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

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

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 : ∀ {ℓ a} {A : Set a} {x y : A} → EqS ℓ 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 : ∀ {ℓ a} {A B : Set a} {x y : A}
(f : A → B) →
EqS ℓ x y → EqS ℓ (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 :
∀ {ℓ a} {A B : Set a} {x y : A} (f : A → B) {x≡y}
(x≈y : EqS ℓ 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.
```