ)) ρ i ⟩
((x , xs) ⟦∷⟧ (ρ , Ρ)) *⟨ ρ ⟩^ i + ((y , ys) ⟦∷⟧ (ρ , Ρ)) *⟨ ρ ⟩^ i
∎
⊞-zip-hom (ℕ.less i k) x xs y ys (ρ , Ρ) = ⊞-zip-r-step-hom i k y ys x xs (ρ , Ρ) ⊙ +-comm _ _
⊞-zip-hom (ℕ.greater j k) = ⊞-zip-r-step-hom j k

⊞-zip-r-step-hom : ∀ {n} j k
→ (x : NonZero n)
→ (xs : Coeff n *)
→ (y : NonZero n)
→ (ys : Coeff n *)
→ ∀ ρ → ⅀⟦ y Δ j & ⊞-zip-r x k xs ys ⟧ ρ ≈ ⅀⟦ x Δ suc (j ℕ.+ k) & xs ⟧ ρ + ⅀⟦ y Δ j & ys ⟧ ρ
⊞-zip-r-step-hom j k x xs y ys (ρ , Ρ) =
let x′  = ⟦ NonZero.poly x ⟧ Ρ
y′  = ⟦ NonZero.poly y ⟧ Ρ
xs′ = ⅀?⟦ xs ⟧ (ρ , Ρ)
ys′ = ⅀?⟦ ys ⟧ (ρ , Ρ)
in
begin
((poly y , ⊞-zip-r x k xs ys) ⟦∷⟧ (ρ , Ρ)) *⟨ ρ ⟩^ j
≈⟨ pow-mul-cong (⟦∷⟧-hom (poly y) (⊞-zip-r x k xs ys) ρ Ρ) ρ j ⟩
(ρ * ⅀?⟦ ⊞-zip-r x k xs ys ⟧ (ρ , Ρ) + y′) *⟨ ρ ⟩^ j
≈⟨ pow-opt _ ρ j ⟩
ρ ^ j * (ρ * ⅀?⟦ ⊞-zip-r x k xs ys ⟧ (ρ , Ρ) + y′)
≈⟨ *≫ ≪+ *≫ (⊞-zip-r-hom k x xs ys (ρ , Ρ) ⟨ trans ⟩ (≪+ pow-mul-cong (⟦∷⟧-hom (poly x) xs ρ Ρ) ρ k)) ⟩
ρ ^ j * (ρ * ((ρ * xs′ + x′) *⟨ ρ ⟩^ k + ys′) + y′)
≈⟨ *≫ ≪+ distribˡ ρ _ _ ⟩
ρ ^ j * ((ρ * (ρ * xs′ + x′) *⟨ ρ ⟩^ k + ρ * ys′) + y′)
≈⟨ *≫ +-assoc _ _ _ ⟩
ρ ^ j * (ρ * (ρ * xs′ + x′) *⟨ ρ ⟩^ k + (ρ * ys′ + y′))
≈⟨ distribˡ _ _ _ ⟩
ρ ^ j * (ρ * (ρ * xs′ + x′) *⟨ ρ ⟩^ k) + ρ ^ j * (ρ * ys′ + y′)
≈⟨ sym (pow-opt _ ρ j) ⟨ flip +-cong ⟩
(begin
ρ ^ j * (ρ * ((ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ k))
≈⟨ (sym (*-assoc _ _ _) ⟨ trans ⟩ (≪* sym (pow-sucʳ ρ j))) ⟩
ρ ^ suc j * ((ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ k)
≈⟨ pow-add _ _ k j ⟩
(ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ (k ℕ.+ suc j)
≡⟨ ≡.cong (λ i → (ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ i) (ℕₚ.+-comm k (suc j)) ⟩
(ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ (suc j ℕ.+ k)
∎)
⟩
(ρ * xs′ + x′) *⟨ ρ ⟩^ suc (j ℕ.+ k) + (ρ * ys′ + y′) *⟨ ρ ⟩^ j
≈⟨ pow-mul-cong (sym (⟦∷⟧-hom (poly x) xs ρ Ρ)) ρ (suc (j ℕ.+ k)) ⟨ +-cong ⟩ pow-mul-cong (sym (⟦∷⟧-hom (poly y) ys ρ Ρ)) ρ j ⟩
((poly x , xs) ⟦∷⟧ (ρ , Ρ)) *⟨ ρ ⟩^ suc (j ℕ.+ k) + ((poly y , ys) ⟦∷⟧ (ρ , Ρ)) *⟨ ρ ⟩^ j
∎

⊞-zip-r-hom : ∀ {n} i
→ (x : NonZero n)
→ (xs ys : Coeff n *)
→ (Ρ : Carrier × Vec Carrier n)
→ ⅀?⟦ ⊞-zip-r x i xs ys ⟧ (Ρ) ≈ ⅀⟦ x Δ i & xs ⟧ ( Ρ) + ⅀?⟦ ys ⟧ ( Ρ)
⊞-zip-r-hom i x xs []                (ρ , Ρ) = sym (+-identityʳ _)
⊞-zip-r-hom i x xs (∹ (y Δ j) & ys )         = ⊞-zip-hom (compare i j) x xs y ys
```