------------------------------------------------------------------------
-- A simple tactic for proving equality of equality proofs
------------------------------------------------------------------------

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

open import Equality

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

open Derived-definitions-and-properties eq
open import Prelude hiding (Level; lift; lower)

------------------------------------------------------------------------
-- Equality expressions

-- Equality expressions.
--
-- Note that the presence of the Refl constructor means that Eq is a
-- definition of equality with a concrete, evaluating eliminator.

data Eq {a} {A : Set a} : A  A  Set (lsuc a) where
  Lift  :  {x y} (x≡y : x  y)  Eq x y
  Refl  :  {x}  Eq x x
  Sym   :  {x y} (x≈y : Eq x y)  Eq y x
  Trans :  {x y z} (x≈y : Eq x y) (y≈z : Eq y z)  Eq x z
  Cong  :  {B : Set a} {x y}
          (f : B  A) (x≈y : Eq x y)  Eq (f x) (f y)

-- Semantics.

⟦_⟧ :  {a} {A : Set a} {x y : A}  Eq x y  x  y
 Lift x≡y       = x≡y
 Refl           = refl _
 Sym x≈y        = sym  x≈y 
 Trans x≈y y≈z  = trans  x≈y   y≈z 
 Cong f x≈y     = cong f  x≈y 

-- A derived combinator.

Cong₂ :  {a} {A B C : Set a} (f : A  B  C) {x y : A} {u v : B} 
        Eq x y  Eq u v  Eq (f x u) (f y v)
Cong₂ f {y = y} {u} x≈y u≈v =
  Trans (Cong (flip f u) x≈y) (Cong (f y) u≈v)

private

  Cong₂-correct :
     {a} {A B C : Set a} (f : A  B  C) {x y : A} {u v : B}
    (x≈y : Eq x y) (u≈v : Eq u v) 
     Cong₂ f x≈y u≈v   cong₂ f  x≈y   u≈v 
  Cong₂-correct f x≈y u≈v = refl _

------------------------------------------------------------------------
-- Simplified expressions

private

  -- The simplified expressions are stratified into three levels.

  data Level : Set where
    upper middle lower : Level

  data EqS {a} {A : Set a} : Level  A  A  Set (lsuc a) where

    -- Bottom layer: a single use of congruence applied to an actual
    -- equality.

    Cong : {B : Set a} {x y : B} (f : B  A) (x≡y : x  y) 
           EqS lower (f x) (f y)

    -- Middle layer: at most one use of symmetry.

    No-Sym :  {x y} (x≈y : EqS lower x y)  EqS middle x y
    Sym    :  {x y} (x≈y : EqS lower x y)  EqS middle y x

    -- Uppermost layer: a sequence of equalities, combined using
    -- transitivity and a single use of reflexivity.

    Refl :  {x}  EqS upper x x
    Cons :  {x y z} (x≈y : EqS middle x y) (y≈z : EqS upper y z) 
           EqS upper x z

  -- Semantics of simplified expressions.

  ⟦_⟧S :  { a} {A : Set a} {x y : A}  EqS  x y  x  y
   Cong f x≡y   ⟧S = cong f x≡y
   No-Sym x≈y   ⟧S =      x≈y ⟧S
   Sym    x≈y   ⟧S = sym  x≈y ⟧S
   Refl         ⟧S = refl _
   Cons x≈y y≈z ⟧S = trans  x≈y ⟧S  y≈z ⟧S

------------------------------------------------------------------------
-- Manipulation of expressions

private

  lift :  {a} {A : Set a} {x y : A} 
         x  y  EqS upper x y
  lift x≡y = Cons (No-Sym (Cong id x≡y)) Refl

  abstract

    lift-correct :  {a} {A : Set a} {x y : A}
                   (x≡y : x  y)  x≡y   lift x≡y ⟧S
    lift-correct x≡y =
      x≡y                           ≡⟨ cong-id _ 
      cong id x≡y                   ≡⟨ sym (trans-reflʳ _) ⟩∎
      trans (cong id x≡y) (refl _)  

  snoc :  {a} {A : Set a} {x y z : A} 
         EqS upper x y  EqS middle y z  EqS upper x z
  snoc Refl           y≈z = Cons y≈z Refl
  snoc (Cons x≈y y≈z) z≈u = Cons x≈y (snoc y≈z z≈u)

  abstract

    snoc-correct :
       {a} {A : Set a} {x y z : A} {x≡y y≡z}
      (z≈y : EqS upper z y) (y≈x : EqS middle y x) 
      sym y≡z   z≈y ⟧S  sym x≡y   y≈x ⟧S 
      sym (trans x≡y y≡z)   snoc z≈y y≈x ⟧S
    snoc-correct {x≡y = x≡y} {y≡z} Refl y≈z h₁ h₂ =
      sym (trans x≡y y≡z)        ≡⟨ sym-trans _ _ 
      trans (sym y≡z) (sym x≡y)  ≡⟨ cong₂ trans h₁ h₂ 
      trans (refl _)  y≈z ⟧S    ≡⟨ trans-reflˡ _ 
       y≈z ⟧S                   ≡⟨ sym (trans-reflʳ _) ⟩∎
      trans  y≈z ⟧S (refl _)    
    snoc-correct {x≡y = x≡y} {y≡z} (Cons x≈y y≈z) z≈u h₁ h₂ =
      sym (trans x≡y y≡z)                                    ≡⟨ sym-trans _ _ 
      trans (sym y≡z) (sym x≡y)                              ≡⟨ cong₂ trans h₁ (refl (sym x≡y)) 
      trans (trans  x≈y ⟧S  y≈z ⟧S) (sym x≡y)              ≡⟨ trans-assoc _ _ _ 
      trans  x≈y ⟧S (trans  y≈z ⟧S (sym x≡y))              ≡⟨ cong (trans  x≈y ⟧S) $
                                                                  cong₂ trans (sym (sym-sym  y≈z ⟧S)) (refl (sym x≡y)) 
      trans  x≈y ⟧S (trans (sym (sym  y≈z ⟧S)) (sym x≡y))  ≡⟨ cong (trans _) $ sym (sym-trans x≡y (sym  y≈z ⟧S)) 
      trans  x≈y ⟧S (sym (trans x≡y (sym  y≈z ⟧S)))        ≡⟨ cong (trans _) $ snoc-correct y≈z z≈u (sym-sym _) h₂ ⟩∎
      trans  x≈y ⟧S  snoc y≈z z≈u ⟧S                       

  append :  {a} {A : Set a} {x y z : A} 
           EqS upper x y  EqS upper y z  EqS upper x z
  append Refl           x≈y = x≈y
  append (Cons x≈y y≈z) z≈u = Cons x≈y (append y≈z z≈u)

  abstract

    append-correct :
       {a} {A : Set a} {x y z : A} {x≡y y≡z}
      (x≈y : EqS upper x y) (y≈z : EqS upper y z) 
      x≡y   x≈y ⟧S  y≡z   y≈z ⟧S 
      trans x≡y y≡z   append x≈y y≈z ⟧S
    append-correct {x≡y = x≡y} {y≡z} Refl x≈y h₁ h₂ =
      trans x≡y y≡z            ≡⟨ cong₂ trans h₁ h₂ 
      trans (refl _)  x≈y ⟧S  ≡⟨ trans-reflˡ _ ⟩∎
       x≈y ⟧S                 
    append-correct {x≡y = x≡z} {z≡u} (Cons x≈y y≈z) z≈u h₁ h₂ =
      trans x≡z z≡u                        ≡⟨ cong₂ trans h₁ (refl z≡u) 
      trans (trans  x≈y ⟧S  y≈z ⟧S) z≡u  ≡⟨ trans-assoc _ _ _ 
      trans  x≈y ⟧S (trans  y≈z ⟧S z≡u)  ≡⟨ cong (trans _) $ append-correct y≈z z≈u (refl _) h₂ ⟩∎
      trans  x≈y ⟧S  append y≈z z≈u ⟧S   

  map-sym :  {a} {A : Set a} {x y : A} 
            EqS middle x y  EqS middle y x
  map-sym (No-Sym x≈y) = Sym    x≈y
  map-sym (Sym    x≈y) = No-Sym x≈y

  abstract

    map-sym-correct :
       {a} {A : Set a} {x y : A} {x≡y}
      (x≈y : EqS middle x y) 
      x≡y   x≈y ⟧S  sym x≡y   map-sym x≈y ⟧S
    map-sym-correct {x≡y = x≡y} (No-Sym x≈y) h =
      sym x≡y       ≡⟨ cong sym h ⟩∎
      sym  x≈y ⟧S  
    map-sym-correct {x≡y = x≡y} (Sym x≈y) h =
      sym x≡y             ≡⟨ cong sym h 
      sym (sym  x≈y ⟧S)  ≡⟨ sym-sym _ ⟩∎
       x≈y ⟧S            

  reverse :  {a} {A : Set a} {x y : A} 
            EqS upper x y  EqS upper y x
  reverse Refl           = Refl
  reverse (Cons x≈y y≈z) = snoc (reverse y≈z) (map-sym x≈y)

  abstract

    reverse-correct :
       {a} {A : Set a} {x y : A} {x≡y}
      (x≈y : EqS upper x y) 
      x≡y   x≈y ⟧S  sym x≡y   reverse x≈y ⟧S
    reverse-correct {x≡y = x≡y} Refl h =
      sym x≡y       ≡⟨ cong sym h 
      sym (refl _)  ≡⟨ sym-refl ⟩∎
      refl _        
    reverse-correct {x≡y = x≡y} (Cons x≈y y≈z) h =
      sym x≡y                                ≡⟨ cong sym h 
      sym (trans  x≈y ⟧S  y≈z ⟧S)          ≡⟨ snoc-correct (reverse y≈z) _
                                                             (reverse-correct y≈z (refl _))
                                                             (map-sym-correct x≈y (refl _)) ⟩∎
       snoc (reverse y≈z) (map-sym x≈y) ⟧S  

  map-cong :  { a} {A B : Set a} {x y : A}
             (f : A  B) 
             EqS  x y  EqS  (f x) (f y)
  map-cong {lower}  f (Cong g x≡y)   = Cong (f  g) x≡y
  map-cong {middle} f (No-Sym x≈y)   = No-Sym (map-cong f x≈y)
  map-cong {middle} f (Sym    y≈x)   = Sym (map-cong f y≈x)
  map-cong {upper}  f Refl           = Refl
  map-cong {upper}  f (Cons x≈y y≈z) =
    Cons (map-cong f x≈y) (map-cong f y≈z)

  abstract

    map-cong-correct :
       { a} {A B : Set a} {x y : A} (f : A  B) {x≡y}
      (x≈y : EqS  x y) 
      x≡y   x≈y ⟧S  cong f x≡y   map-cong f x≈y ⟧S
    map-cong-correct {lower}  f {gx≡gy} (Cong g x≡y)   h = cong f gx≡gy         ≡⟨ cong (cong f) h 
                                                           cong f (cong g x≡y)  ≡⟨ cong-∘ f g _ ⟩∎
                                                           cong (f  g) x≡y     
    map-cong-correct {middle} f {x≡y}   (No-Sym x≈y)   h = cong f x≡y           ≡⟨ map-cong-correct f x≈y h ⟩∎
                                                            map-cong f x≈y ⟧S  
    map-cong-correct {middle} f {x≡y}   (Sym    y≈x)   h = cong f x≡y               ≡⟨ cong (cong f) h 
                                                           cong f (sym  y≈x ⟧S)     cong-sym f _ 
                                                           sym (cong f  y≈x ⟧S)    ≡⟨ cong sym (map-cong-correct f y≈x (refl _)) ⟩∎
                                                           sym  map-cong f y≈x ⟧S  
    map-cong-correct {upper}  f {x≡y}    Refl          h = cong f x≡y       ≡⟨ cong (cong f) h 
                                                           cong f (refl _)  ≡⟨ cong-refl f ⟩∎
                                                           refl _           
    map-cong-correct {upper}  f {x≡y}   (Cons x≈y y≈z) h =
      cong f x≡y                                     ≡⟨ cong (cong f) h 
      cong f (trans  x≈y ⟧S  y≈z ⟧S)               ≡⟨ cong-trans f _ _ 
      trans (cong f  x≈y ⟧S) (cong f  y≈z ⟧S)      ≡⟨ cong₂ trans (map-cong-correct f x≈y (refl _))
                                                                    (map-cong-correct f y≈z (refl _)) ⟩∎
      trans  map-cong f x≈y ⟧S  map-cong f y≈z ⟧S  

-- Equality-preserving simplifier.

simplify :  {a} {A : Set a} {x y : A} 
           Eq x y  EqS upper x y
simplify (Lift x≡y)      = lift x≡y
simplify Refl            = Refl
simplify (Sym x≡y)       = reverse (simplify x≡y)
simplify (Trans x≡y y≡z) = append (simplify x≡y) (simplify y≡z)
simplify (Cong f x≡y)    = map-cong f (simplify x≡y)

abstract

  simplify-correct :
     {a} {A : Set a} {x y : A}
    (x≈y : Eq x y)   x≈y    simplify x≈y ⟧S
  simplify-correct (Lift x≡y)      = lift-correct x≡y
  simplify-correct Refl            = refl _
  simplify-correct (Sym x≈y)       = reverse-correct (simplify x≈y)
                                       (simplify-correct x≈y)
  simplify-correct (Trans x≈y y≈z) = append-correct (simplify x≈y) _
                                       (simplify-correct x≈y)
                                       (simplify-correct y≈z)
  simplify-correct (Cong f x≈y)    = map-cong-correct f (simplify x≈y)
                                       (simplify-correct x≈y)

------------------------------------------------------------------------
-- Tactic

abstract

  -- Simple tactic for proving equality of equality proofs.

  prove :  {a} {A : Set a} {x y : A} (x≡y x≡y′ : Eq x y) 
           simplify x≡y ⟧S   simplify x≡y′ ⟧S 
           x≡y    x≡y′ 
  prove x≡y x≡y′ hyp =
     x≡y              ≡⟨ simplify-correct x≡y 
     simplify x≡y  ⟧S  ≡⟨ hyp 
     simplify x≡y′ ⟧S  ≡⟨ sym (simplify-correct x≡y′) ⟩∎
     x≡y′             

------------------------------------------------------------------------
-- Some examples

private
  module Examples {A : Set} {x y : A} (x≡y : x  y) where

    ex₁ : trans (refl x) (sym (sym x≡y))  x≡y
    ex₁ = prove (Trans Refl (Sym (Sym (Lift x≡y)))) (Lift x≡y) (refl _)

    ex₂ : cong proj₂ (sym (cong (_,_ x) x≡y))  sym x≡y
    ex₂ = prove (Cong proj₂ (Sym (Cong (_,_ x) (Lift x≡y))))
                (Sym (Lift x≡y))
                (refl _)

-- Non-examples: The tactic cannot prove trans-symˡ or trans-symʳ.