------------------------------------------------------------------------
-- Some decision procedures for equality
------------------------------------------------------------------------

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

open import Equality

module Equality.Decision-procedures
  {reflexive} (eq :  {a p}  Equality-with-J a p reflexive) where

open Derived-definitions-and-properties eq
open import Equality.Decidable-UIP eq
open import Prelude
  hiding ( module; module; module Bool; module; module Σ
         ; module List
         )

------------------------------------------------------------------------
-- The unit type

module  where

  -- Equality of values of the unit type is decidable.

  _≟_ : Decidable-equality 
  _  _ = inj₁ (refl _)

------------------------------------------------------------------------
-- The empty type

module  {} where

  -- Equality of values of the empty type is decidable.

  _≟_ : Decidable-equality ( { = })
  ()  ()

------------------------------------------------------------------------
-- Booleans

module Bool where

  -- The values true and false are distinct.

  abstract

    true≢false : true  false
    true≢false true≡false = subst T true≡false _

  -- Equality of booleans is decidable.

  _≟_ : Decidable-equality Bool
  true   true  = inj₁ (refl _)
  false  false = inj₁ (refl _)
  true   false = inj₂ true≢false
  false  true  = inj₂ (true≢false  sym)

------------------------------------------------------------------------
-- Natural numbers

module  where

  -- Inhabited only for zero.

  Zero :   Set
  Zero zero    = 
  Zero (suc n) = 

  -- Predecessor (except if the argument is zero).

  pred :   
  pred zero    = zero
  pred (suc n) = n

  abstract

    -- Zero is not equal to the successor of any number.

    0≢+ : {n : }  zero  suc n
    0≢+ 0≡+ = subst Zero 0≡+ tt

  -- The suc constructor is cancellative.

  cancel-suc : {m n : }  suc m  suc n  m  n
  cancel-suc = cong pred

  -- Equality of natural numbers is decidable.

  _≟_ : Decidable-equality 
  zero   zero  = inj₁ (refl _)
  suc m  suc n = ⊎-map (cong suc)  m≢n  m≢n  cancel-suc) (m  n)
  zero   suc n = inj₂ 0≢+
  suc m  zero  = inj₂ (0≢+  sym)

------------------------------------------------------------------------
-- Σ-types

module Σ {a b} {A : Set a} {B : A  Set b} where

  -- Σ preserves decidability of equality.

  module Dec (_≟A_ : Decidable-equality A)
             (_≟B_ : {x : A}  Decidable-equality (B x)) where

    _≟_ : Decidable-equality (Σ A B)
    (x₁ , y₁)  (x₂ , y₂) with x₁ ≟A x₂
    ... | inj₂ x₁≢x₂ = inj₂ (x₁≢x₂  cong proj₁)
    ... | inj₁ x₁≡x₂ with subst B x₁≡x₂ y₁ ≟B y₂
    ...   | inj₁ cast-y₁≡y₂ = inj₁ (Σ-≡,≡→≡ x₁≡x₂ cast-y₁≡y₂)
    ...   | inj₂ cast-y₁≢y₂ =
      inj₂ (cast-y₁≢y₂ 
            subst  p  subst B p y₁  y₂) (decidable⇒UIP _≟A_ _ _) 
            proj₂  Σ-≡,≡←≡)

------------------------------------------------------------------------
-- Binary products

module × {a b} {A : Set a} {B : Set b} where

  -- _×_ preserves decidability of equality.

  module Dec (_≟A_ : Decidable-equality A)
             (_≟B_ : Decidable-equality B) where

    _≟_ : Decidable-equality (A × B)
    _≟_ = Σ.Dec._≟_ _≟A_ _≟B_

------------------------------------------------------------------------
-- Binary sums

module  {a b} {A : Set a} {B : Set b} where

  abstract

    -- The values inj₁ x and inj₂ y are never equal.

    inj₁≢inj₂ : {x : A} {y : B}  inj₁ x  inj₂ y
    inj₁≢inj₂ = Bool.true≢false 
                cong {A = A  B} {B = Bool} [ const true , const false ]

  -- The inj₁ and inj₂ constructors are cancellative.

  cancel-inj₁ : {x y : A}  _≡_ {A = A  B} (inj₁ x) (inj₁ y)  x  y
  cancel-inj₁ {x = x} = cong {A = A  B} {B = A} [ id , const x ]

  cancel-inj₂ : {x y : B}  _≡_ {A = A  B} (inj₂ x) (inj₂ y)  x  y
  cancel-inj₂ {x = x} = cong {A = A  B} {B = B} [ const x , id ]

  -- _⊎_ preserves decidability of equality.

  module Dec (_≟A_ : Decidable-equality A)
             (_≟B_ : Decidable-equality B) where

    _≟_ : Decidable-equality (A  B)
    inj₁ x  inj₁ y = ⊎-map (cong (inj₁ {B = B}))  x≢y  x≢y  cancel-inj₁) (x ≟A y)
    inj₂ x  inj₂ y = ⊎-map (cong (inj₂ {A = A}))  x≢y  x≢y  cancel-inj₂) (x ≟B y)
    inj₁ x  inj₂ y = inj₂ inj₁≢inj₂
    inj₂ x  inj₁ y = inj₂ (inj₁≢inj₂  sym)

------------------------------------------------------------------------
-- Lists

module List {a} {A : Set a} where

  abstract

    -- The values [] and x ∷ xs are never equal.

    []≢∷ :  {x : A} {xs}  []  x  xs
    []≢∷ = Bool.true≢false 
             cong  { []  true; (_  _)  false })

  -- The _∷_ constructor is cancellative in both arguments.

  private

    head? : A  List A  A
    head? x []      = x
    head? _ (x  _) = x

    tail? : List A  List A
    tail? []       = []
    tail? (_  xs) = xs

  cancel-∷-head :  {x y : A} {xs ys}  x  xs  y  ys  x  y
  cancel-∷-head {x} = cong (head? x)

  cancel-∷-tail :  {x y : A} {xs ys}  x  xs  y  ys  xs  ys
  cancel-∷-tail = cong tail?

  abstract

    -- An η-like result for the cancellation lemmas.

    unfold-∷ :  {x y : A} {xs ys} (eq : x  xs  y  ys) 
               eq  cong₂ _∷_ (cancel-∷-head eq) (cancel-∷-tail eq)
    unfold-∷ {x} eq =
      eq                                             ≡⟨ sym $ trans-reflʳ eq 
      trans eq (refl _)                              ≡⟨ sym $ cong (trans eq) sym-refl 
      trans eq (sym (refl _))                        ≡⟨ sym $ trans-reflˡ _ 
      trans (refl _) (trans eq (sym (refl _)))       ≡⟨ sym $ cong-roughly-id  xs  head? x xs  tail? xs)
                                                                              non-empty eq tt tt lemma₁ 
      cong  xs  head? x xs  tail? xs) eq         ≡⟨ lemma₂ _∷_ (head? x) tail? eq ⟩∎
      cong₂ _∷_ (cong (head? x) eq) (cong tail? eq)  
      where
      non-empty : List A  Bool
      non-empty []      = false
      non-empty (_  _) = true

      lemma₁ : (xs : List A) 
               if non-empty xs then  else  
               head? x xs  tail? xs  xs
      lemma₁ []      ()
      lemma₁ (_  _) _ = refl _

      lemma₂ : {A B C D : Set a} {x y : A}
               (f : B  C  D) (g : A  B) (h : A  C) 
               (eq : x  y) 
               cong  x  f (g x) (h x)) eq 
               cong₂ f (cong g eq) (cong h eq)
      lemma₂ f g h =
        elim  eq  cong  x  f (g x) (h x)) eq 
                     cong₂ f (cong g eq) (cong h eq))
              x  cong  x  f (g x) (h x)) (refl x)          ≡⟨ cong-refl  x  f (g x) (h x)) 
                    refl (f (g x) (h x))                         ≡⟨ sym $ cong₂-refl f 
                    cong₂ f (refl (g x)) (refl (h x))            ≡⟨ sym $ cong₂ (cong₂ f) (cong-refl g) (cong-refl h) ⟩∎
                    cong₂ f (cong g (refl x)) (cong h (refl x))  )

  -- List preserves decidability of equality.

  module Dec (_≟A_ : Decidable-equality A) where

    _≟_ : Decidable-equality (List A)
    []        []       = inj₁ (refl [])
    []        (_  _)  = inj₂ []≢∷
    (_  _)   []       = inj₂ ([]≢∷  sym)
    (x  xs)  (y  ys) with x ≟A y
    ... | inj₂ x≢y = inj₂ (x≢y  cancel-∷-head)
    ... | inj₁ x≡y with xs  ys
    ...   | inj₁ xs≡ys = inj₁ (cong₂ _∷_ x≡y xs≡ys)
    ...   | inj₂ xs≢ys = inj₂ (xs≢ys  cancel-∷-tail)

------------------------------------------------------------------------
-- Finite sets

module Fin where

  -- Equality of finite numbers is decidable.

  _≟_ :  {n}  Decidable-equality (Fin n)
  _≟_ {zero}  = λ ()
  _≟_ {suc n} = ⊎.Dec._≟_  _ _  inj₁ (refl tt)) (_≟_ {n})