{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Lists.Lists
open import Rings.Homomorphisms.Definition

module Rings.Polynomial.Evaluation {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where

open Ring R
open Setoid S
open Equivalence eq
open Group additiveGroup
open import Groups.Polynomials.Definition additiveGroup
open import Groups.Polynomials.Addition additiveGroup
open import Rings.Polynomial.Ring R
open import Rings.Polynomial.Multiplication R

inducedFunction : NaivePoly → A → A
inducedFunction [] a = 0R
inducedFunction (x :: p) a = x + (a * inducedFunction p a)

inducedFunctionMult : (as : NaivePoly) (b c : A) → inducedFunction (map (_*_ b) as) c ∼ (inducedFunction as c) * b
inducedFunctionMult [] b c = symmetric (transitive *Commutative timesZero)
inducedFunctionMult (x :: as) b c = transitive (transitive (+WellDefined reflexive (transitive (transitive (*WellDefined reflexive (inducedFunctionMult as b c)) *Associative) *Commutative)) (symmetric *DistributesOver+)) *Commutative

inducedFunctionWellDefined : {a b : NaivePoly} → polysEqual a b → (c : A) → inducedFunction a c ∼ inducedFunction b c
inducedFunctionWellDefined {[]} {[]} a=b c = reflexive
inducedFunctionWellDefined {[]} {x :: b} (fst ,, snd) c = symmetric (transitive (+WellDefined fst (transitive (*WellDefined reflexive (symmetric (inducedFunctionWellDefined {[]} {b} snd c))) (timesZero {c}))) identRight)
inducedFunctionWellDefined {a :: as} {[]} (fst ,, snd) c = transitive (+WellDefined fst reflexive) (transitive identLeft (transitive (*WellDefined reflexive (inducedFunctionWellDefined {as} {[]} snd c)) (timesZero {c})))
inducedFunctionWellDefined {a :: as} {b :: bs} (fst ,, snd) c = +WellDefined fst (*WellDefined reflexive (inducedFunctionWellDefined {as} {bs} snd c))

inducedFunctionGroupHom : {x y : NaivePoly} → (a : A) → inducedFunction (x +P y) a ∼ (inducedFunction x a + inducedFunction y a)
inducedFunctionGroupHom {[]} {[]} a = symmetric identLeft
inducedFunctionGroupHom {[]} {x :: y} a rewrite mapId y = symmetric identLeft
inducedFunctionGroupHom {x :: xs} {[]} a rewrite mapId xs = symmetric identRight
inducedFunctionGroupHom {x :: xs} {y :: ys} a = transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (transitive (+WellDefined reflexive (transitive (*WellDefined reflexive (transitive (inducedFunctionGroupHom {xs} {ys} a) groupIsAbelian)) *DistributesOver+)) +Associative) groupIsAbelian)) +Associative)

inducedFunctionRingHom : (r s : NaivePoly) (a : A) → inducedFunction (r *P s) a ∼ (inducedFunction r a * inducedFunction s a)
inducedFunctionRingHom [] s a = symmetric (transitive *Commutative timesZero)
inducedFunctionRingHom (x :: xs) [] a = symmetric timesZero
inducedFunctionRingHom (b :: bs) (c :: cs) a = transitive (+WellDefined reflexive (*WellDefined reflexive (inducedFunctionGroupHom {map (_*_ b) cs +P map (_*_ c) bs} {0G :: (bs *P cs)} a))) (transitive (+WellDefined reflexive (*WellDefined reflexive (+WellDefined (inducedFunctionGroupHom {map (_*_ b) cs} {map (_*_ c) bs} a) identLeft))) (transitive (transitive (transitive (+WellDefined reflexive (transitive (transitive (transitive (*WellDefined reflexive (transitive (+WellDefined (transitive groupIsAbelian (+WellDefined (inducedFunctionMult bs c a) (inducedFunctionMult cs b a))) (transitive (transitive (*WellDefined reflexive (transitive (inducedFunctionRingHom bs cs a) *Commutative)) *Associative) *Commutative)) (symmetric +Associative))) *DistributesOver+) (+WellDefined reflexive *DistributesOver+)) (+WellDefined *Associative (+WellDefined (transitive *Associative *Commutative) *Associative)))) +Associative) (+WellDefined (symmetric *DistributesOver+') (symmetric *DistributesOver+'))) (symmetric *DistributesOver+)))

inducedFunctionIsHom : (a : A) → RingHom polyRing R (λ p → inducedFunction p a)
RingHom.preserves1 (inducedFunctionIsHom a) = transitive (+WellDefined reflexive (timesZero {a})) identRight
RingHom.ringHom (inducedFunctionIsHom a) {r} {s} = inducedFunctionRingHom r s a
GroupHom.groupHom (RingHom.groupHom (inducedFunctionIsHom a)) {x} {y} = inducedFunctionGroupHom {x} {y} a
GroupHom.wellDefined (RingHom.groupHom (inducedFunctionIsHom a)) x=y = inducedFunctionWellDefined x=y a