------------------------------------------------------------------------
-- The Agda standard library
--
-- Old 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 (in particular, our
-- Horner normal forms are not sparse).
--
-- At first the `WeaklyDecidable` type may at first glance look useless
-- as there is no guarantee that it doesn't always return `nothing`.
-- However the implementation of it affects the power of the solver. The
-- more equalities it returns, the more expressions the ring solver can
-- solve.

{-# OPTIONS --without-K --safe #-}

open import Algebra.Bundles
open import Algebra.Solver.Ring.AlmostCommutativeRing
open import Relation.Binary.Definitions using (WeaklyDecidable)

module Algebra.Solver.Ring
  {r₁ r₂ r₃ r₄}
  (Coeff : RawRing r₁ r₄)            -- Coefficient "ring".
  (R : AlmostCommutativeRing r₂ r₃)  -- Main "ring".
  (morphism : Coeff -Raw-AlmostCommutative⟶ R)
  (_coeff≟_ : WeaklyDecidable (Induced-equivalence morphism))
  where

open import Algebra.Core
open import Algebra.Solver.Ring.Lemmas Coeff R morphism
private module C = RawRing Coeff
open AlmostCommutativeRing R
  renaming (zero to *-zero; zeroˡ to *-zeroˡ; zeroʳ to *-zeroʳ)
open import Algebra.Definitions _≈_
open import Algebra.Morphism
open _-Raw-AlmostCommutative⟶_ morphism renaming (⟦_⟧ to ⟦_⟧′)
open import Algebra.Properties.Semiring.Exp semiring

open import Relation.Binary
open import Relation.Nullary using (yes; no)
open import Relation.Binary.Reasoning.Setoid setoid
import Relation.Binary.PropositionalEquality as PropEq
import Relation.Binary.Reflection as Reflection

open import Data.Nat.Base using (; suc; zero)
open import Data.Fin.Base using (Fin; zero; suc)
open import Data.Vec.Base using (Vec; []; _∷_; lookup)
open import Data.Maybe.Base using (just; nothing)
open import Function
open import Level using (_⊔_)

infix  9 :-_ -H_ -N_
infixr 9 _:×_ _:^_ _^N_
infix  8 _*x+_ _*x+HN_ _*x+H_
infixl 8 _:*_ _*N_ _*H_ _*NH_ _*HN_
infixl 7 _:+_ _:-_ _+H_ _+N_
infix  4 _≈H_ _≈N_

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

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

-- The polynomials are indexed by 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

_:×_ :  {n}    Polynomial n  Polynomial n
zero  p = con C.0#
suc m  p = p :+ m  p

-- 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 ρ x
 p :^ n      ρ =  p  ρ ^ n
 :- p        ρ = -  p  ρ

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

-- A univariate polynomial of degree d,
--
--     p = a_d x^d + a_{d-1}x^{d-1} + … + a_0,
--
-- is represented in Horner normal form by
--
--     p = ((a_d x + a_{d-1})x + …)x + a_0.
--
-- Note that Horner normal forms can be represented as lists, with the
-- empty list standing for the zero polynomial of degree "-1".
--
-- Given this representation of univariate polynomials over an
-- arbitrary ring, polynomials in any number of variables over the
-- ring C can be represented via the isomorphisms
--
--     C[] ≅ C
--
-- and
--
--     C[X_0,...X_{n+1}] ≅ C[X_0,...,X_n][X_{n+1}].

mutual

  -- The polynomial representations are indexed by the polynomial's
  -- degree.

  data HNF :   Set r₁ where
         :  {n}  HNF (suc n)
    _*x+_ :  {n}  HNF (suc n)  Normal n  HNF (suc n)

  data Normal :   Set r₁ where
    con  : C.Carrier  Normal zero
    poly :  {n}  HNF (suc n)  Normal (suc n)

  -- Note that the data types above do /not/ ensure uniqueness of
  -- normal forms: the zero polynomial of degree one can be
  -- represented using both ∅ and ∅ *x+ con C.0#.

mutual

  -- Semantics.

  ⟦_⟧H :  {n}  HNF (suc n)  Vec Carrier (suc n)  Carrier
          ⟧H _       = 0#
   p *x+ c ⟧H (x  ρ) =  p ⟧H (x  ρ) * x +  c ⟧N ρ

  ⟦_⟧N :  {n}  Normal n  Vec Carrier n  Carrier
   con c  ⟧N _ =  c ⟧′
   poly p ⟧N ρ =  p ⟧H ρ

------------------------------------------------------------------------
-- Equality and decidability

mutual

  -- Equality.

  data _≈H_ :  {n}  HNF n  HNF n  Set (r₁  r₃) where
         :  {n}  _≈H_ {suc n}  
    _*x+_ :  {n} {p₁ p₂ : HNF (suc n)} {c₁ c₂ : Normal n} 
            p₁ ≈H p₂  c₁ ≈N c₂  (p₁ *x+ c₁) ≈H (p₂ *x+ c₂)

  data _≈N_ :  {n}  Normal n  Normal n  Set (r₁  r₃) where
    con  :  {c₁ c₂}   c₁ ⟧′   c₂ ⟧′  con c₁ ≈N con c₂
    poly :  {n} {p₁ p₂ : HNF (suc n)}  p₁ ≈H p₂  poly p₁ ≈N poly p₂

mutual

  -- Equality is weakly decidable.

  _≟H_ :  {n}  WeaklyDecidable (_≈H_ {n = n})
             ≟H            = just 
             ≟H (_ *x+ _)   = nothing
  (_ *x+ _)   ≟H            = nothing
  (p₁ *x+ c₁) ≟H (p₂ *x+ c₂) with p₁ ≟H p₂ | c₁ ≟N c₂
  ... | just p₁≈p₂ | just c₁≈c₂ = just (p₁≈p₂ *x+ c₁≈c₂)
  ... | _          | nothing    = nothing
  ... | nothing    | _          = nothing

  _≟N_ :  {n}  WeaklyDecidable (_≈N_ {n = n})
  con c₁ ≟N con c₂ with c₁ coeff≟ c₂
  ... | just c₁≈c₂ = just (con c₁≈c₂)
  ... | nothing    = nothing
  poly p₁ ≟N poly p₂ with p₁ ≟H p₂
  ... | just p₁≈p₂ = just (poly p₁≈p₂)
  ... | nothing    = nothing

mutual

  -- The semantics respect the equality relations defined above.

  ⟦_⟧H-cong :  {n} {p₁ p₂ : HNF (suc n)} 
              p₁ ≈H p₂   ρ   p₁ ⟧H ρ   p₂ ⟧H ρ
                  ⟧H-cong _       = refl
   p₁≈p₂ *x+ c₁≈c₂ ⟧H-cong (x  ρ) =
    ( p₁≈p₂ ⟧H-cong (x  ρ)  *-cong  refl)
       +-cong 
     c₁≈c₂ ⟧N-cong ρ

  ⟦_⟧N-cong :
     {n} {p₁ p₂ : Normal n} 
    p₁ ≈N p₂   ρ   p₁ ⟧N ρ   p₂ ⟧N ρ
   con c₁≈c₂  ⟧N-cong _ = c₁≈c₂
   poly p₁≈p₂ ⟧N-cong ρ =  p₁≈p₂ ⟧H-cong ρ

------------------------------------------------------------------------
-- Ring operations on Horner normal forms

-- Zero.

0H :  {n}  HNF (suc n)
0H = 

0N :  {n}  Normal n
0N {zero}  = con C.0#
0N {suc n} = poly 0H

mutual

  -- One.

  1H :  {n}  HNF (suc n)
  1H {n} =  *x+ 1N {n}

  1N :  {n}  Normal n
  1N {zero}  = con C.1#
  1N {suc n} = poly 1H

-- A simplifying variant of _*x+_.

_*x+HN_ :  {n}  HNF (suc n)  Normal n  HNF (suc n)
(p *x+ c′) *x+HN c = (p *x+ c′) *x+ c
          *x+HN c with c ≟N 0N
... | just c≈0 = 
... | nothing  =  *x+ c

mutual

  -- Addition.

  _+H_ :  {n}  HNF (suc n)  HNF (suc n)  HNF (suc n)
             +H p           = p
  p           +H            = p
  (p₁ *x+ c₁) +H (p₂ *x+ c₂) = (p₁ +H p₂) *x+HN (c₁ +N c₂)

  _+N_ :  {n}  Normal n  Normal n  Normal n
  con c₁  +N con c₂  = con (c₁ C.+ c₂)
  poly p₁ +N poly p₂ = poly (p₁ +H p₂)

-- Multiplication.

_*x+H_ :  {n}  HNF (suc n)  HNF (suc n)  HNF (suc n)
p₁         *x+H (p₂ *x+ c) = (p₁ +H p₂) *x+HN c
          *x+H           = 
(p₁ *x+ c) *x+H           = (p₁ *x+ c) *x+ 0N

mutual

  _*NH_ :  {n}  Normal n  HNF (suc n)  HNF (suc n)
  c *NH           = 
  c *NH (p *x+ c′) with c ≟N 0N
  ... | just c≈0 = 
  ... | nothing  = (c *NH p) *x+ (c *N c′)

  _*HN_ :  {n}  HNF (suc n)  Normal n  HNF (suc n)
            *HN c = 
  (p *x+ c′) *HN c with c ≟N 0N
  ... | just c≈0 = 
  ... | nothing  = (p *HN c) *x+ (c′ *N c)

  _*H_ :  {n}  HNF (suc n)  HNF (suc n)  HNF (suc n)
             *H _           = 
  (_ *x+ _)   *H            = 
  (p₁ *x+ c₁) *H (p₂ *x+ c₂) =
    ((p₁ *H p₂) *x+H (p₁ *HN c₂ +H c₁ *NH p₂)) *x+HN (c₁ *N c₂)

  _*N_ :  {n}  Normal n  Normal n  Normal n
  con c₁  *N con c₂  = con (c₁ C.* c₂)
  poly p₁ *N poly p₂ = poly (p₁ *H p₂)

-- Exponentiation.

_^N_ :  {n}  Normal n    Normal n
p ^N zero  = 1N
p ^N suc n = p *N (p ^N n)

mutual

  -- Negation.

  -H_ :  {n}  HNF (suc n)  HNF (suc n)
  -H p = (-N 1N) *NH p

  -N_ :  {n}  Normal n  Normal n
  -N con c  = con (C.- c)
  -N poly p = poly (-H p)

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

normalise-con :  {n}  C.Carrier  Normal n
normalise-con {zero}  c = con c
normalise-con {suc n} c = poly ( *x+HN normalise-con c)

normalise-var :  {n}  Fin n  Normal n
normalise-var zero    = poly (( *x+ 1N) *x+ 0N)
normalise-var (suc i) = poly ( *x+HN normalise-var i)

normalise :  {n}  Polynomial n  Normal n
normalise (op [+] t₁ t₂) = normalise t₁ +N normalise t₂
normalise (op [*] t₁ t₂) = normalise t₁ *N normalise t₂
normalise (con c)        = normalise-con c
normalise (var i)        = normalise-var i
normalise (t :^ k)       = normalise t ^N k
normalise (:- t)         = -N normalise t

-- Evaluation after normalisation.

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

------------------------------------------------------------------------
-- Homomorphism lemmas

0N-homo :  {n} ρ   0N {n} ⟧N ρ  0#
0N-homo []      = 0-homo
0N-homo (x  ρ) = refl

-- If c is equal to 0N, then c is semantically equal to 0#.

0≈⟦0⟧ :  {n} {c : Normal n}  c ≈N 0N   ρ  0#   c ⟧N ρ
0≈⟦0⟧ {c = c} c≈0 ρ = sym (begin
   c  ⟧N ρ  ≈⟨  c≈0 ⟧N-cong ρ 
   0N ⟧N ρ  ≈⟨ 0N-homo ρ 
  0#         )

1N-homo :  {n} ρ   1N {n} ⟧N ρ  1#
1N-homo []      = 1-homo
1N-homo (x  ρ) = begin
  0# * x +  1N ⟧N ρ  ≈⟨ refl  +-cong  1N-homo ρ 
  0# * x + 1#         ≈⟨ lemma₆ _ _ 
  1#                  

-- _*x+HN_ is equal to _*x+_.

*x+HN≈*x+ :  {n} (p : HNF (suc n)) (c : Normal n) 
             ρ   p *x+HN c ⟧H ρ   p *x+ c ⟧H ρ
*x+HN≈*x+ (p *x+ c′) c ρ       = refl
*x+HN≈*x+           c (x  ρ) with c ≟N 0N
... | just c≈0 = begin
  0#                 ≈⟨ 0≈⟦0⟧ c≈0 ρ 
   c ⟧N ρ           ≈⟨ sym $ lemma₆ _ _ 
  0# * x +  c ⟧N ρ  
... | nothing = refl

∅*x+HN-homo :  {n} (c : Normal n) x ρ 
                *x+HN c ⟧H (x  ρ)   c ⟧N ρ
∅*x+HN-homo c x ρ with c ≟N 0N
... | just c≈0 = 0≈⟦0⟧ c≈0 ρ
... | nothing = lemma₆ _ _

mutual

  +H-homo :  {n} (p₁ p₂ : HNF (suc n)) 
             ρ   p₁ +H p₂ ⟧H ρ   p₁ ⟧H ρ +  p₂ ⟧H ρ
  +H-homo            p₂          ρ       = sym (+-identityˡ _)
  +H-homo (p₁ *x+ x₁)            ρ       = sym (+-identityʳ _)
  +H-homo (p₁ *x+ c₁) (p₂ *x+ c₂) (x  ρ) = begin
     (p₁ +H p₂) *x+HN (c₁ +N c₂) ⟧H (x  ρ)                           ≈⟨ *x+HN≈*x+ (p₁ +H p₂) (c₁ +N c₂) (x  ρ) 

     p₁ +H p₂ ⟧H (x  ρ) * x +  c₁ +N c₂ ⟧N ρ                        ≈⟨ (+H-homo p₁ p₂ (x  ρ)  *-cong  refl)  +-cong  +N-homo c₁ c₂ ρ 

    ( p₁ ⟧H (x  ρ) +  p₂ ⟧H (x  ρ)) * x + ( c₁ ⟧N ρ +  c₂ ⟧N ρ)  ≈⟨ lemma₁ _ _ _ _ _ 

    ( p₁ ⟧H (x  ρ) * x +  c₁ ⟧N ρ) +
    ( p₂ ⟧H (x  ρ) * x +  c₂ ⟧N ρ)                                  

  +N-homo :  {n} (p₁ p₂ : Normal n) 
             ρ   p₁ +N p₂ ⟧N ρ   p₁ ⟧N ρ +  p₂ ⟧N ρ
  +N-homo (con c₁)  (con c₂)  _ = +-homo _ _
  +N-homo (poly p₁) (poly p₂) ρ = +H-homo p₁ p₂ ρ

*x+H-homo :
   {n} (p₁ p₂ : HNF (suc n)) x ρ 
   p₁ *x+H p₂ ⟧H (x  ρ) 
   p₁ ⟧H (x  ρ) * x +  p₂ ⟧H (x  ρ)
*x+H-homo           _ _ = sym $ lemma₆ _ _
*x+H-homo (p *x+ c)  x ρ = begin
   p *x+ c ⟧H (x  ρ) * x +  0N ⟧N ρ  ≈⟨ refl  +-cong  0N-homo ρ 
   p *x+ c ⟧H (x  ρ) * x + 0#         
*x+H-homo p₁         (p₂ *x+ c₂) x ρ = begin
   (p₁ +H p₂) *x+HN c₂ ⟧H (x  ρ)                         ≈⟨ *x+HN≈*x+ (p₁ +H p₂) c₂ (x  ρ) 
   p₁ +H p₂ ⟧H (x  ρ) * x +  c₂ ⟧N ρ                    ≈⟨ (+H-homo p₁ p₂ (x  ρ)  *-cong  refl)  +-cong  refl 
  ( p₁ ⟧H (x  ρ) +  p₂ ⟧H (x  ρ)) * x +  c₂ ⟧N ρ      ≈⟨ lemma₀ _ _ _ _ 
   p₁ ⟧H (x  ρ) * x + ( p₂ ⟧H (x  ρ) * x +  c₂ ⟧N ρ)  

mutual

  *NH-homo :
     {n} (c : Normal n) (p : HNF (suc n)) x ρ 
     c *NH p ⟧H (x  ρ)   c ⟧N ρ *  p ⟧H (x  ρ)
  *NH-homo c           x ρ = sym (*-zeroʳ _)
  *NH-homo c (p *x+ c′) x ρ with c ≟N 0N
  ... | just c≈0 = begin
    0#                                            ≈⟨ sym (*-zeroˡ _) 
    0# * ( p ⟧H (x  ρ) * x +  c′ ⟧N ρ)         ≈⟨ 0≈⟦0⟧ c≈0 ρ  *-cong  refl 
     c ⟧N ρ  * ( p ⟧H (x  ρ) * x +  c′ ⟧N ρ)  
  ... | nothing = begin
     c *NH p ⟧H (x  ρ) * x +  c *N c′ ⟧N ρ                 ≈⟨ (*NH-homo c p x ρ  *-cong  refl)  +-cong  *N-homo c c′ ρ 
    ( c ⟧N ρ *  p ⟧H (x  ρ)) * x + ( c ⟧N ρ *  c′ ⟧N ρ)  ≈⟨ lemma₃ _ _ _ _ 
     c ⟧N ρ * ( p ⟧H (x  ρ) * x +  c′ ⟧N ρ)               

  *HN-homo :
     {n} (p : HNF (suc n)) (c : Normal n) x ρ 
     p *HN c ⟧H (x  ρ)   p ⟧H (x  ρ) *  c ⟧N ρ
  *HN-homo           c x ρ = sym (*-zeroˡ _)
  *HN-homo (p *x+ c′) c x ρ with c ≟N 0N
  ... | just c≈0 = begin
    0#                                           ≈⟨ sym (*-zeroʳ _) 
    ( p ⟧H (x  ρ) * x +  c′ ⟧N ρ) * 0#        ≈⟨ refl  *-cong  0≈⟦0⟧ c≈0 ρ 
    ( p ⟧H (x  ρ) * x +  c′ ⟧N ρ) *  c ⟧N ρ  
  ... | nothing = begin
     p *HN c ⟧H (x  ρ) * x +  c′ *N c ⟧N ρ                 ≈⟨ (*HN-homo p c x ρ  *-cong  refl)  +-cong  *N-homo c′ c ρ 
    ( p ⟧H (x  ρ) *  c ⟧N ρ) * x + ( c′ ⟧N ρ *  c ⟧N ρ)  ≈⟨ lemma₂ _ _ _ _ 
    ( p ⟧H (x  ρ) * x +  c′ ⟧N ρ) *  c ⟧N ρ               

  *H-homo :  {n} (p₁ p₂ : HNF (suc n)) 
             ρ   p₁ *H p₂ ⟧H ρ   p₁ ⟧H ρ *  p₂ ⟧H ρ
  *H-homo            p₂          ρ       = sym $ *-zeroˡ _
  *H-homo (p₁ *x+ c₁)            ρ       = sym $ *-zeroʳ _
  *H-homo (p₁ *x+ c₁) (p₂ *x+ c₂) (x  ρ) = begin
     ((p₁ *H p₂) *x+H ((p₁ *HN c₂) +H (c₁ *NH p₂))) *x+HN
      (c₁ *N c₂) ⟧H (x  ρ)                                              ≈⟨ *x+HN≈*x+ ((p₁ *H p₂) *x+H ((p₁ *HN c₂) +H (c₁ *NH p₂)))
                                                                                      (c₁ *N c₂) (x  ρ) 
     (p₁ *H p₂) *x+H
      ((p₁ *HN c₂) +H (c₁ *NH p₂)) ⟧H (x  ρ) * x +
     c₁ *N c₂ ⟧N ρ                                                      ≈⟨ (*x+H-homo (p₁ *H p₂) ((p₁ *HN c₂) +H (c₁ *NH p₂)) x ρ
                                                                                *-cong 
                                                                             refl)
                                                                               +-cong 
                                                                            *N-homo c₁ c₂ ρ 
    ( p₁ *H p₂ ⟧H (x  ρ) * x +
      (p₁ *HN c₂) +H (c₁ *NH p₂) ⟧H (x  ρ)) * x +
     c₁ ⟧N ρ *  c₂ ⟧N ρ                                                ≈⟨ (((*H-homo p₁ p₂ (x  ρ)  *-cong  refl)
                                                                                 +-cong 
                                                                              (+H-homo (p₁ *HN c₂) (c₁ *NH p₂) (x  ρ)))
                                                                                *-cong 
                                                                             refl)
                                                                               +-cong 
                                                                            refl 
    ( p₁ ⟧H (x  ρ) *  p₂ ⟧H (x  ρ) * x +
     ( p₁ *HN c₂ ⟧H (x  ρ) +  c₁ *NH p₂ ⟧H (x  ρ))) * x +
     c₁ ⟧N ρ *  c₂ ⟧N ρ                                                ≈⟨ ((refl  +-cong  (*HN-homo p₁ c₂ x ρ  +-cong  *NH-homo c₁ p₂ x ρ))
                                                                                *-cong 
                                                                             refl)
                                                                               +-cong 
                                                                            refl 
    ( p₁ ⟧H (x  ρ) *  p₂ ⟧H (x  ρ) * x +
     ( p₁ ⟧H (x  ρ) *  c₂ ⟧N ρ +  c₁ ⟧N ρ *  p₂ ⟧H (x  ρ))) * x +
    ( c₁ ⟧N ρ *  c₂ ⟧N ρ)                                              ≈⟨ lemma₄ _ _ _ _ _ 

    ( p₁ ⟧H (x  ρ) * x +  c₁ ⟧N ρ) *
    ( p₂ ⟧H (x  ρ) * x +  c₂ ⟧N ρ)                                    

  *N-homo :  {n} (p₁ p₂ : Normal n) 
             ρ   p₁ *N p₂ ⟧N ρ   p₁ ⟧N ρ *  p₂ ⟧N ρ
  *N-homo (con c₁)  (con c₂)  _ = *-homo _ _
  *N-homo (poly p₁) (poly p₂) ρ = *H-homo p₁ p₂ ρ

^N-homo :  {n} (p : Normal n) (k : ) 
           ρ   p ^N k ⟧N ρ   p ⟧N ρ ^ k
^N-homo p zero    ρ = 1N-homo ρ
^N-homo p (suc k) ρ = begin
   p *N (p ^N k) ⟧N ρ       ≈⟨ *N-homo p (p ^N k) ρ 
   p ⟧N ρ *  p ^N k ⟧N ρ   ≈⟨ refl  *-cong  ^N-homo p k ρ 
   p ⟧N ρ * ( p ⟧N ρ ^ k)  

mutual

  -H‿-homo :  {n} (p : HNF (suc n)) 
              ρ   -H p ⟧H ρ  -  p ⟧H ρ
  -H‿-homo p (x  ρ) = begin
     (-N 1N) *NH p ⟧H (x  ρ)     ≈⟨ *NH-homo (-N 1N) p x ρ 
     -N 1N ⟧N ρ *  p ⟧H (x  ρ)  ≈⟨ trans (-N‿-homo 1N ρ) (-‿cong (1N-homo ρ))  *-cong  refl 
    - 1# *  p ⟧H (x  ρ)          ≈⟨ lemma₇ _ 
    -  p ⟧H (x  ρ)               

  -N‿-homo :  {n} (p : Normal n) 
              ρ   -N p ⟧N ρ  -  p ⟧N ρ
  -N‿-homo (con c)  _ = -‿homo _
  -N‿-homo (poly p) ρ = -H‿-homo p ρ

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

correct-con :  {n} (c : C.Carrier) (ρ : Vec Carrier n) 
               normalise-con c ⟧N ρ   c ⟧′
correct-con c []      = refl
correct-con c (x  ρ) = begin
    *x+HN normalise-con c ⟧H (x  ρ)  ≈⟨ ∅*x+HN-homo (normalise-con c) x ρ 
   normalise-con c ⟧N ρ            ≈⟨ correct-con c ρ 
   c ⟧′                                   

correct-var :  {n} (i : Fin n) 
               ρ   normalise-var i ⟧N ρ  lookup ρ i
correct-var (suc i) (x  ρ) = begin
    *x+HN normalise-var i ⟧H (x  ρ)  ≈⟨ ∅*x+HN-homo (normalise-var i) x ρ 
   normalise-var i ⟧N ρ                ≈⟨ correct-var i ρ 
  lookup ρ i                            
correct-var zero (x  ρ) = begin
  (0# * x +  1N ⟧N ρ) * x +  0N ⟧N ρ  ≈⟨ ((refl  +-cong  1N-homo ρ)  *-cong  refl)  +-cong  0N-homo ρ 
  (0# * x + 1#) * x + 0#                ≈⟨ lemma₅ _ 
  x                                     

correct :  {n} (p : Polynomial n)   ρ   p ⟧↓ ρ   p  ρ
correct (op [+] p₁ p₂) ρ = begin
   normalise p₁ +N normalise p₂ ⟧N ρ  ≈⟨ +N-homo (normalise p₁) (normalise p₂) ρ 
   p₁ ⟧↓ ρ +  p₂ ⟧↓ ρ                ≈⟨ correct p₁ ρ  +-cong  correct p₂ ρ 
   p₁  ρ +  p₂  ρ                  
correct (op [*] p₁ p₂) ρ = begin
   normalise p₁ *N normalise p₂ ⟧N ρ  ≈⟨ *N-homo (normalise p₁) (normalise p₂) ρ 
   p₁ ⟧↓ ρ *  p₂ ⟧↓ ρ                ≈⟨ correct p₁ ρ  *-cong  correct p₂ ρ 
   p₁  ρ *  p₂  ρ                  
correct (con c)  ρ = correct-con c ρ
correct (var i)  ρ = correct-var i ρ
correct (p :^ k) ρ = begin
   normalise p ^N k ⟧N ρ  ≈⟨ ^N-homo (normalise p) k ρ 
   p ⟧↓ ρ ^ k             ≈⟨ correct p ρ  ^-cong  PropEq.refl {x = k} 
   p  ρ ^ k              
correct (:- p) ρ = begin
   -N normalise p ⟧N ρ  ≈⟨ -N‿-homo (normalise p) ρ 
  -  p ⟧↓ ρ             ≈⟨ -‿cong (correct p ρ) 
  -  p  ρ              

------------------------------------------------------------------------
-- "Tactic.

open Reflection setoid var ⟦_⟧ ⟦_⟧↓ correct 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.