------------------------------------------------------------------------
-- The Agda standard library
--
-- Solver for commutative ring or semiring equalities
------------------------------------------------------------------------

-- Uses ideas from the Coq ring tactic. See "Proving Equalities in a
-- Commutative Ring Done Right in Coq" by Grégoire and Mahboubi. The
-- code below is not optimised like theirs, though.

open import Algebra
open import Algebra.RingSolver.AlmostCommutativeRing

module Algebra.RingSolver
  {r₁ r₂ r₃}
  (Coeff : RawRing r₁)               -- Coefficient "ring".
  (R : AlmostCommutativeRing r₂ r₃)  -- Main "ring".
  (morphism : Coeff -Raw-AlmostCommutative⟶ R)
  where

import Algebra.RingSolver.Lemmas as L; open L Coeff R morphism
private module C = RawRing Coeff
open AlmostCommutativeRing R hiding (zero)
import Algebra.FunctionProperties as P; open P _≈_
open import Algebra.Morphism
open _-Raw-AlmostCommutative⟶_ morphism renaming (⟦_⟧ to ⟦_⟧')
import Algebra.Operations as Ops; open Ops semiring

open import Relation.Binary
import Relation.Binary.PropositionalEquality as PropEq
import Relation.Binary.Reflection as Reflection

open import Data.Nat using (; suc; zero) renaming (_+_ to _ℕ-+_)
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Vec
open import Function hiding (type-signature)
open import Level using (_⊔_)

infix  9 _↑ :-_ -‿NF_
infixr 9 _:^_ _^-NF_ _:↑_
infix  8 _*x _*x+_
infixl 8 _:*_ _*-NF_ _↑-*-NF_
infixl 7 _:+_ _+-NF_ _:-_
infixl 0 _∶_

------------------------------------------------------------------------
-- Polynomials

data Op : Set where
  [+] : Op
  [*] : Op

-- The polynomials are indexed over the number of variables.

data Polynomial (m : ) : Set r₁ where
  op   : (o : Op) (p₁ : Polynomial m) (p₂ : Polynomial m)  Polynomial m
  con  : (c : C.Carrier)  Polynomial m
  var  : (x : Fin m)  Polynomial m
  _:^_ : (p : Polynomial m) (n : )  Polynomial m
  :-_  : (p : Polynomial m)  Polynomial m

-- Short-hand notation.

_:+_ :  {n}  Polynomial n  Polynomial n  Polynomial n
_:+_ = op [+]

_:*_ :  {n}  Polynomial n  Polynomial n  Polynomial n
_:*_ = op [*]

_:-_ :  {n}  Polynomial n  Polynomial n  Polynomial n
x :- y = x :+ :- y

-- Semantics.

sem : Op  Op₂ Carrier
sem [+] = _+_
sem [*] = _*_

⟦_⟧ :  {n}  Polynomial n  Vec Carrier n  Carrier
 op o p₁ p₂  ρ =  p₁  ρ  sem o   p₂  ρ
 con c       ρ =  c ⟧'
 var x       ρ = lookup x ρ
 p :^ n      ρ =  p  ρ ^ n
 :- p        ρ = -  p  ρ

-- Equality.

_≛_ :  {n}  Polynomial n  Polynomial n  Set _
p₁  p₂ =  {ρ}   p₁  ρ   p₂  ρ

private

  -- Reindexing.

  _:↑_ :  {n}  Polynomial n  (m : )  Polynomial (m ℕ-+ n)
  op o p₁ p₂ :↑ m = op o (p₁ :↑ m) (p₂ :↑ m)
  con c      :↑ m = con c
  var x      :↑ m = var (Fin.raise m x)
  (p :^ n)   :↑ m = (p :↑ m) :^ n
  (:- p)     :↑ m = :- (p :↑ m)

------------------------------------------------------------------------
-- Normal forms of polynomials

-- The normal forms (Horner forms) are indexed over
-- * the number of variables in the polynomial, and
-- * an equivalent polynomial.

data Normal : (n : )  Polynomial n  Set (r₁  r₂  r₃) where
  con   : (c : C.Carrier)  Normal 0 (con c)
  _↑    :  {n p'} (p : Normal n p')  Normal (suc n) (p' :↑ 1)
  _*x+_ :  {n p' c'} (p : Normal (suc n) p') (c : Normal n c') 
          Normal (suc n) (p' :* var zero :+ c' :↑ 1)
  _∶_   :  {n p₁ p₂} (p : Normal n p₁) (eq : p₁  p₂)  Normal n p₂

⟦_⟧-Normal :  {n p}  Normal n p  Vec Carrier n  Carrier
 p  _   ⟧-Normal ρ       =  p ⟧-Normal ρ
 con c   ⟧-Normal ρ       =  c ⟧'
 p      ⟧-Normal (x  ρ) =  p ⟧-Normal ρ
 p *x+ c ⟧-Normal (x  ρ) = ( p ⟧-Normal (x  ρ) * x) +  c ⟧-Normal ρ

------------------------------------------------------------------------
-- Normalisation

private

  con-NF :  {n}  (c : C.Carrier)  Normal n (con c)
  con-NF {zero}  c = con c
  con-NF {suc _} c = con-NF c 

  _+-NF_ :  {n p₁ p₂}  Normal n p₁  Normal n p₂  Normal n (p₁ :+ p₂)
  (p₁  eq₁) +-NF (p₂  eq₂) = p₁ +-NF p₂                     eq₁   +-cong  eq₂
  (p₁  eq)  +-NF p₂         = p₁ +-NF p₂                     eq    +-cong  refl
  p₁         +-NF (p₂  eq)  = p₁ +-NF p₂                     refl  +-cong  eq
  con c₁     +-NF con c₂     = con (C._+_ c₁ c₂)              +-homo _ _
  p₁        +-NF p₂        = (p₁ +-NF p₂)                  refl
  p₁ *x+ c₁  +-NF p₂        = p₁ *x+ (c₁ +-NF p₂)            sym (+-assoc _ _ _)
  p₁ *x+ c₁  +-NF p₂ *x+ c₂  = (p₁ +-NF p₂) *x+ (c₁ +-NF c₂)  lemma₁ _ _ _ _ _
  p₁        +-NF p₂ *x+ c₂  = p₂ *x+ (p₁ +-NF c₂)            lemma₂ _ _ _

  _*x :  {n p}  Normal (suc n) p  Normal (suc n) (p :* var zero)
  p *x = p *x+ con-NF C.0#  lemma₀ _

  mutual

    -- The first function is just a variant of _*-NF_ which I used to
    -- make the termination checker believe that the code is
    -- terminating.

    _↑-*-NF_ :  {n p₁ p₂} 
               Normal n p₁  Normal (suc n) p₂ 
               Normal (suc n) (p₁ :↑ 1 :* p₂)
    p₁ ↑-*-NF (p₂  eq)   = p₁ ↑-*-NF p₂                     refl  *-cong  eq
    p₁ ↑-*-NF p₂         = (p₁ *-NF p₂)                    refl
    p₁ ↑-*-NF (p₂ *x+ c₂) = (p₁ ↑-*-NF p₂) *x+ (p₁ *-NF c₂)  lemma₄ _ _ _ _

    _*-NF_ :  {n p₁ p₂} 
             Normal n p₁  Normal n p₂  Normal n (p₁ :* p₂)
    (p₁  eq₁)  *-NF (p₂  eq₂)  = p₁ *-NF p₂                       eq₁   *-cong  eq₂
    (p₁  eq)   *-NF p₂          = p₁ *-NF p₂                       eq    *-cong  refl
    p₁          *-NF (p₂  eq)   = p₁ *-NF p₂                       refl  *-cong  eq
    con c₁      *-NF con c₂      = con (C._*_ c₁ c₂)                *-homo _ _
    p₁         *-NF p₂         = (p₁ *-NF p₂)                    refl
    (p₁ *x+ c₁) *-NF p₂         = (p₁ *-NF p₂ ) *x+ (c₁ *-NF p₂)  lemma₃ _ _ _ _
    p₁         *-NF (p₂ *x+ c₂) = (p₁  *-NF p₂) *x+ (p₁ *-NF c₂)  lemma₄ _ _ _ _
    (p₁ *x+ c₁) *-NF (p₂ *x+ c₂) =
      (p₁ *-NF p₂) *x *x +-NF
      (p₁ *-NF c₂  +-NF c₁ ↑-*-NF p₂) *x+ (c₁ *-NF c₂)             lemma₅ _ _ _ _ _

  -‿NF_ :  {n p}  Normal n p  Normal n (:- p)
  -‿NF (p  eq)  = -‿NF p  -‿cong eq
  -‿NF con c     = con (C.-_ c)  -‿homo _
  -‿NF (p )     = (-‿NF p) 
  -‿NF (p *x+ c) = -‿NF p *x+ -‿NF c  lemma₆ _ _ _

  var-NF :  {n}  (i : Fin n)  Normal n (var i)
  var-NF zero    = con-NF C.1# *x+ con-NF C.0#  lemma₇ _
  var-NF (suc i) = var-NF i 

  _^-NF_ :  {n p}  Normal n p  (i : )  Normal n (p :^ i)
  p ^-NF zero  = con-NF C.1#      1-homo
  p ^-NF suc n = p *-NF p ^-NF n  refl

  normaliseOp :  (o : Op) {n p₁ p₂} 
                Normal n p₁  Normal n p₂  Normal n (p₁  op o  p₂)
  normaliseOp [+] = _+-NF_
  normaliseOp [*] = _*-NF_

normalise :  {n} (p : Polynomial n)  Normal n p
normalise (op o p₁ p₂) = normalise p₁  normaliseOp o  normalise p₂
normalise (con c)      = con-NF c
normalise (var i)      = var-NF i
normalise (p :^ n)     = normalise p ^-NF n
normalise (:- p)       = -‿NF normalise p

⟦_⟧↓ :  {n}  Polynomial n  Vec Carrier n  Carrier
 p ⟧↓ ρ =  normalise p ⟧-Normal ρ

------------------------------------------------------------------------
-- Correctness

private
  sem-cong :  op  sem op Preserves₂ _≈_  _≈_  _≈_
  sem-cong [+] = +-cong
  sem-cong [*] = *-cong

  raise-sem :  {n x} (p : Polynomial n) ρ 
               p :↑ 1  (x  ρ)   p  ρ
  raise-sem (op o p₁ p₂) ρ = raise-sem p₁ ρ  sem-cong o 
                             raise-sem p₂ ρ
  raise-sem (con c)      ρ = refl
  raise-sem (var x)      ρ = refl
  raise-sem (p :^ n)     ρ = raise-sem p ρ  ^-cong 
                             PropEq.refl {x = n}
  raise-sem (:- p)       ρ = -‿cong (raise-sem p ρ)

  nf-sound :  {n p} (nf : Normal n p) ρ 
              nf ⟧-Normal ρ   p  ρ
  nf-sound (nf  eq)         ρ       = nf-sound nf ρ  trans  eq
  nf-sound (con c)           ρ       = refl
  nf-sound (_↑ {p' = p'} nf) (x  ρ) =
    nf-sound nf ρ  trans  sym (raise-sem p' ρ)
  nf-sound (_*x+_ {c' = c'} nf₁ nf₂) (x  ρ) =
    (nf-sound nf₁ (x  ρ)  *-cong  refl)
       +-cong 
    (nf-sound nf₂ ρ  trans  sym (raise-sem c' ρ))

-- Completeness can presumably also be proved (i.e. the normal forms
-- should be unique, if the casts are ignored).

------------------------------------------------------------------------
-- "Tactics"

open Reflection setoid var ⟦_⟧ ⟦_⟧↓ (nf-sound  normalise)
  public using (prove; solve) renaming (_⊜_ to _:=_)

-- For examples of how solve and _:=_ can be used to
-- semi-automatically prove ring equalities, see, for instance,
-- Data.Digit or Data.Nat.DivMod.