```------------------------------------------------------------------------
-- Two logically equivalent axiomatisations of equality
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Equality where

open import Logical-equivalence hiding (id; _∘_)
open import Prelude

private
variable
ℓ                                      : Level
A B C D                                : Type ℓ
P                                      : A → Type ℓ
a a₁ a₂ a₃ b c p u v x x₁ x₂ y y₁ y₂ z : A
f g                                    : (x : A) → P x

------------------------------------------------------------------------
-- Reflexive relations

record Reflexive-relation a : Type (lsuc a) where
no-eta-equality
infix 4 _≡_
field

-- "Equality".

_≡_ : {A : Type a} → A → A → Type a

-- Reflexivity.

refl : (x : A) → x ≡ x

-- Some definitions.

module Reflexive-relation′
(reflexive : ∀ ℓ → Reflexive-relation ℓ) where

private
open module R {ℓ} = Reflexive-relation (reflexive ℓ) public

-- Non-equality.

infix 4 _≢_

_≢_ : {A : Type a} → A → A → Type a
x ≢ y = ¬ (x ≡ y)

-- The property of having decidable equality.

Decidable-equality : Type ℓ → Type ℓ
Decidable-equality A = Decidable (_≡_ {A = A})

-- A type is contractible if it is inhabited and all elements are
-- equal.

Contractible : Type ℓ → Type ℓ
Contractible A = ∃ λ (x : A) → ∀ y → x ≡ y

-- The property of being a proposition.

Is-proposition : Type ℓ → Type ℓ
Is-proposition A = (x y : A) → x ≡ y

-- The property of being a set.

Is-set : Type ℓ → Type ℓ
Is-set A = {x y : A} → Is-proposition (x ≡ y)

-- Uniqueness of identity proofs (for a specific universe level).

Uniqueness-of-identity-proofs : ∀ ℓ → Type (lsuc ℓ)
Uniqueness-of-identity-proofs ℓ = {A : Type ℓ} → Is-set A

-- The K rule (without computational content).

K-rule : ∀ a p → Type (lsuc (a ⊔ p))
K-rule a p = {A : Type a} (P : {x : A} → x ≡ x → Type p) →
(∀ x → P (refl x)) →
∀ {x} (x≡x : x ≡ x) → P x≡x

-- Singleton x is a set which contains all elements which are equal
-- to x.

Singleton : {A : Type a} → A → Type a
Singleton x = ∃ λ y → y ≡ x

-- A variant of Singleton.

Other-singleton : {A : Type a} → A → Type a
Other-singleton x = ∃ λ y → x ≡ y

-- The inspect idiom.

inspect : (x : A) → Other-singleton x
inspect x = x , refl x

------------------------------------------------------------------------
-- Abstract definition of equality based on the J rule

-- Parametrised by a reflexive relation.

record Equality-with-J₀
a p (reflexive : ∀ ℓ → Reflexive-relation ℓ) :
Type (lsuc (a ⊔ p)) where

open Reflexive-relation′ reflexive

field

-- The J rule.

elim : ∀ {A : Type a} {x y}
(P : {x y : A} → x ≡ y → Type p) →
(∀ x → P (refl x)) →
(x≡y : x ≡ y) → P x≡y

-- The usual computational behaviour of the J rule.

elim-refl : ∀ {A : Type a} {x}
(P : {x y : A} → x ≡ y → Type p)
(r : ∀ x → P (refl x)) →
elim P r (refl x) ≡ r x

-- Extended variants of Reflexive-relation and Equality-with-J₀ with
-- some extra fields. These fields can be derived from
-- Equality-with-J₀ (see J₀⇒Equivalence-relation⁺ and J₀⇒J below), but
-- those derived definitions may not have the intended computational
-- behaviour (in particular, not when the paths of Cubical Agda are
-- used).

-- A variant of Reflexive-relation: equivalence relations with some
-- extra structure.

record Equivalence-relation⁺ a : Type (lsuc a) where
no-eta-equality
field
reflexive-relation : Reflexive-relation a

open Reflexive-relation reflexive-relation

field

-- Symmetry.

sym      : {A : Type a} {x y : A} → x ≡ y → y ≡ x
sym-refl : {A : Type a} {x : A} → sym (refl x) ≡ refl x

-- Transitivity.

trans           : {A : Type a} {x y z : A} → x ≡ y → y ≡ z → x ≡ z
trans-refl-refl : {A : Type a} {x : A} →
trans (refl x) (refl x) ≡ refl x

-- A variant of Equality-with-J₀.

record Equality-with-J
a b (e⁺ : ∀ ℓ → Equivalence-relation⁺ ℓ) :
Type (lsuc (a ⊔ b)) where
no-eta-equality

private
open module R  {ℓ} = Equivalence-relation⁺ (e⁺ ℓ)
open module R₀ {ℓ} = Reflexive-relation (reflexive-relation {ℓ})

field
equality-with-J₀ : Equality-with-J₀ a b (λ _ → reflexive-relation)

open Equality-with-J₀ equality-with-J₀

field

-- Congruence.

cong      : {A : Type a} {B : Type b} {x y : A}
(f : A → B) → x ≡ y → f x ≡ f y
cong-refl : {A : Type a} {B : Type b} {x : A} (f : A → B) →
cong f (refl x) ≡ refl (f x)

-- Substitutivity.

subst      : {A : Type a} {x y : A} (P : A → Type b) →
x ≡ y → P x → P y
subst-refl : ∀ {A : Type a} {x} (P : A → Type b) p →
subst P (refl x) p ≡ p

-- A dependent variant of cong.

dcong :
∀ {A : Type a} {P : A → Type b} {x y}
(f : (x : A) → P x) (x≡y : x ≡ y) →
subst P x≡y (f x) ≡ f y
dcong-refl :
∀ {A : Type a} {P : A → Type b} {x} (f : (x : A) → P x) →
dcong f (refl x) ≡ subst-refl _ _

-- Equivalence-relation⁺ can be derived from Equality-with-J₀.

J₀⇒Equivalence-relation⁺ :
∀ {ℓ reflexive} →
Equality-with-J₀ ℓ ℓ reflexive →
Equivalence-relation⁺ ℓ
J₀⇒Equivalence-relation⁺ {ℓ} {r} eq = record
{ reflexive-relation = r ℓ
; sym                = sym
; sym-refl           = sym-refl
; trans              = trans
; trans-refl-refl    = trans-refl-refl
}
where
open Reflexive-relation (r ℓ)
open Equality-with-J₀ eq

cong : (f : A → B) → x ≡ y → f x ≡ f y
cong f = elim (λ {u v} _ → f u ≡ f v) (λ x → refl (f x))

subst : (P : A → Type ℓ) → x ≡ y → P x → P y
subst P = elim (λ {u v} _ → P u → P v) (λ _ p → p)

subst-refl : (P : A → Type ℓ) (p : P x) → subst P (refl x) p ≡ p
subst-refl P p = cong (_\$ p) \$ elim-refl (λ {u} _ → P u → _) _

sym : x ≡ y → y ≡ x
sym {x = x} x≡y = subst (λ z → x ≡ z → z ≡ x) x≡y id x≡y

abstract

sym-refl : sym (refl x) ≡ refl x
sym-refl = cong (_\$ _) \$ subst-refl (λ z → _ ≡ z → z ≡ _) _

trans : x ≡ y → y ≡ z → x ≡ z
trans {x = x} = flip (subst (x ≡_))

abstract

trans-refl-refl : trans (refl x) (refl x) ≡ refl x
trans-refl-refl = subst-refl _ _

-- Equality-with-J (for arbitrary universe levels) can be derived from
-- Equality-with-J₀ (for arbitrary universe levels).

J₀⇒J :
∀ {reflexive} →
(eq : ∀ {a p} → Equality-with-J₀ a p reflexive) →
∀ {a p} → Equality-with-J a p (λ _ → J₀⇒Equivalence-relation⁺ eq)
J₀⇒J {r} eq {a} {b} = record
{ equality-with-J₀ = eq
; cong             = cong
; cong-refl        = cong-refl
; subst            = subst
; subst-refl       = subst-refl
; dcong            = dcong
; dcong-refl       = dcong-refl
}
where
open module R {ℓ}     = Reflexive-relation (r ℓ)
open module E {a} {b} = Equality-with-J₀ (eq {a} {b})

cong : (f : A → B) → x ≡ y → f x ≡ f y
cong f = elim (λ {u v} _ → f u ≡ f v) (λ x → refl (f x))

abstract

cong-refl : (f : A → B) → cong f (refl x) ≡ refl (f x)
cong-refl _ = elim-refl _ _

subst : (P : A → Type b) → x ≡ y → P x → P y
subst P = elim (λ {u v} _ → P u → P v) (λ _ p → p)

subst-refl≡id : (P : A → Type b) → subst P (refl x) ≡ id
subst-refl≡id P = elim-refl (λ {u v} _ → P u → P v) (λ _ p → p)

subst-refl : ∀ (P : A → Type b) p → subst P (refl x) p ≡ p
subst-refl P p = cong (_\$ p) (subst-refl≡id P)

dcong : (f : (x : A) → P x) (x≡y : x ≡ y) →
subst P x≡y (f x) ≡ f y
dcong {A = A} {P = P} f x≡y = elim
(λ {x y} (x≡y : x ≡ y) → (f : (x : A) → P x) →
subst P x≡y (f x) ≡ f y)
(λ _ _ → subst-refl _ _)
x≡y
f

abstract

dcong-refl : (f : (x : A) → P x) →
dcong f (refl x) ≡ subst-refl _ _
dcong-refl {P = P} f =
cong (_\$ f) \$ elim-refl (λ _ → (_ : ∀ x → P x) → _) _

-- Some derived properties.

module Equality-with-J′
{e⁺ : ∀ ℓ → Equivalence-relation⁺ ℓ}
(eq : ∀ {a p} → Equality-with-J a p e⁺)
where

private
open module E⁺ {ℓ}   = Equivalence-relation⁺ (e⁺ ℓ) public
open module E  {a b} = Equality-with-J (eq {a} {b}) public
hiding (subst; subst-refl)
open module E₀ {a p} = Equality-with-J₀ (equality-with-J₀ {a} {p})
public
open Reflexive-relation′ (λ ℓ → reflexive-relation {ℓ}) public

-- Substitutivity.

subst : (P : A → Type p) → x ≡ y → P x → P y
subst = E.subst

subst-refl : (P : A → Type p) (p : P x) → subst P (refl x) p ≡ p
subst-refl = E.subst-refl

-- Singleton types are contractible.

private

irr : (p : Singleton x) → (x , refl x) ≡ p
irr p =
elim (λ {u v} u≡v → (v , refl v) ≡ (u , u≡v))
(λ _ → refl _)
(proj₂ p)

singleton-contractible : (x : A) → Contractible (Singleton x)
singleton-contractible x = ((x , refl x) , irr)

abstract

-- "Evaluation rule" for singleton-contractible.

singleton-contractible-refl :
(x : A) →
proj₂ (singleton-contractible x) (x , refl x) ≡ refl (x , refl x)
singleton-contractible-refl _ = elim-refl _ _

------------------------------------------------------------------------
-- Abstract definition of equality based on substitutivity and
-- contractibility of singleton types

record Equality-with-substitutivity-and-contractibility
a p (reflexive : ∀ ℓ → Reflexive-relation ℓ) :
Type (lsuc (a ⊔ p)) where
no-eta-equality

open Reflexive-relation′ reflexive

field

-- Substitutivity.

subst : {A : Type a} {x y : A} (P : A → Type p) → x ≡ y → P x → P y

-- The usual computational behaviour of substitutivity.

subst-refl : {A : Type a} {x : A} (P : A → Type p) (p : P x) →
subst P (refl x) p ≡ p

-- Singleton types are contractible.

singleton-contractible :
{A : Type a} (x : A) → Contractible (Singleton x)

-- Some derived properties.

module Equality-with-substitutivity-and-contractibility′
{reflexive : ∀ ℓ → Reflexive-relation ℓ}
(eq : ∀ {a p} → Equality-with-substitutivity-and-contractibility
a p reflexive)
where

private
open Reflexive-relation′ reflexive public
open module E {a p} =
Equality-with-substitutivity-and-contractibility (eq {a} {p}) public
hiding (singleton-contractible)
open module E′ {a} =
Equality-with-substitutivity-and-contractibility (eq {a} {a}) public
using (singleton-contractible)

abstract

-- Congruence.

cong : (f : A → B) → x ≡ y → f x ≡ f y
cong {x = x} f x≡y =
subst (λ y → x ≡ y → f x ≡ f y) x≡y (λ _ → refl (f x)) x≡y

-- Symmetry.

sym : x ≡ y → y ≡ x
sym {x = x} x≡y = subst (λ z → x ≡ z → z ≡ x) x≡y id x≡y

abstract

-- "Evaluation rule" for sym.

sym-refl : sym (refl x) ≡ refl x
sym-refl {x = x} =
cong (λ f → f (refl x)) \$
subst-refl (λ z → x ≡ z → z ≡ x) _

-- Transitivity.

trans : x ≡ y → y ≡ z → x ≡ z
trans {x = x} = flip (subst (_≡_ x))

abstract

-- "Evaluation rule" for trans.

trans-refl-refl : trans (refl x) (refl x) ≡ refl x
trans-refl-refl = subst-refl _ _

abstract

-- The J rule.

elim : (P : {x y : A} → x ≡ y → Type p) →
(∀ x → P (refl x)) →
(x≡y : x ≡ y) → P x≡y
elim {x = x} {y = y} P p x≡y =
let lemma = proj₂ (singleton-contractible y) in
subst (P ∘ proj₂)
(trans (sym (lemma (y , refl y))) (lemma (x , x≡y)))
(p y)

-- Transitivity and symmetry sometimes cancel each other out.

trans-sym : (x≡y : x ≡ y) → trans (sym x≡y) x≡y ≡ refl y
trans-sym =
elim (λ {x y} (x≡y : x ≡ y) → trans (sym x≡y) x≡y ≡ refl y)
(λ _ → trans (cong (λ p → trans p _) sym-refl)
trans-refl-refl)

-- "Evaluation rule" for elim.

elim-refl : (P : {x y : A} → x ≡ y → Type p)
(p : ∀ x → P (refl x)) →
elim P p (refl x) ≡ p x
elim-refl {x = x} _ _ =
let lemma = proj₂ (singleton-contractible x) (x , refl x) in
trans (cong (λ q → subst _ q _) (trans-sym lemma))
(subst-refl _ _)

------------------------------------------------------------------------
-- The two abstract definitions are logically equivalent

J⇒subst+contr :
∀ {reflexive} →
(∀ {a p} → Equality-with-J₀ a p reflexive) →
∀ {a p} → Equality-with-substitutivity-and-contractibility
a p reflexive
J⇒subst+contr eq = record
{ subst                  = subst
; subst-refl             = subst-refl
; singleton-contractible = singleton-contractible
}
where open Equality-with-J′ (J₀⇒J eq)

subst+contr⇒J :
∀ {reflexive} →
(∀ {a p} → Equality-with-substitutivity-and-contractibility
a p reflexive) →
∀ {a p} → Equality-with-J₀ a p reflexive
subst+contr⇒J eq = record
{ elim      = elim
; elim-refl = elim-refl
}
where open Equality-with-substitutivity-and-contractibility′ eq

------------------------------------------------------------------------
-- Some derived definitions and properties

module Derived-definitions-and-properties
{e⁺}
(equality-with-J : ∀ {a p} → Equality-with-J a p e⁺)
where

-- This module reexports most of the definitions and properties
-- introduced above.

open Equality-with-J′ equality-with-J public

private
variable
eq u≡v v≡w x≡y y≡z x₁≡x₂ : x ≡ y

-- Equational reasoning combinators.

infix  -1 finally _∎
infixr -2 step-≡ _≡⟨⟩_

_∎ : (x : A) → x ≡ x
x ∎ = refl x

-- It can be easier for Agda to type-check typical equational
-- reasoning chains if the transitivity proof gets the equality
-- arguments in the opposite order, because then the y argument is
-- (perhaps more) known once the proof of x ≡ y is type-checked.
--
-- The idea behind this optimisation came up in discussions with Ulf
-- Norell.

step-≡ : ∀ x → y ≡ z → x ≡ y → x ≡ z
step-≡ _ y≡z x≡y = trans x≡y y≡z

syntax step-≡ x y≡z x≡y = x ≡⟨ x≡y ⟩ y≡z

_≡⟨⟩_ : ∀ x → x ≡ y → x ≡ y
_ ≡⟨⟩ x≡y = x≡y

finally : (x y : A) → x ≡ y → x ≡ y
finally _ _ x≡y = x≡y

syntax finally x y x≡y = x ≡⟨ x≡y ⟩∎ y ∎

-- The equational reasoning combinators are inlined to avoid

{-# INLINE _∎      #-}
{-# INLINE step-≡  #-}
{-# INLINE _≡⟨⟩_   #-}
{-# INLINE finally #-}

-- A minor variant of Christine Paulin-Mohring's version of the J
-- rule.
--
-- This definition is based on Martin Hofmann's (see the addendum
-- to Thomas Streicher's Habilitation thesis). Note that it is
-- also very similar to the definition of
-- Equality-with-substitutivity-and-contractibility.elim.

elim₁ : (P : ∀ {x} → x ≡ y → Type p) →
P (refl y) →
(x≡y : x ≡ y) → P x≡y
elim₁ {y = y} {x = x} P p x≡y =
subst (P ∘ proj₂)
(proj₂ (singleton-contractible y) (x , x≡y))
p

abstract

-- "Evaluation rule" for elim₁.

elim₁-refl : (P : ∀ {x} → x ≡ y → Type p) (p : P (refl y)) →
elim₁ P p (refl y) ≡ p
elim₁-refl {y = y} P p =
subst (P ∘ proj₂)
(proj₂ (singleton-contractible y) (y , refl y))
p                                                ≡⟨ cong (λ q → subst (P ∘ proj₂) q _) (singleton-contractible-refl _) ⟩

subst (P ∘ proj₂) (refl (y , refl y)) p                ≡⟨ subst-refl _ _ ⟩∎

p                                                      ∎

-- A variant of singleton-contractible.

private

irr : (p : Other-singleton x) → (x , refl x) ≡ p
irr p =
elim (λ {u v} u≡v → (u , refl u) ≡ (v , u≡v))
(λ _ → refl _)
(proj₂ p)

other-singleton-contractible :
(x : A) → Contractible (Other-singleton x)
other-singleton-contractible x = ((x , refl x) , irr)

abstract

-- "Evaluation rule" for other-singleton-contractible.

other-singleton-contractible-refl :
(x : A) →
proj₂ (other-singleton-contractible x) (x , refl x) ≡
refl (x , refl x)
other-singleton-contractible-refl _ = elim-refl _ _

-- Christine Paulin-Mohring's version of the J rule.

elim¹ : (P : ∀ {y} → x ≡ y → Type p) →
P (refl x) →
(x≡y : x ≡ y) → P x≡y
elim¹ {x = x} {y = y} P p x≡y =
subst (P ∘ proj₂)
(proj₂ (other-singleton-contractible x) (y , x≡y))
p

abstract

-- "Evaluation rule" for elim¹.

elim¹-refl : (P : ∀ {y} → x ≡ y → Type p) (p : P (refl x)) →
elim¹ P p (refl x) ≡ p
elim¹-refl {x = x} P p =
subst (P ∘ proj₂)
(proj₂ (other-singleton-contractible x) (x , refl x)) p    ≡⟨ cong (λ q → subst (P ∘ proj₂) q _)
(other-singleton-contractible-refl _) ⟩
subst (P ∘ proj₂) (refl (x , refl x)) p                          ≡⟨ subst-refl _ _ ⟩∎
p                                                                ∎

-- Every conceivable alternative implementation of cong (for two
-- specific types) is pointwise equal to cong.

monomorphic-cong-canonical :
(cong′ : {x y : A} (f : A → B) → x ≡ y → f x ≡ f y) →
({x : A} (f : A → B) → cong′ f (refl x) ≡ refl (f x)) →
cong′ f x≡y ≡ cong f x≡y
monomorphic-cong-canonical {f = f} cong′ cong′-refl = elim
(λ x≡y → cong′ f x≡y ≡ cong f x≡y)
(λ x →
cong′ f (refl x)  ≡⟨ cong′-refl _ ⟩
refl (f x)        ≡⟨ sym \$ cong-refl _ ⟩∎
cong f (refl x)   ∎)
_

-- Every conceivable alternative implementation of cong (for
-- arbitrary types) is pointwise equal to cong.

cong-canonical :
(cong′ :
∀ {a b} {A : Type a} {B : Type b} {x y : A}
(f : A → B) → x ≡ y → f x ≡ f y) →
(∀ {a b} {A : Type a} {B : Type b} {x : A}
(f : A → B) → cong′ f (refl x) ≡ refl (f x)) →
cong′ f x≡y ≡ cong f x≡y
cong-canonical cong′ cong′-refl =
monomorphic-cong-canonical cong′ cong′-refl

-- A generalisation of dcong.

dcong′ :
(f : (x : A) → x ≡ y → P x) (x≡y : x ≡ y) →
subst P x≡y (f x x≡y) ≡ f y (refl y)
dcong′ {y = y} {P = P} f x≡y = elim₁
(λ {x} (x≡y : x ≡ y) →
(f : ∀ x → x ≡ y → P x) →
subst P x≡y (f x x≡y) ≡ f y (refl y))
(λ f → subst P (refl y) (f y (refl y))  ≡⟨ subst-refl _ _ ⟩∎
f y (refl y)                     ∎)
x≡y f

abstract

-- "Evaluation rule" for dcong′.

dcong′-refl :
(f : (x : A) → x ≡ y → P x) →
dcong′ f (refl y) ≡ subst-refl _ _
dcong′-refl {y = y} {P = P} f =
cong (_\$ f) \$ elim₁-refl (λ _ → (f : ∀ x → x ≡ y → P x) → _) _

-- Binary congruence.

cong₂ : (f : A → B → C) → x ≡ y → u ≡ v → f x u ≡ f y v
cong₂ {x = x} {y = y} {u = u} {v = v} f x≡y u≡v =
f x u  ≡⟨ cong (flip f u) x≡y ⟩
f y u  ≡⟨ cong (f y)      u≡v ⟩∎
f y v  ∎

abstract

-- "Evaluation rule" for cong₂.

cong₂-refl : (f : A → B → C) →
cong₂ f (refl x) (refl y) ≡ refl (f x y)
cong₂-refl {x = x} {y = y} f =
trans (cong (flip f y) (refl x)) (cong (f x) (refl y))  ≡⟨ cong₂ trans (cong-refl _) (cong-refl _) ⟩
trans (refl (f x y)) (refl (f x y))                     ≡⟨ trans-refl-refl ⟩∎
refl (f x y)                                            ∎

-- The K rule is logically equivalent to uniqueness of identity
-- proofs (at least for certain combinations of levels).

K⇔UIP : K-rule ℓ ℓ ⇔ Uniqueness-of-identity-proofs ℓ
K⇔UIP = record
{ from = λ UIP P r {x} x≡x → subst P (UIP (refl x) x≡x) (r x)
; to   = λ K →
elim (λ p → ∀ q → p ≡ q)
(λ x → K (λ {x} p → refl x ≡ p) (λ x → refl (refl x)))
}

-- A bunch of lemmas that can be used to rearrange equalities.

abstract

trans-reflʳ : (x≡y : x ≡ y) → trans x≡y (refl y) ≡ x≡y
trans-reflʳ =
elim (λ {u v} u≡v → trans u≡v (refl v) ≡ u≡v)
(λ _ → trans-refl-refl)

trans-reflˡ : (x≡y : x ≡ y) → trans (refl x) x≡y ≡ x≡y
trans-reflˡ =
elim (λ {u v} u≡v → trans (refl u) u≡v ≡ u≡v)
(λ _ → trans-refl-refl)

trans-assoc : (x≡y : x ≡ y) (y≡z : y ≡ z) (z≡u : z ≡ u) →
trans (trans x≡y y≡z) z≡u ≡ trans x≡y (trans y≡z z≡u)
trans-assoc =
elim (λ x≡y → ∀ y≡z z≡u → trans (trans x≡y y≡z) z≡u ≡
trans x≡y (trans y≡z z≡u))
(λ y y≡z z≡u →
trans (trans (refl y) y≡z) z≡u ≡⟨ cong₂ trans (trans-reflˡ _) (refl _) ⟩
trans y≡z z≡u                  ≡⟨ sym \$ trans-reflˡ _ ⟩∎
trans (refl y) (trans y≡z z≡u) ∎)

sym-sym : (x≡y : x ≡ y) → sym (sym x≡y) ≡ x≡y
sym-sym = elim (λ {u v} u≡v → sym (sym u≡v) ≡ u≡v)
(λ x → sym (sym (refl x))  ≡⟨ cong sym sym-refl ⟩
sym (refl x)        ≡⟨ sym-refl ⟩∎
refl x              ∎)

sym-trans : (x≡y : x ≡ y) (y≡z : y ≡ z) →
sym (trans x≡y y≡z) ≡ trans (sym y≡z) (sym x≡y)
sym-trans =
elim (λ x≡y → ∀ y≡z → sym (trans x≡y y≡z) ≡ trans (sym y≡z) (sym x≡y))
(λ y y≡z → sym (trans (refl y) y≡z)        ≡⟨ cong sym (trans-reflˡ _) ⟩
sym y≡z                         ≡⟨ sym \$ trans-reflʳ _ ⟩
trans (sym y≡z) (refl y)        ≡⟨ cong (trans (sym y≡z)) (sym sym-refl)  ⟩∎
trans (sym y≡z) (sym (refl y))  ∎)

trans-symˡ : (p : x ≡ y) → trans (sym p) p ≡ refl y
trans-symˡ =
elim (λ p → trans (sym p) p ≡ refl _)
(λ x → trans (sym (refl x)) (refl x)  ≡⟨ trans-reflʳ _ ⟩
sym (refl x)                   ≡⟨ sym-refl ⟩∎
refl x                         ∎)

trans-symʳ : (p : x ≡ y) → trans p (sym p) ≡ refl _
trans-symʳ =
elim (λ p → trans p (sym p) ≡ refl _)
(λ x → trans (refl x) (sym (refl x))  ≡⟨ trans-reflˡ _ ⟩
sym (refl x)                   ≡⟨ sym-refl ⟩∎
refl x                         ∎)

cong-trans : (f : A → B) (x≡y : x ≡ y) (y≡z : y ≡ z) →
cong f (trans x≡y y≡z) ≡ trans (cong f x≡y) (cong f y≡z)
cong-trans f =
elim (λ x≡y → ∀ y≡z → cong f (trans x≡y y≡z) ≡
trans (cong f x≡y) (cong f y≡z))
(λ y y≡z → cong f (trans (refl y) y≡z)           ≡⟨ cong (cong f) (trans-reflˡ _) ⟩
cong f y≡z                            ≡⟨ sym \$ trans-reflˡ _ ⟩
trans (refl (f y)) (cong f y≡z)       ≡⟨ cong₂ trans (sym (cong-refl _)) (refl _) ⟩∎
trans (cong f (refl y)) (cong f y≡z)  ∎)

cong-id : (x≡y : x ≡ y) → x≡y ≡ cong id x≡y
cong-id = elim (λ u≡v → u≡v ≡ cong id u≡v)
(λ x → refl x            ≡⟨ sym (cong-refl _) ⟩∎
cong id (refl x)  ∎)

cong-const : (x≡y : x ≡ y) → cong (const z) x≡y ≡ refl z
cong-const {z = z} =
elim (λ u≡v → cong (const z) u≡v ≡ refl z)
(λ x → cong (const z) (refl x)  ≡⟨ cong-refl _ ⟩∎
refl z                   ∎)

cong-∘ : (f : B → C) (g : A → B) (x≡y : x ≡ y) →
cong f (cong g x≡y) ≡ cong (f ∘ g) x≡y
cong-∘ f g = elim (λ x≡y → cong f (cong g x≡y) ≡ cong (f ∘ g) x≡y)
(λ x → cong f (cong g (refl x))  ≡⟨ cong (cong f) (cong-refl _) ⟩
cong f (refl (g x))       ≡⟨ cong-refl _ ⟩
refl (f (g x))            ≡⟨ sym (cong-refl _) ⟩∎
cong (f ∘ g) (refl x)     ∎)

cong-uncurry-cong₂-, :
{x≡y : x ≡ y} {u≡v : u ≡ v} →
cong (uncurry f) (cong₂ _,_ x≡y u≡v) ≡ cong₂ f x≡y u≡v
cong-uncurry-cong₂-, {y = y} {u = u} {f = f} {x≡y = x≡y} {u≡v} =
cong (uncurry f)
(trans (cong (flip _,_ u) x≡y) (cong (_,_ y) u≡v))  ≡⟨ cong-trans _ _ _ ⟩

trans (cong (uncurry f) (cong (flip _,_ u) x≡y))
(cong (uncurry f) (cong (_,_ y) u≡v))              ≡⟨ cong₂ trans (cong-∘ _ _ _) (cong-∘ _ _ _) ⟩∎

trans (cong (flip f u) x≡y) (cong (f y) u≡v)             ∎

cong-proj₁-cong₂-, :
(x≡y : x ≡ y) (u≡v : u ≡ v) →
cong proj₁ (cong₂ _,_ x≡y u≡v) ≡ x≡y
cong-proj₁-cong₂-, {y = y} x≡y u≡v =
cong proj₁ (cong₂ _,_ x≡y u≡v)            ≡⟨ cong-uncurry-cong₂-, ⟩
cong₂ const x≡y u≡v                       ≡⟨⟩
trans (cong id x≡y) (cong (const y) u≡v)  ≡⟨ cong₂ trans (sym \$ cong-id _) (cong-const _) ⟩
trans x≡y (refl y)                        ≡⟨ trans-reflʳ _ ⟩∎
x≡y                                       ∎

cong-proj₂-cong₂-, :
(x≡y : x ≡ y) (u≡v : u ≡ v) →
cong proj₂ (cong₂ _,_ x≡y u≡v) ≡ u≡v
cong-proj₂-cong₂-, {u = u} x≡y u≡v =
cong proj₂ (cong₂ _,_ x≡y u≡v)            ≡⟨ cong-uncurry-cong₂-, ⟩
cong₂ (const id) x≡y u≡v                  ≡⟨⟩
trans (cong (const u) x≡y) (cong id u≡v)  ≡⟨ cong₂ trans (cong-const _) (sym \$ cong-id _) ⟩
trans (refl u) u≡v                        ≡⟨ trans-reflˡ _ ⟩∎
u≡v                                       ∎

cong₂-reflˡ : {u≡v : u ≡ v}
(f : A → B → C) →
cong₂ f (refl x) u≡v ≡ cong (f x) u≡v
cong₂-reflˡ {u = u} {x = x} {u≡v = u≡v} f =
trans (cong (flip f u) (refl x)) (cong (f x) u≡v)  ≡⟨ cong₂ trans (cong-refl _) (refl _) ⟩
trans (refl (f x u)) (cong (f x) u≡v)              ≡⟨ trans-reflˡ _ ⟩∎
cong (f x) u≡v                                     ∎

cong₂-reflʳ : (f : A → B → C) {x≡y : x ≡ y} →
cong₂ f x≡y (refl u) ≡ cong (flip f u) x≡y
cong₂-reflʳ {y = y} {u = u} f {x≡y} =
trans (cong (flip f u) x≡y) (cong (f y) (refl u))  ≡⟨ cong (trans _) (cong-refl _) ⟩
trans (cong (flip f u) x≡y) (refl (f y u))         ≡⟨ trans-reflʳ _ ⟩∎
cong (flip f u) x≡y                                ∎

cong-sym : (f : A → B) (x≡y : x ≡ y) →
cong f (sym x≡y) ≡ sym (cong f x≡y)
cong-sym f = elim (λ x≡y → cong f (sym x≡y) ≡ sym (cong f x≡y))
(λ x → cong f (sym (refl x))  ≡⟨ cong (cong f) sym-refl ⟩
cong f (refl x)        ≡⟨ cong-refl _ ⟩
refl (f x)             ≡⟨ sym sym-refl ⟩
sym (refl (f x))       ≡⟨ cong sym \$ sym (cong-refl _) ⟩∎
sym (cong f (refl x))  ∎)

cong₂-sym :
cong₂ f (sym x≡y) (sym u≡v) ≡ sym (cong₂ f x≡y u≡v)
cong₂-sym {f = f} {x≡y = x≡y} {u≡v = u≡v} = elim¹
(λ u≡v → cong₂ f (sym x≡y) (sym u≡v) ≡ sym (cong₂ f x≡y u≡v))
(cong₂ f (sym x≡y) (sym (refl _))  ≡⟨ cong (cong₂ _ _) sym-refl ⟩
cong₂ f (sym x≡y) (refl _)        ≡⟨ cong₂-reflʳ _ ⟩
cong (flip f _) (sym x≡y)         ≡⟨ cong-sym _ _ ⟩
sym (cong (flip f _) x≡y)         ≡⟨ cong sym \$ sym \$ cong₂-reflʳ _ ⟩∎
sym (cong₂ f x≡y (refl _))        ∎)
u≡v

cong₂-trans :
{f : A → B → C} →
cong₂ f (trans x≡y y≡z) (trans u≡v v≡w) ≡
trans (cong₂ f x≡y u≡v) (cong₂ f y≡z v≡w)
cong₂-trans
{x≡y = x≡y} {y≡z = y≡z} {u≡v = u≡v} {v≡w = v≡w} {f = f} =
elim₁
(λ x≡y →
cong₂ f (trans x≡y y≡z) (trans u≡v v≡w) ≡
trans (cong₂ f x≡y u≡v) (cong₂ f y≡z v≡w))
(elim₁
(λ u≡v →
cong₂ f (trans (refl _) y≡z) (trans u≡v v≡w) ≡
trans (cong₂ f (refl _) u≡v) (cong₂ f y≡z v≡w))
(cong₂ f (trans (refl _) y≡z) (trans (refl _) v≡w)    ≡⟨ cong₂ (cong₂ f) (trans-reflˡ _) (trans-reflˡ _) ⟩
cong₂ f y≡z v≡w                                      ≡⟨ sym \$
trans (cong (flip trans _) \$ cong₂-refl _) \$
trans-reflˡ _ ⟩∎
trans (cong₂ f (refl _) (refl _)) (cong₂ f y≡z v≡w)  ∎)
u≡v)
x≡y

cong₂-∘ˡ :
{f : B → C → D} {g : A → B} {x≡y : x ≡ y} {u≡v : u ≡ v} →
cong₂ (f ∘ g) x≡y u≡v ≡ cong₂ f (cong g x≡y) u≡v
cong₂-∘ˡ {y = y} {u = u} {f = f} {g = g} {x≡y = x≡y} {u≡v} =
trans (cong (flip (f ∘ g) u) x≡y) (cong (f (g y)) u≡v)     ≡⟨ cong (flip trans _) \$ sym \$ cong-∘ _ _ _ ⟩∎
trans (cong (flip f u) (cong g x≡y)) (cong (f (g y)) u≡v)  ∎

cong₂-∘ʳ :
{x≡y : x ≡ y} {u≡v : u ≡ v} →
cong₂ (λ x → f x ∘ g) x≡y u≡v ≡ cong₂ f x≡y (cong g u≡v)
cong₂-∘ʳ {y = y} {u = u} {f = f} {g = g} {x≡y = x≡y} {u≡v} =
trans (cong (flip f (g u)) x≡y) (cong (f y ∘ g) u≡v)       ≡⟨ cong (trans _) \$ sym \$ cong-∘ _ _ _ ⟩∎
trans (cong (flip f (g u)) x≡y) (cong (f y) (cong g u≡v))  ∎

cong₂-cong-cong :
(f : A → B) (g : A → C) (h : B → C → D) →
cong₂ h (cong f eq) (cong g eq) ≡
cong (λ x → h (f x) (g x)) eq
cong₂-cong-cong f g h = elim¹
(λ eq → cong₂ h (cong f eq) (cong g eq) ≡
cong (λ x → h (f x) (g x)) eq)
(cong₂ h (cong f (refl _)) (cong g (refl _))  ≡⟨ cong₂ (cong₂ h) (cong-refl _) (cong-refl _) ⟩
cong₂ h (refl _) (refl _)                    ≡⟨ cong₂-refl h ⟩
refl _                                       ≡⟨ sym \$ cong-refl _ ⟩∎
cong (λ x → h (f x) (g x)) (refl _)          ∎)
_

cong-≡id :
{f : A → A}
(f≡id : f ≡ id) →
cong (λ g → g (f x)) f≡id ≡
cong (λ g → f (g x)) f≡id
cong-≡id = elim₁
(λ {f} p → cong (λ g → g (f _)) p ≡ cong (λ g → f (g _)) p)
(refl _)

cong-≡id-≡-≡id :
(f≡id : ∀ x → f x ≡ x) →
cong f (f≡id x) ≡ f≡id (f x)
cong-≡id-≡-≡id {f = f} {x = x} f≡id =
cong f (f≡id x)                                             ≡⟨ elim¹
(λ {y} (p : f x ≡ y) →
cong f p ≡ trans (f≡id (f x)) (trans p (sym (f≡id y)))) (
cong f (refl _)                                              ≡⟨ cong-refl _ ⟩
refl _                                                       ≡⟨ sym \$ trans-symʳ _ ⟩
trans (f≡id (f x)) (sym (f≡id (f x)))                        ≡⟨ cong (trans (f≡id (f x))) \$ sym \$ trans-reflˡ _ ⟩∎
trans (f≡id (f x)) (trans (refl _) (sym (f≡id (f x))))       ∎)
(f≡id x)⟩
trans (f≡id (f x)) (trans (f≡id x) (sym (f≡id x)))          ≡⟨ cong (trans (f≡id (f x))) \$ trans-symʳ _ ⟩
trans (f≡id (f x)) (refl _)                                 ≡⟨ trans-reflʳ _ ⟩
f≡id (f x)                                                  ∎

elim-∘ :
(P Q : ∀ {x y} → x ≡ y → Type p)
(f : ∀ {x y} {x≡y : x ≡ y} → P x≡y → Q x≡y)
(r : ∀ x → P (refl x)) {x≡y : x ≡ y} →
f (elim P r x≡y) ≡ elim Q (f ∘ r) x≡y
elim-∘ {x = x} P Q f r {x≡y} = elim¹
(λ x≡y → f (elim P r x≡y) ≡
elim Q (f ∘ r) x≡y)
(f (elim P r (refl x))    ≡⟨ cong f \$ elim-refl _ _ ⟩
f (r x)                  ≡⟨ sym \$ elim-refl _ _ ⟩∎
elim Q (f ∘ r) (refl x)  ∎)
x≡y

elim-cong :
(P : B → B → Type p) (f : A → B)
(r : ∀ x → P x x) {x≡y : x ≡ y} →
elim (λ {x y} _ → P x y) r (cong f x≡y) ≡
elim (λ {x y} _ → P (f x) (f y)) (r ∘ f) x≡y
elim-cong {x = x} P f r {x≡y} = elim¹
(λ x≡y → elim (λ {x y} _ → P x y) r (cong f x≡y) ≡
elim (λ {x y} _ → P (f x) (f y)) (r ∘ f) x≡y)
(elim (λ {x y} _ → P x y) r (cong f (refl x))       ≡⟨ cong (elim (λ {x y} _ → P x y) _) \$ cong-refl _ ⟩
elim (λ {x y} _ → P x y) r (refl (f x))            ≡⟨ elim-refl _ _ ⟩
r (f x)                                            ≡⟨ sym \$ elim-refl _ _ ⟩∎
elim (λ {x y} _ → P (f x) (f y)) (r ∘ f) (refl x)  ∎)
x≡y

subst-const : ∀ (x₁≡x₂ : x₁ ≡ x₂) {b} →
subst (const B) x₁≡x₂ b ≡ b
subst-const {B = B} x₁≡x₂ {b} =
elim¹ (λ x₁≡x₂ → subst (const B) x₁≡x₂ b ≡ b)
(subst-refl _ _)
x₁≡x₂

abstract

-- One can express sym in terms of subst.

sym-subst : sym x≡y ≡ subst (λ z → x ≡ z → z ≡ x) x≡y id x≡y
sym-subst = elim
(λ {x} x≡y → sym x≡y ≡ subst (λ z → x ≡ z → z ≡ x) x≡y id x≡y)
(λ x →
sym (refl x)                                      ≡⟨ sym-refl ⟩
refl x                                            ≡⟨ cong (_\$ refl x) \$ sym \$ subst-refl (λ z → x ≡ z → _) _ ⟩∎
subst (λ z → x ≡ z → z ≡ x) (refl x) id (refl x)  ∎)
_

-- One can express trans in terms of subst (in several ways).

trans-subst :
{x≡y : x ≡ y} {y≡z : y ≡ z} →
trans x≡y y≡z ≡ subst (x ≡_) y≡z x≡y
trans-subst {z = z} = elim
(λ {x y} x≡y → (y≡z : y ≡ z) → trans x≡y y≡z ≡ subst (x ≡_) y≡z x≡y)
(λ y → elim
(λ {y} y≡z → trans (refl y) y≡z ≡ subst (y ≡_) y≡z (refl y))
(λ x →
trans (refl x) (refl x)         ≡⟨ trans-refl-refl ⟩
refl x                          ≡⟨ sym \$ subst-refl _ _ ⟩∎
subst (x ≡_) (refl x) (refl x)  ∎))
_
_

subst-trans :
(x≡y : x ≡ y) {y≡z : y ≡ z} →
subst (_≡ z) (sym x≡y) y≡z ≡ trans x≡y y≡z
subst-trans {y = y} {z} x≡y {y≡z} =
elim₁ (λ x≡y → subst (λ x → x ≡ z) (sym x≡y) y≡z ≡
trans x≡y y≡z)
(subst (λ x → x ≡ z) (sym (refl y)) y≡z  ≡⟨ cong (λ eq → subst (λ x → x ≡ z) eq _) sym-refl ⟩
subst (λ x → x ≡ z) (refl y) y≡z        ≡⟨ subst-refl _ _ ⟩
y≡z                                     ≡⟨ sym \$ trans-reflˡ _ ⟩∎
trans (refl y) y≡z                      ∎)
x≡y

subst-trans-sym :
{y≡x : y ≡ x} {y≡z : y ≡ z} →
subst (_≡ z) y≡x y≡z ≡ trans (sym y≡x) y≡z
subst-trans-sym {z = z} {y≡x = y≡x} {y≡z = y≡z} =
subst (_≡ z) y≡x y≡z              ≡⟨ cong (flip (subst (_≡ z)) _) \$ sym \$ sym-sym _ ⟩
subst (_≡ z) (sym (sym y≡x)) y≡z  ≡⟨ subst-trans _ ⟩∎
trans (sym y≡x) y≡z               ∎

-- One can express subst in terms of elim.

subst-elim :
subst P x≡y p ≡ elim (λ {u v} _ → P u → P v) (λ _ → id) x≡y p
subst-elim {P = P} = elim
(λ x≡y → ∀ p →
subst P x≡y p ≡ elim (λ {u v} _ → P u → P v) (λ _ → id) x≡y p)
(λ x p →
subst P (refl x) p                                  ≡⟨ subst-refl _ _ ⟩
p                                                   ≡⟨ cong (_\$ p) \$ sym \$ elim-refl (λ {u} _ → P u → _) _ ⟩∎
elim (λ {u v} _ → P u → P v) (λ _ → id) (refl x) p  ∎)
_
_

subst-∘ : (P : B → Type p) (f : A → B) (x≡y : x ≡ y) {p : P (f x)} →
subst (P ∘ f) x≡y p ≡ subst P (cong f x≡y) p
subst-∘ P f _ {p} = elim¹
(λ x≡y → subst (P ∘ f) x≡y p ≡ subst P (cong f x≡y) p)
(subst (P ∘ f) (refl _) p     ≡⟨ subst-refl _ _ ⟩
p                            ≡⟨ sym \$ subst-refl _ _ ⟩
subst P (refl _) p           ≡⟨ cong (flip (subst _) _) \$ sym \$ cong-refl _ ⟩∎
subst P (cong f (refl _)) p  ∎)
_

subst-↑ : (P : A → Type p) {p : ↑ ℓ (P x)} →
subst (↑ ℓ ∘ P) x≡y p ≡ lift (subst P x≡y (lower p))
subst-↑ {ℓ = ℓ} P {p} = elim¹
(λ x≡y → subst (↑ ℓ ∘ P) x≡y p ≡ lift (subst P x≡y (lower p)))
(subst (↑ ℓ ∘ P) (refl _) p         ≡⟨ subst-refl _ _ ⟩
p                                  ≡⟨ cong lift \$ sym \$ subst-refl _ _ ⟩∎
lift (subst P (refl _) (lower p))  ∎)
_

-- A fusion law for subst.

subst-subst :
(P : A → Type p) (x≡y : x ≡ y) (y≡z : y ≡ z) (p : P x) →
subst P y≡z (subst P x≡y p) ≡ subst P (trans x≡y y≡z) p
subst-subst P x≡y y≡z p =
elim (λ {x y} x≡y → ∀ {z} (y≡z : y ≡ z) p →
subst P y≡z (subst P x≡y p) ≡ subst P (trans x≡y y≡z) p)
(λ x y≡z p →
subst P y≡z (subst P (refl x) p)  ≡⟨ cong (subst P _) \$ subst-refl _ _ ⟩
subst P y≡z p                     ≡⟨ cong (λ q → subst P q _) (sym \$ trans-reflˡ _) ⟩∎
subst P (trans (refl x) y≡z) p    ∎)
x≡y y≡z p

-- "Computation rules" for subst-subst.

subst-subst-reflˡ :
∀ (P : A → Type p) {p} →
subst-subst P (refl x) x≡y p ≡
cong₂ (flip (subst P)) (subst-refl _ _) (sym \$ trans-reflˡ x≡y)
subst-subst-reflˡ P =
cong (λ f → f _ _) \$
elim-refl (λ {x y} x≡y → ∀ {z} (y≡z : y ≡ z) p →
subst P y≡z (subst P x≡y p) ≡ _)
_

subst-subst-refl-refl :
∀ (P : A → Type p) {p} →
subst-subst P (refl x) (refl x) p ≡
cong₂ (flip (subst P)) (subst-refl _ _) (sym trans-refl-refl)
subst-subst-refl-refl {x = x} P {p} =
subst-subst P (refl x) (refl x) p                              ≡⟨ subst-subst-reflˡ _ ⟩

cong₂ (flip (subst P)) (subst-refl _ _)
(sym \$ trans-reflˡ (refl x))            ≡⟨ cong (cong₂ (flip (subst P)) (subst-refl _ _) ∘ sym) \$
elim-refl _ _ ⟩∎
cong₂ (flip (subst P)) (subst-refl _ _) (sym trans-refl-refl)  ∎

-- Substitutivity and symmetry sometimes cancel each other out.

subst-subst-sym :
(P : A → Type p) (x≡y : x ≡ y) (p : P y) →
subst P x≡y (subst P (sym x≡y) p) ≡ p
subst-subst-sym P =
elim¹
(λ x≡y → ∀ p → subst P x≡y (subst P (sym x≡y) p) ≡ p)
(λ p →
subst P (refl _) (subst P (sym (refl _)) p)  ≡⟨ subst-refl _ _ ⟩
subst P (sym (refl _)) p                     ≡⟨ cong (flip (subst P) _) sym-refl ⟩
subst P (refl _) p                           ≡⟨ subst-refl _ _ ⟩∎
p                                            ∎)

subst-sym-subst :
(P : A → Type p) {x≡y : x ≡ y} {p : P x} →
subst P (sym x≡y) (subst P x≡y p) ≡ p
subst-sym-subst P {x≡y = x≡y} {p = p} =
elim¹
(λ x≡y → ∀ p → subst P (sym x≡y) (subst P x≡y p) ≡ p)
(λ p →
subst P (sym (refl _)) (subst P (refl _) p)  ≡⟨ cong (flip (subst P) _) sym-refl ⟩
subst P (refl _) (subst P (refl _) p)        ≡⟨ subst-refl _ _ ⟩
subst P (refl _) p                           ≡⟨ subst-refl _ _ ⟩∎
p                                            ∎)
x≡y p

-- Some "computation rules".

subst-subst-sym-refl :
(P : A → Type p) {p : P x} →
subst-subst-sym P (refl x) p ≡
trans (subst-refl _ _)
(trans (cong (flip (subst P) _) sym-refl)
(subst-refl _ _))
subst-subst-sym-refl P {p = p} =
cong (_\$ _) \$
elim¹-refl
(λ x≡y → ∀ p → subst P x≡y (subst P (sym x≡y) p) ≡ p)
_

subst-sym-subst-refl :
(P : A → Type p) {p : P x} →
subst-sym-subst P {x≡y = refl x} {p = p} ≡
trans (cong (flip (subst P) _) sym-refl)
(trans (subst-refl _ _) (subst-refl _ _))
subst-sym-subst-refl P =
cong (_\$ _) \$
elim¹-refl
(λ x≡y → ∀ p → subst P (sym x≡y) (subst P x≡y p) ≡ p)
_

-- Some corollaries and variants.

trans-[trans-sym]- : (a≡b : a ≡ b) (c≡b : c ≡ b) →
trans (trans a≡b (sym c≡b)) c≡b ≡ a≡b
trans-[trans-sym]- a≡b c≡b =
trans (trans a≡b (sym c≡b)) c≡b                ≡⟨ trans-subst ⟩
subst (_ ≡_) c≡b (trans a≡b (sym c≡b))         ≡⟨ cong (subst _ _) trans-subst ⟩
subst (_ ≡_) c≡b (subst (_ ≡_) (sym c≡b) a≡b)  ≡⟨ subst-subst-sym _ _ _ ⟩∎
a≡b                                            ∎

trans-[trans]-sym : (a≡b : a ≡ b) (b≡c : b ≡ c) →
trans (trans a≡b b≡c) (sym b≡c) ≡ a≡b
trans-[trans]-sym a≡b b≡c =
trans (trans a≡b b≡c)             (sym b≡c)  ≡⟨ sym \$ cong (λ eq → trans (trans _ eq) (sym b≡c)) \$ sym-sym _ ⟩
trans (trans a≡b (sym (sym b≡c))) (sym b≡c)  ≡⟨ trans-[trans-sym]- _ _ ⟩∎
a≡b                                          ∎

trans--[trans-sym] : (b≡a : b ≡ a) (b≡c : b ≡ c) →
trans b≡a (trans (sym b≡a) b≡c) ≡ b≡c
trans--[trans-sym] b≡a b≡c =
trans b≡a (trans (sym b≡a) b≡c)  ≡⟨ sym \$ trans-assoc _ _ _ ⟩
trans (trans b≡a (sym b≡a)) b≡c  ≡⟨ cong (flip trans _) \$ trans-symʳ _ ⟩
trans (refl _) b≡c               ≡⟨ trans-reflˡ _ ⟩∎
b≡c                              ∎

trans-sym-[trans] : (a≡b : a ≡ b) (b≡c : b ≡ c) →
trans (sym a≡b) (trans a≡b b≡c) ≡ b≡c
trans-sym-[trans] a≡b b≡c =
trans (sym a≡b) (trans a≡b b≡c)              ≡⟨ cong (λ p → trans (sym _) (trans p _)) \$ sym \$ sym-sym _ ⟩
trans (sym a≡b) (trans (sym (sym a≡b)) b≡c)  ≡⟨ trans--[trans-sym] _ _ ⟩∎
b≡c                                          ∎

-- The lemmas subst-refl and subst-const can cancel each other
-- out.

subst-refl-subst-const :
trans (sym \$ subst-refl (λ _ → B) b) (subst-const (refl x)) ≡
refl b
subst-refl-subst-const {b = b} {x = x} =
trans (sym \$ subst-refl _ _)
(elim¹ (λ eq → subst (λ _ → _) eq b ≡ b)
(subst-refl _ _)
(refl _))                          ≡⟨ cong (trans _) (elim¹-refl _ _) ⟩
trans (sym \$ subst-refl _ _) (subst-refl _ _)   ≡⟨ trans-symˡ _ ⟩∎
refl _                                          ∎

-- In non-dependent cases one can express dcong using subst-const
-- and cong.
--
-- This is (similar to) Lemma 2.3.8 in the HoTT book.

dcong-subst-const-cong :
(f : A → B) (x≡y : x ≡ y) →
dcong f x≡y ≡
(subst (const B) x≡y (f x)  ≡⟨ subst-const _ ⟩
f x                        ≡⟨ cong f x≡y ⟩∎
f y                        ∎)
dcong-subst-const-cong f = elim
(λ {x y} x≡y → dcong f x≡y ≡
trans (subst-const x≡y) (cong f x≡y))
(λ x →
dcong f (refl x)                         ≡⟨ dcong-refl _ ⟩
subst-refl _ _                           ≡⟨ sym \$ trans-reflʳ _ ⟩
trans (subst-refl _ _) (refl (f x))      ≡⟨ cong₂ trans
(sym \$ elim¹-refl _ _)
(sym \$ cong-refl _) ⟩∎
trans (subst-const _) (cong f (refl x))  ∎)

-- A corollary.

dcong≡→cong≡ :
{x≡y : x ≡ y} {fx≡fy : f x ≡ f y} →
dcong f x≡y ≡ trans (subst-const _) fx≡fy →
cong f x≡y ≡ fx≡fy
dcong≡→cong≡ {f = f} {x≡y = x≡y} {fx≡fy} hyp =
cong f x≡y                                                        ≡⟨ sym \$ trans-sym-[trans] _ _ ⟩
trans (sym \$ subst-const _) (trans (subst-const _) \$ cong f x≡y)  ≡⟨ cong (trans (sym \$ subst-const _)) \$ sym \$
dcong-subst-const-cong _ _ ⟩
trans (sym \$ subst-const _) (dcong f x≡y)                         ≡⟨ cong (trans (sym \$ subst-const _)) hyp ⟩
trans (sym \$ subst-const _) (trans (subst-const _) fx≡fy)         ≡⟨ trans-sym-[trans] _ _ ⟩∎
fx≡fy                                                             ∎

-- A kind of symmetry for "dependent paths".

dsym :
{x≡y : x ≡ y} {P : A → Type p} {p : P x} {q : P y} →
subst P x≡y p ≡ q → subst P (sym x≡y) q ≡ p
dsym {x≡y = x≡y} {P = P} p≡q = elim
(λ {x y} x≡y →
∀ {p : P x} {q : P y} →
subst P x≡y p ≡ q →
subst P (sym x≡y) q ≡ p)
(λ _ {p q} p≡q →
subst P (sym (refl _)) q  ≡⟨ cong (flip (subst P) _) sym-refl ⟩
subst P (refl _) q        ≡⟨ subst-refl _ _ ⟩
q                         ≡⟨ sym p≡q ⟩
subst P (refl _) p        ≡⟨ subst-refl _ _ ⟩∎
p                         ∎)
x≡y
p≡q

-- A "computation rule" for dsym.

dsym-subst-refl :
{P : A → Type p} {p : P x} →
dsym (subst-refl P p) ≡
trans (cong (flip (subst P) _) sym-refl) (subst-refl _ _)
dsym-subst-refl {P = P} =
dsym (subst-refl _ _)                                      ≡⟨ cong (λ f → f (subst-refl _ _)) \$
elim-refl
(λ {x y} x≡y →
∀ {p : P x} {q : P y} →
subst P x≡y p ≡ q →
subst P (sym x≡y) q ≡ p)
_ ⟩
trans (cong (flip (subst P) _) sym-refl)
(trans (subst-refl _ _)
(trans (sym (subst-refl P _)) (subst-refl _ _)))      ≡⟨ cong (trans (cong (flip (subst P) _) sym-refl)) \$ trans--[trans-sym] _ _ ⟩∎

trans (cong (flip (subst P) _) sym-refl) (subst-refl _ _)  ∎

-- A kind of transitivity for "dependent paths".
--
-- This lemma is suggested in the HoTT book (first edition,
-- Exercise 6.1).

dtrans :
{x≡y : x ≡ y} {y≡z : y ≡ z}
(P : A → Type p) {p : P x} {q : P y} {r : P z} →
subst P x≡y p ≡ q →
subst P y≡z q ≡ r →
subst P (trans x≡y y≡z) p ≡ r
dtrans {x≡y = x≡y} {y≡z = y≡z} P {p = p} {q = q} {r = r} p≡q q≡r =
subst P (trans x≡y y≡z) p    ≡⟨ sym \$ subst-subst _ _ _ _ ⟩
subst P y≡z (subst P x≡y p)  ≡⟨ cong (subst P y≡z) p≡q ⟩
subst P y≡z q                ≡⟨ q≡r ⟩∎
r                            ∎

-- "Computation rules" for dtrans.

dtrans-reflˡ :
{x≡y : x ≡ y} {y≡z : y ≡ z}
{P : A → Type p} {p : P x} {r : P z}
{p≡r : subst P y≡z (subst P x≡y p) ≡ r} →
dtrans P (refl _) p≡r ≡
trans (sym \$ subst-subst _ _ _ _) p≡r
dtrans-reflˡ {y≡z = y≡z} {P = P} {p≡r = p≡r} =
trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst P y≡z) (refl _)) p≡r)             ≡⟨ cong (trans (sym \$ subst-subst _ _ _ _) ∘ flip trans _) \$ cong-refl _ ⟩

trans (sym \$ subst-subst _ _ _ _) (trans (refl _) p≡r)  ≡⟨ cong (trans (sym \$ subst-subst _ _ _ _)) \$ trans-reflˡ _ ⟩∎

trans (sym \$ subst-subst _ _ _ _) p≡r                   ∎

dtrans-reflʳ :
{x≡y : x ≡ y} {y≡z : y ≡ z}
{P : A → Type p} {p : P x} {q : P y}
{p≡q : subst P x≡y p ≡ q} →
dtrans P p≡q (refl (subst P y≡z q)) ≡
trans (sym \$ subst-subst _ _ _ _) (cong (subst P y≡z) p≡q)
dtrans-reflʳ {x≡y = x≡y} {y≡z = y≡z} {P = P} {p≡q = p≡q} =
trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst P y≡z) p≡q) (refl _))                 ≡⟨ cong (trans _) \$ trans-reflʳ _ ⟩∎

trans (sym \$ subst-subst _ _ _ _) (cong (subst P y≡z) p≡q)  ∎

dtrans-subst-reflˡ :
{x≡y : x ≡ y} {P : A → Type p} {p : P x} {q : P y}
{p≡q : subst P x≡y p ≡ q} →
dtrans P (subst-refl _ _) p≡q ≡
trans (cong (flip (subst P) _) (trans-reflˡ _)) p≡q
dtrans-subst-reflˡ {x≡y = x≡y} {P = P} {p≡q = p≡q} =
trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst P x≡y) (subst-refl _ _)) p≡q)              ≡⟨ cong (λ eq → trans (sym eq)
(trans (cong (subst P x≡y) (subst-refl _ _)) _)) \$
subst-subst-reflˡ _ ⟩
trans (sym \$ trans (cong (subst P _) (subst-refl _ _))
(cong (flip (subst P) _) (sym \$ trans-reflˡ _)))
(trans (cong (subst P _) (subst-refl _ _)) p≡q)                ≡⟨ cong (flip trans _) \$ sym-trans _ _ ⟩

trans (trans
(sym \$ cong (flip (subst P) _) (sym \$ trans-reflˡ _))
(sym \$ cong (subst P _) (subst-refl _ _)))
(trans (cong (subst P _) (subst-refl _ _)) p≡q)                ≡⟨ trans-assoc _ _ _ ⟩

trans (sym \$ cong (flip (subst P) _) (sym \$ trans-reflˡ _))
(trans (sym \$ cong (subst P _) (subst-refl _ _))
(trans (cong (subst P _) (subst-refl _ _)) p≡q))            ≡⟨ cong (trans _) \$ trans-sym-[trans] _ _ ⟩

trans (sym \$ cong (flip (subst P) _) (sym \$ trans-reflˡ _)) p≡q  ≡⟨ cong (flip trans _ ∘ sym) \$ cong-sym _ _ ⟩

trans (sym \$ sym \$ cong (flip (subst P) _) (trans-reflˡ _)) p≡q  ≡⟨ cong (flip trans _) \$ sym-sym _ ⟩∎

trans (cong (flip (subst P) _) (trans-reflˡ _)) p≡q              ∎

dtrans-subst-reflʳ :
{x≡y : x ≡ y} {P : A → Type p} {p : P x} {q : P y}
{p≡q : subst P x≡y p ≡ q} →
dtrans P p≡q (subst-refl _ _) ≡
trans (cong (flip (subst P) _) (trans-reflʳ _)) p≡q
dtrans-subst-reflʳ {x≡y = x≡y} {P = P} {p = p} {p≡q = p≡q} = elim¹
(λ x≡y → ∀ {q} (p≡q : subst P x≡y p ≡ q) →
dtrans P p≡q (subst-refl _ _) ≡
trans (cong (flip (subst P) _) (trans-reflʳ _)) p≡q)
(λ p≡q →
trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst P (refl _)) p≡q) (subst-refl _ _))         ≡⟨ cong (λ eq → trans (sym eq) (trans (cong (subst P (refl _)) _)
(subst-refl _ _))) \$
subst-subst-refl-refl _ ⟩
trans (sym \$ cong₂ (flip (subst P)) (subst-refl _ _) \$
sym trans-refl-refl)
(trans (cong (subst P (refl _)) p≡q) (subst-refl _ _))         ≡⟨⟩

trans (sym \$ trans (cong (subst P _) (subst-refl _ _))
(cong (flip (subst P) _) (sym trans-refl-refl)))
(trans (cong (subst P (refl _)) p≡q) (subst-refl _ _))         ≡⟨ cong (flip trans _) \$ sym-trans _ _ ⟩

trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P _) (subst-refl _ _)))
(trans (cong (subst P (refl _)) p≡q) (subst-refl _ _))         ≡⟨ lemma₁ p≡q ⟩

trans (cong (flip (subst P) _) trans-refl-refl) p≡q              ≡⟨ cong (λ eq → trans (cong (flip (subst P) _) eq) _) \$ sym \$
elim-refl _ _ ⟩∎
trans (cong (flip (subst P) _) (trans-reflʳ _)) p≡q              ∎)
x≡y
p≡q
where
lemma₂ :
cong (subst P (refl _)) (subst-refl P p) ≡
cong id (subst-refl P (subst P (refl _) p))
lemma₂ =
cong (subst P (refl _)) (subst-refl P p)     ≡⟨ cong-≡id-≡-≡id (subst-refl P) ⟩
subst-refl P (subst P (refl _) p)            ≡⟨ cong-id _ ⟩∎
cong id (subst-refl P (subst P (refl _) p))  ∎

lemma₁ :
∀ {q} (p≡q : subst P (refl _) p ≡ q) →
trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P (refl _)) (subst-refl _ _)))
(trans (cong (subst P (refl _)) p≡q) (subst-refl _ _)) ≡
trans (cong (flip (subst P) _) trans-refl-refl) p≡q
lemma₁ = elim¹
(λ p≡q →
trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P (refl _)) (subst-refl _ _)))
(trans (cong (subst P (refl _)) p≡q) (subst-refl _ _)) ≡
trans (cong (flip (subst P) _) trans-refl-refl) p≡q)
(trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P (refl _)) (subst-refl _ _)))
(trans (cong (subst P (refl _)) (refl _)) (subst-refl _ _))  ≡⟨ cong (λ eq → trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P _) (subst-refl _ _)))
(trans eq (subst-refl _ _))) \$
cong-refl (subst P (refl _)) ⟩
trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P (refl _)) (subst-refl _ _)))
(trans (refl _) (subst-refl _ _))                            ≡⟨ cong (trans _) \$ trans-reflˡ _ ⟩

trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong (subst P (refl _)) (subst-refl _ _)))
(subst-refl _ _)                                             ≡⟨ cong (λ eq → trans (trans (sym \$ cong (flip (subst P) _) _) (sym eq))
(subst-refl _ _))
lemma₂ ⟩
trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ cong id (subst-refl _ _)))
(subst-refl _ _)                                             ≡⟨ cong (λ eq → trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl)) (sym eq))
(subst-refl _ _)) \$ sym \$
cong-id _ ⟩
trans (trans (sym \$ cong (flip (subst P) _)
(sym trans-refl-refl))
(sym \$ subst-refl _ _))
(subst-refl _ _)                                             ≡⟨ trans-[trans-sym]- _ _ ⟩

sym (cong (flip (subst P) _) (sym trans-refl-refl))            ≡⟨ cong sym \$ cong-sym _ _ ⟩

sym (sym (cong (flip (subst P) _) trans-refl-refl))            ≡⟨ sym-sym _ ⟩

cong (flip (subst P) _) trans-refl-refl                        ≡⟨ sym \$ trans-reflʳ _ ⟩∎

trans (cong (flip (subst P) _) trans-refl-refl) (refl _)       ∎)

-- A lemma relating dcong, trans and dtrans.
--
-- This lemma is suggested in the HoTT book (first edition,
-- Exercise 6.1).

dcong-trans :
{f : (x : A) → P x} {x≡y : x ≡ y} {y≡z : y ≡ z} →
dcong f (trans x≡y y≡z) ≡ dtrans P (dcong f x≡y) (dcong f y≡z)
dcong-trans {P = P} {f = f} {x≡y = x≡y} {y≡z = y≡z} = elim₁
(λ x≡y → dcong f (trans x≡y y≡z) ≡ dtrans P (dcong f x≡y) (dcong f y≡z))
(dcong f (trans (refl _) y≡z)                                   ≡⟨ elim₁ (λ {p} eq → dcong f p ≡
trans (cong (flip (subst P) _) eq) (dcong f y≡z)) (
dcong f y≡z                                                     ≡⟨ sym \$ trans-reflˡ _ ⟩
trans (refl _) (dcong f y≡z)                                    ≡⟨ cong (flip trans _) \$ sym \$ cong-refl _ ⟩∎
trans (cong (flip (subst P) _) (refl _)) (dcong f y≡z)          ∎)
(trans-reflˡ _) ⟩
trans (cong (flip (subst P) _) (trans-reflˡ _)) (dcong f y≡z)  ≡⟨ sym dtrans-subst-reflˡ ⟩
dtrans P (subst-refl _ _) (dcong f y≡z)                        ≡⟨ cong (λ eq → dtrans P eq (dcong f y≡z)) \$ sym \$ dcong-refl f ⟩∎
dtrans P (dcong f (refl _)) (dcong f y≡z)                      ∎)
x≡y

-- An equality between pairs can be proved using a pair of
-- equalities.

Σ-≡,≡→≡ : {B : A → Type b} {p₁ p₂ : Σ A B} →
(p : proj₁ p₁ ≡ proj₁ p₂) →
subst B p (proj₂ p₁) ≡ proj₂ p₂ →
p₁ ≡ p₂
Σ-≡,≡→≡ {B = B} p q = elim
(λ {x₁ y₁} (p : x₁ ≡ y₁) → ∀ {x₂ y₂} →
subst B p x₂ ≡ y₂ → (x₁ , x₂) ≡ (y₁ , y₂))
(λ z₁ {x₂} {y₂} x₂≡y₂ → cong (_,_ z₁) (
x₂                    ≡⟨ sym \$ subst-refl _ _ ⟩
subst B (refl z₁) x₂  ≡⟨ x₂≡y₂ ⟩∎
y₂                    ∎))
p q

-- The uncurried form of Σ-≡,≡→≡ has an inverse, Σ-≡,≡←≡. (For a
-- proof, see Bijection.Σ-≡,≡↔≡.)

Σ-≡,≡←≡ : {B : A → Type b} {p₁ p₂ : Σ A B} →
p₁ ≡ p₂ →
∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
subst B p (proj₂ p₁) ≡ proj₂ p₂
Σ-≡,≡←≡ {A = A} {B = B} = elim
(λ {p₁ p₂ : Σ A B} _ →
∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂)
(λ p → refl _ , subst-refl _ _)

abstract

-- "Evaluation rules" for Σ-≡,≡→≡.

Σ-≡,≡→≡-reflˡ :
∀ {B : A → Type b} {y₁ y₂} →
(y₁≡y₂ : subst B (refl x) y₁ ≡ y₂) →
Σ-≡,≡→≡ (refl x) y₁≡y₂ ≡
cong (x ,_) (trans (sym \$ subst-refl _ _) y₁≡y₂)
Σ-≡,≡→≡-reflˡ {B = B} y₁≡y₂ =
cong (λ f → f y₁≡y₂) \$
elim-refl (λ {x₁ y₁} (p : x₁ ≡ y₁) → ∀ {x₂ y₂} →
subst B p x₂ ≡ y₂ → (x₁ , x₂) ≡ (y₁ , y₂))
_

Σ-≡,≡→≡-refl-refl :
∀ {B : A → Type b} {y} →
Σ-≡,≡→≡ (refl x) (refl (subst B (refl x) y)) ≡
cong (x ,_) (sym (subst-refl _ _))
Σ-≡,≡→≡-refl-refl {x = x} =
Σ-≡,≡→≡ (refl x) (refl _)                            ≡⟨ Σ-≡,≡→≡-reflˡ (refl _) ⟩
cong (x ,_) (trans (sym \$ subst-refl _ _) (refl _))  ≡⟨ cong (cong (x ,_)) (trans-reflʳ _) ⟩∎
cong (x ,_) (sym (subst-refl _ _))                   ∎

Σ-≡,≡→≡-refl-subst-refl :
{B : A → Type b} {p : Σ A B} →
Σ-≡,≡→≡ (refl _) (subst-refl _ _) ≡ refl p
Σ-≡,≡→≡-refl-subst-refl {B = B} =
Σ-≡,≡→≡ (refl _) (subst-refl B _)                            ≡⟨ Σ-≡,≡→≡-reflˡ _ ⟩
cong (_ ,_) (trans (sym \$ subst-refl _ _) (subst-refl _ _))  ≡⟨ cong (cong _) (trans-symˡ _) ⟩
cong (_ ,_) (refl _)                                         ≡⟨ cong-refl _ ⟩∎
refl _                                                       ∎

Σ-≡,≡→≡-refl-subst-const :
{p : A × B} →
Σ-≡,≡→≡ (refl _) (subst-const _) ≡ refl p
Σ-≡,≡→≡-refl-subst-const =
Σ-≡,≡→≡ (refl _) (subst-const _)                            ≡⟨ Σ-≡,≡→≡-reflˡ _ ⟩
cong (_ ,_) (trans (sym \$ subst-refl _ _) (subst-const _))  ≡⟨ cong (cong _) subst-refl-subst-const ⟩
cong (_ ,_) (refl _)                                        ≡⟨ cong-refl _ ⟩∎
refl _                                                      ∎

-- "Evaluation rule" for Σ-≡,≡←≡.

Σ-≡,≡←≡-refl :
{B : A → Type b} {p : Σ A B} →
Σ-≡,≡←≡ (refl p) ≡ (refl _ , subst-refl _ _)
Σ-≡,≡←≡-refl = elim-refl _ _

-- Proof transformation rules for Σ-≡,≡→≡.

proj₁-Σ-≡,≡→≡ :
∀ {B : A → Type b} {y₁ y₂}
(x₁≡x₂ : x₁ ≡ x₂) (y₁≡y₂ : subst B x₁≡x₂ y₁ ≡ y₂) →
cong proj₁ (Σ-≡,≡→≡ x₁≡x₂ y₁≡y₂) ≡ x₁≡x₂
proj₁-Σ-≡,≡→≡ {B = B} {y₁ = y₁} x₁≡x₂ y₁≡y₂ = elim¹
(λ x₁≡x₂ → ∀ {y₂} (y₁≡y₂ : subst B x₁≡x₂ y₁ ≡ y₂) →
cong proj₁ (Σ-≡,≡→≡ x₁≡x₂ y₁≡y₂) ≡ x₁≡x₂)
(λ y₁≡y₂ →
cong proj₁ (Σ-≡,≡→≡ (refl _) y₁≡y₂)                             ≡⟨ cong (cong proj₁) \$ Σ-≡,≡→≡-reflˡ _ ⟩
cong proj₁ (cong (_,_ _) (trans (sym \$ subst-refl _ _) y₁≡y₂))  ≡⟨ cong-∘ _ (_,_ _) _ ⟩
cong (const _) (trans (sym \$ subst-refl _ _) y₁≡y₂)             ≡⟨ cong-const _ ⟩∎
refl _                                                          ∎)
x₁≡x₂ y₁≡y₂

Σ-≡,≡→≡-cong :
{B : A → Type b} {p₁ p₂ : Σ A B}
{q₁ q₂ : proj₁ p₁ ≡ proj₁ p₂}
(q₁≡q₂ : q₁ ≡ q₂)
{r₁ : subst B q₁ (proj₂ p₁) ≡ proj₂ p₂}
{r₂ : subst B q₂ (proj₂ p₁) ≡ proj₂ p₂}
(r₁≡r₂ : (subst B q₂ (proj₂ p₁)  ≡⟨ cong (flip (subst B) _) (sym q₁≡q₂) ⟩
subst B q₁ (proj₂ p₁)  ≡⟨ r₁ ⟩∎
proj₂ p₂               ∎)
≡
r₂) →
Σ-≡,≡→≡ q₁ r₁ ≡ Σ-≡,≡→≡ q₂ r₂
Σ-≡,≡→≡-cong {B = B} = elim
(λ {q₁ q₂} q₁≡q₂ →
∀ {r₁ r₂}
(r₁≡r₂ : trans (cong (flip (subst B) _) (sym q₁≡q₂)) r₁ ≡ r₂) →
Σ-≡,≡→≡ q₁ r₁ ≡ Σ-≡,≡→≡ q₂ r₂)
(λ q {r₁ r₂} r₁≡r₂ → cong (Σ-≡,≡→≡ q) (
r₁                                                 ≡⟨ sym \$ trans-reflˡ _ ⟩
trans (refl (subst B q _)) r₁                      ≡⟨ cong (flip trans _) \$ sym \$ cong-refl _ ⟩
trans (cong (flip (subst B) _) (refl q)) r₁        ≡⟨ cong (λ e → trans (cong (flip (subst B) _) e) _) \$ sym sym-refl ⟩
trans (cong (flip (subst B) _) (sym (refl q))) r₁  ≡⟨ r₁≡r₂ ⟩∎
r₂                                                 ∎))

trans-Σ-≡,≡→≡ :
{B : A → Type b} {p₁ p₂ p₃ : Σ A B} →
(q₁₂ : proj₁ p₁ ≡ proj₁ p₂) (q₂₃ : proj₁ p₂ ≡ proj₁ p₃)
(r₁₂ : subst B q₁₂ (proj₂ p₁) ≡ proj₂ p₂)
(r₂₃ : subst B q₂₃ (proj₂ p₂) ≡ proj₂ p₃) →
trans (Σ-≡,≡→≡ q₁₂ r₁₂) (Σ-≡,≡→≡ q₂₃ r₂₃) ≡
Σ-≡,≡→≡ (trans q₁₂ q₂₃)
(subst B (trans q₁₂ q₂₃) (proj₂ p₁)    ≡⟨ sym \$ subst-subst _ _ _ _ ⟩
subst B q₂₃ (subst B q₁₂ (proj₂ p₁))  ≡⟨ cong (subst _ _) r₁₂ ⟩
subst B q₂₃ (proj₂ p₂)                ≡⟨ r₂₃ ⟩∎
proj₂ p₃                              ∎)
trans-Σ-≡,≡→≡ {B = B} q₁₂ q₂₃ r₁₂ r₂₃ = elim
(λ {p₂₁ p₃₁} q₂₃ → ∀ {p₁₁} (q₁₂ : p₁₁ ≡ p₂₁)
{p₁₂ p₂₂} (r₁₂ : subst B q₁₂ p₁₂ ≡ p₂₂)
{p₃₂} (r₂₃ : subst B q₂₃ p₂₂ ≡ p₃₂) →
trans (Σ-≡,≡→≡ q₁₂ r₁₂) (Σ-≡,≡→≡ q₂₃ r₂₃) ≡
Σ-≡,≡→≡ (trans q₁₂ q₂₃)
(trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst _ _) r₁₂) r₂₃)))
(λ x → elim₁
(λ q₁₂ →
∀ {p₁₂ p₂₂} (r₁₂ : subst B q₁₂ p₁₂ ≡ p₂₂)
{p₃₂} (r₂₃ : subst B (refl _) p₂₂ ≡ p₃₂) →
trans (Σ-≡,≡→≡ q₁₂ r₁₂) (Σ-≡,≡→≡ (refl _) r₂₃) ≡
Σ-≡,≡→≡ (trans q₁₂ (refl _))
(trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst _ _) r₁₂) r₂₃)))
(λ {y} → elim¹
(λ {p₂₂} r₁₂ →
∀ {p₃₂} (r₂₃ : subst B (refl _) p₂₂ ≡ p₃₂) →
trans (Σ-≡,≡→≡ (refl _) r₁₂) (Σ-≡,≡→≡ (refl _) r₂₃) ≡
Σ-≡,≡→≡ (trans (refl _) (refl _))
(trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst _ _) r₁₂) r₂₃)))
(elim¹
(λ r₂₃ →
trans (Σ-≡,≡→≡ (refl _) (refl _))
(Σ-≡,≡→≡ (refl _) r₂₃) ≡
Σ-≡,≡→≡ (trans (refl _) (refl _))
(trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst _ _) (refl _))
r₂₃)))
(let lemma₁ =
sym (cong (subst B _) (subst-refl _ _))          ≡⟨ sym \$ trans-sym-[trans] _ _ ⟩

trans (sym \$ cong (flip (subst B) _)
trans-refl-refl)
(trans (cong (flip (subst B) _)
trans-refl-refl)
(sym (cong (subst B _)
(subst-refl _ _))))      ≡⟨ cong (flip trans _) \$ sym \$ cong-sym _ _ ⟩∎

trans (cong (flip (subst B) _)
(sym trans-refl-refl))
(trans (cong (flip (subst B) _)
trans-refl-refl)
(sym (cong (subst B _)
(subst-refl _ _))))      ∎

lemma₂ =
trans (cong (flip (subst B) _) trans-refl-refl)
(sym (cong (subst B _) (subst-refl _ _)))     ≡⟨ cong (λ e → trans (cong (flip (subst B) _) e)
(sym \$ cong (subst B _) (subst-refl _ _))) \$
sym \$ sym-sym _ ⟩
trans (cong (flip (subst B) _)
(sym \$ sym trans-refl-refl))
(sym (cong (subst B _) (subst-refl _ _)))     ≡⟨ cong (flip trans _) \$ cong-sym _ _ ⟩

trans (sym (cong (flip (subst B) _)
(sym trans-refl-refl)))
(sym (cong (subst B _) (subst-refl _ _)))     ≡⟨ sym \$ sym-trans _ _ ⟩

sym (trans (cong (subst B _) (subst-refl _ _))
(cong (flip (subst B) _)
(sym trans-refl-refl)))            ≡⟨⟩

sym (cong₂ (flip (subst B)) (subst-refl _ _)
(sym trans-refl-refl))  ≡⟨ cong sym \$ sym \$ subst-subst-refl-refl _ ⟩

sym (subst-subst _ _ _ _)                           ≡⟨ sym \$ trans-reflʳ _ ⟩

trans (sym \$ subst-subst _ _ _ _) (refl _)          ≡⟨ cong (trans (sym \$ subst-subst _ _ _ _)) \$ sym trans-refl-refl ⟩

trans (sym \$ subst-subst _ _ _ _)
(trans (refl _) (refl _))                     ≡⟨ cong (λ x → trans (sym \$ subst-subst _ _ _ _) (trans x (refl _))) \$
sym \$ cong-refl _ ⟩∎
trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst _ _) (refl _)) (refl _))  ∎
in
trans (Σ-≡,≡→≡ (refl _) (refl _))
(Σ-≡,≡→≡ (refl _) (refl _))                          ≡⟨ cong₂ trans Σ-≡,≡→≡-refl-refl Σ-≡,≡→≡-refl-refl ⟩

trans (cong (_ ,_) (sym (subst-refl _ _)))
(cong (_ ,_) (sym (subst-refl B _)))                 ≡⟨ sym \$ cong-trans _ _ _ ⟩

cong (_ ,_) (trans (sym (subst-refl _ _))
(sym (subst-refl _ _)))                 ≡⟨ cong (cong (_ ,_) ∘ trans (sym (subst-refl _ _)) ∘ sym) \$ sym \$
cong-≡id-≡-≡id (subst-refl B) ⟩
cong (_ ,_)
(trans (sym (subst-refl _ _))
(sym (cong (subst B _) (subst-refl _ _))))     ≡⟨ sym \$ Σ-≡,≡→≡-reflˡ _ ⟩

Σ-≡,≡→≡ (refl _)
(sym (cong (subst B _) (subst-refl _ _)))          ≡⟨ cong (Σ-≡,≡→≡ _) lemma₁ ⟩

Σ-≡,≡→≡ (refl _)
(trans (cong (flip (subst B) _) (sym trans-refl-refl))
(trans (cong (flip (subst B) _) trans-refl-refl)
(sym (cong (subst B _) (subst-refl _ _)))))        ≡⟨ sym \$ Σ-≡,≡→≡-cong _ (refl _) ⟩

Σ-≡,≡→≡ (trans (refl _) (refl _))
(trans (cong (flip (subst B) _) trans-refl-refl)
(sym (cong (subst B _) (subst-refl _ _))))  ≡⟨ cong (Σ-≡,≡→≡ (trans (refl _) (refl _))) lemma₂ ⟩∎

Σ-≡,≡→≡ (trans (refl _) (refl _))
(trans (sym \$ subst-subst _ _ _ _)
(trans (cong (subst _ _) (refl _))
(refl _)))                           ∎))))
q₂₃ q₁₂ r₁₂ r₂₃

Σ-≡,≡→≡-subst-const :
{p₁ p₂ : A × B} →
(p : proj₁ p₁ ≡ proj₁ p₂) (q : proj₂ p₁ ≡ proj₂ p₂) →
Σ-≡,≡→≡ p (trans (subst-const _) q) ≡ cong₂ _,_ p q
Σ-≡,≡→≡-subst-const p q = elim
(λ {x₁ y₁} (p : x₁ ≡ y₁) →
Σ-≡,≡→≡ p (trans (subst-const _) q) ≡ cong₂ _,_ p q)
(λ x →
let lemma =
trans (sym \$ subst-refl _ _) (trans (subst-const _) q)  ≡⟨ sym \$ trans-assoc _ _ _ ⟩
trans (trans (sym \$ subst-refl _ _) (subst-const _)) q  ≡⟨ cong₂ trans subst-refl-subst-const (refl _) ⟩
trans (refl _) q                                        ≡⟨ trans-reflˡ _ ⟩∎
q                                                       ∎ in

Σ-≡,≡→≡ (refl x) (trans (subst-const _) q)     ≡⟨ Σ-≡,≡→≡-reflˡ _ ⟩
cong (x ,_) (trans (sym \$ subst-refl _ _)
(trans (subst-const _) q))  ≡⟨ cong (cong (x ,_)) lemma ⟩
cong (x ,_) q                                  ≡⟨ sym \$ cong₂-reflˡ _,_ ⟩∎
cong₂ _,_ (refl x) q                           ∎)
p

Σ-≡,≡→≡-subst-const-refl :
Σ-≡,≡→≡ x₁≡x₂ (subst-const _) ≡ cong₂ _,_ x₁≡x₂ (refl y)
Σ-≡,≡→≡-subst-const-refl {x₁≡x₂ = x₁≡x₂} {y = y} =
Σ-≡,≡→≡ x₁≡x₂ (subst-const _)                   ≡⟨ cong (Σ-≡,≡→≡ x₁≡x₂) \$ sym \$ trans-reflʳ _ ⟩
Σ-≡,≡→≡ x₁≡x₂ (trans (subst-const _) (refl _))  ≡⟨ Σ-≡,≡→≡-subst-const _ _ ⟩∎
cong₂ _,_ x₁≡x₂ (refl y)                        ∎

-- Proof simplification rule for Σ-≡,≡←≡.

proj₁-Σ-≡,≡←≡ :
{B : A → Type b} {p₁ p₂ : Σ A B}
(p₁≡p₂ : p₁ ≡ p₂) →
proj₁ (Σ-≡,≡←≡ p₁≡p₂) ≡ cong proj₁ p₁≡p₂
proj₁-Σ-≡,≡←≡ = elim
(λ p₁≡p₂ → proj₁ (Σ-≡,≡←≡ p₁≡p₂) ≡ cong proj₁ p₁≡p₂)
(λ p →
proj₁ (Σ-≡,≡←≡ (refl p))  ≡⟨ cong proj₁ \$ Σ-≡,≡←≡-refl ⟩
refl (proj₁ p)            ≡⟨ sym \$ cong-refl _ ⟩∎
cong proj₁ (refl p)       ∎)

-- A binary variant of subst.

subst₂ : ∀ {B : A → Type b} (P : Σ A B → Type p) {x₁ x₂ y₁ y₂} →
(x₁≡x₂ : x₁ ≡ x₂) → subst B x₁≡x₂ y₁ ≡ y₂ →
P (x₁ , y₁) → P (x₂ , y₂)
subst₂ P x₁≡x₂ y₁≡y₂ = subst P (Σ-≡,≡→≡ x₁≡x₂ y₁≡y₂)

abstract

-- "Evaluation rule" for subst₂.

subst₂-refl-refl :
∀ {B : A → Type b} (P : Σ A B → Type p) {y p} →
subst₂ P (refl _) (refl _) p ≡
subst (curry P x) (sym \$ subst-refl B y) p
subst₂-refl-refl {x = x} P {p = p} =
subst P (Σ-≡,≡→≡ (refl _) (refl _)) p           ≡⟨ cong (λ eq₁ → subst P eq₁ _) Σ-≡,≡→≡-refl-refl ⟩
subst P (cong (x ,_) (sym (subst-refl _ _))) p  ≡⟨ sym \$ subst-∘ _ _ _ ⟩∎
subst (curry P x) (sym \$ subst-refl _ _) p      ∎

-- The subst function can be "pushed" inside pairs.

push-subst-pair :
∀ (B : A → Type b) (C : Σ A B → Type c) {p} →
subst (λ x → Σ (B x) (curry C x)) y≡z p ≡
(subst B y≡z (proj₁ p) , subst₂ C y≡z (refl _) (proj₂ p))
push-subst-pair {y≡z = y≡z} B C {p} = elim¹
(λ y≡z →
subst (λ x → Σ (B x) (curry C x)) y≡z p ≡
(subst B y≡z (proj₁ p) , subst₂ C y≡z (refl _) (proj₂ p)))
(subst (λ x → Σ (B x) (curry C x)) (refl _) p  ≡⟨ subst-refl _ _ ⟩
p                                             ≡⟨ Σ-≡,≡→≡ (sym (subst-refl _ _)) (sym (subst₂-refl-refl _)) ⟩∎
(subst B (refl _) (proj₁ p) ,
subst₂ C (refl _) (refl _) (proj₂ p))        ∎)
y≡z

-- A proof transformation rule for push-subst-pair.

proj₁-push-subst-pair-refl :
∀ {A : Type a} {y : A} (B : A → Type b) (C : Σ A B → Type c) {p} →
cong proj₁ (push-subst-pair {y≡z = refl y} B C {p = p}) ≡
trans (cong proj₁ (subst-refl (λ _ → Σ _ _) _))
(sym \$ subst-refl _ _)
proj₁-push-subst-pair-refl B C =
cong proj₁ (push-subst-pair _ _)                                   ≡⟨ cong (cong proj₁) \$
elim¹-refl
(λ y≡z →
subst (λ x → Σ (B x) (curry C x)) y≡z _ ≡
(subst B y≡z _ , subst₂ C y≡z (refl _) _))
_ ⟩
cong proj₁
(trans (subst-refl (λ _ → Σ _ _) _)
(Σ-≡,≡→≡ (sym \$ subst-refl B _) (sym (subst₂-refl-refl _))))  ≡⟨ cong-trans _ _ _ ⟩

trans (cong proj₁ (subst-refl _ _))
(cong proj₁
(Σ-≡,≡→≡ (sym \$ subst-refl _ _) (sym (subst₂-refl-refl _))))  ≡⟨ cong (trans _) \$
proj₁-Σ-≡,≡→≡ _ _ ⟩∎
trans (cong proj₁ (subst-refl _ _)) (sym \$ subst-refl _ _)         ∎

-- Corollaries of push-subst-pair.

push-subst-pair′ :
∀ (B : A → Type b) (C : Σ A B → Type c) {p p₁} →
(p₁≡p₁ : subst B y≡z (proj₁ p) ≡ p₁) →
subst (λ x → Σ (B x) (curry C x)) y≡z p ≡
(p₁ , subst₂ C y≡z p₁≡p₁ (proj₂ p))
push-subst-pair′ {y≡z = y≡z} B C {p} =
elim¹ (λ {p₁} p₁≡p₁ →
subst (λ x → Σ (B x) (curry C x)) y≡z p ≡
(p₁ , subst₂ C y≡z p₁≡p₁ (proj₂ p)))
(push-subst-pair _ _)

push-subst-pair-× :
∀ {y≡z : y ≡ z} (B : Type b) (C : A × B → Type c) {p} →
subst (λ x → Σ B (curry C x)) y≡z p ≡
(proj₁ p , subst (λ x → C (x , proj₁ p)) y≡z (proj₂ p))
push-subst-pair-× {y≡z = y≡z} B C {p} =
subst (λ x → Σ B (curry C x)) y≡z p                       ≡⟨ push-subst-pair′ _ C (subst-const _) ⟩
(proj₁ p , subst₂ C y≡z (subst-const _) (proj₂ p))        ≡⟨ cong (_ ,_) \$
elim¹
(λ y≡z → subst₂ C y≡z (subst-const y≡z) (proj₂ p) ≡
subst (λ x → C (x , proj₁ p)) y≡z (proj₂ p))
(
subst₂ C (refl _) (subst-const _) (proj₂ p)                 ≡⟨⟩
subst C (Σ-≡,≡→≡ (refl _) (subst-const _)) (proj₂ p)        ≡⟨ cong (λ eq → subst C eq _) Σ-≡,≡→≡-refl-subst-const ⟩
subst C (refl _) (proj₂ p)                                  ≡⟨ subst-refl _ _ ⟩
proj₂ p                                                     ≡⟨ sym \$ subst-refl _ _ ⟩∎
subst (λ x → C (x , _)) (refl _) (proj₂ p)                  ∎)
y≡z ⟩
(proj₁ p , subst (λ x → C (x , _)) y≡z (proj₂ p))         ∎

-- A proof simplification rule for subst₂.

subst₂-proj₁ :
∀ {B : A → Type b} {y₁ y₂}
{x₁≡x₂ : x₁ ≡ x₂} {y₁≡y₂ : subst B x₁≡x₂ y₁ ≡ y₂}
(P : A → Type p) {p} →
subst₂ {B = B} (P ∘ proj₁) x₁≡x₂ y₁≡y₂ p ≡ subst P x₁≡x₂ p
subst₂-proj₁ {x₁≡x₂ = x₁≡x₂} {y₁≡y₂} P {p} =
subst₂ (P ∘ proj₁) x₁≡x₂ y₁≡y₂ p              ≡⟨ subst-∘ _ _ _ ⟩
subst P (cong proj₁ (Σ-≡,≡→≡ x₁≡x₂ y₁≡y₂)) p  ≡⟨ cong (λ eq → subst P eq _) (proj₁-Σ-≡,≡→≡ _ _) ⟩∎
subst P x₁≡x₂ p                               ∎

-- The subst function can be "pushed" inside non-dependent pairs.

push-subst-, :
∀ (B : A → Type b) (C : A → Type c) {p} →
subst (λ x → B x × C x) y≡z p ≡
(subst B y≡z (proj₁ p) , subst C y≡z (proj₂ p))
push-subst-, {y≡z = y≡z} B C {x , y} =
subst (λ x → B x × C x) y≡z (x , y)                           ≡⟨ push-subst-pair _ _ ⟩
(subst B y≡z x , subst (C ∘ proj₁) (Σ-≡,≡→≡ y≡z (refl _)) y)  ≡⟨ cong (_,_ _) \$ subst₂-proj₁ _ ⟩∎
(subst B y≡z x , subst C y≡z y)                               ∎

-- A proof transformation rule for push-subst-,.

proj₁-push-subst-,-refl :
∀ {A : Type a} {y : A} (B : A → Type b) (C : A → Type c) {p} →
cong proj₁ (push-subst-, {y≡z = refl y} B C {p = p}) ≡
trans (cong proj₁ (subst-refl (λ _ → _ × _) _))
(sym \$ subst-refl _ _)
proj₁-push-subst-,-refl _ _ =
cong proj₁ (trans (push-subst-pair _ _)
(cong (_,_ _) \$ subst₂-proj₁ _))              ≡⟨ cong-trans _ _ _ ⟩

trans (cong proj₁ (push-subst-pair _ _))
(cong proj₁ (cong (_,_ _) \$ subst₂-proj₁ _))              ≡⟨ cong (trans _) \$
cong-∘ _ _ _ ⟩
trans (cong proj₁ (push-subst-pair _ _))
(cong (const _) \$ subst₂-proj₁ _)                         ≡⟨ trans (cong (trans _) (cong-const _)) \$
trans-reflʳ _ ⟩

cong proj₁ (push-subst-pair _ _)                            ≡⟨ proj₁-push-subst-pair-refl _ _ ⟩∎

trans (cong proj₁ (subst-refl _ _)) (sym \$ subst-refl _ _)  ∎

-- The subst function can be "pushed" inside inj₁ and inj₂.

push-subst-inj₁ :
∀ (B : A → Type b) (C : A → Type c) {x} →
subst (λ x → B x ⊎ C x) y≡z (inj₁ x) ≡ inj₁ (subst B y≡z x)
push-subst-inj₁ {y≡z = y≡z} B C {x} = elim¹
(λ y≡z → subst (λ x → B x ⊎ C x) y≡z (inj₁ x) ≡
inj₁ (subst B y≡z x))
(subst (λ x → B x ⊎ C x) (refl _) (inj₁ x)  ≡⟨ subst-refl _ _ ⟩
inj₁ x                                     ≡⟨ cong inj₁ \$ sym \$ subst-refl _ _ ⟩∎
inj₁ (subst B (refl _) x)                  ∎)
y≡z

push-subst-inj₂ :
∀ (B : A → Type b) (C : A → Type c) {x} →
subst (λ x → B x ⊎ C x) y≡z (inj₂ x) ≡ inj₂ (subst C y≡z x)
push-subst-inj₂ {y≡z = y≡z} B C {x} = elim¹
(λ y≡z → subst (λ x → B x ⊎ C x) y≡z (inj₂ x) ≡
inj₂ (subst C y≡z x))
(subst (λ x → B x ⊎ C x) (refl _) (inj₂ x)  ≡⟨ subst-refl _ _ ⟩
inj₂ x                                     ≡⟨ cong inj₂ \$ sym \$ subst-refl _ _ ⟩∎
inj₂ (subst C (refl _) x)                  ∎)
y≡z

-- The subst function can be "pushed" inside applications.

push-subst-application :
{B : A → Type b}
(x₁≡x₂ : x₁ ≡ x₂) (C : (x : A) → B x → Type c)
{f : (x : A) → B x} {g : (y : B x₁) → C x₁ y} →
subst (λ x → C x (f x)) x₁≡x₂ (g (f x₁)) ≡
subst (λ x → (y : B x) → C x y) x₁≡x₂ g (f x₂)
push-subst-application {x₁ = x₁} x₁≡x₂ C {f} {g} = elim¹
(λ {x₂} x₁≡x₂ →
subst (λ x → C x (f x)) x₁≡x₂ (g (f x₁)) ≡
subst (λ x → ∀ y → C x y) x₁≡x₂ g (f x₂))
(subst (λ x → C x (f x)) (refl _) (g (f x₁))  ≡⟨ subst-refl _ _ ⟩
g (f x₁)                                     ≡⟨ cong (_\$ f x₁) \$ sym \$ subst-refl (λ x → ∀ y → C x y) _ ⟩∎
subst (λ x → ∀ y → C x y) (refl _) g (f x₁)  ∎)
x₁≡x₂

push-subst-implicit-application :
{B : A → Type b}
(x₁≡x₂ : x₁ ≡ x₂) (C : (x : A) → B x → Type c)
{f : (x : A) → B x} {g : {y : B x₁} → C x₁ y} →
subst (λ x → C x (f x)) x₁≡x₂ (g {y = f x₁}) ≡
subst (λ x → {y : B x} → C x y) x₁≡x₂ g {y = f x₂}
push-subst-implicit-application {x₁ = x₁} x₁≡x₂ C {f} {g} = elim¹
(λ {x₂} x₁≡x₂ →
subst (λ x → C x (f x)) x₁≡x₂ (g {y = f x₁}) ≡
subst (λ x → ∀ {y} → C x y) x₁≡x₂ g {y = f x₂})
(subst (λ x → C x (f x)) (refl _) (g {y = f x₁})    ≡⟨ subst-refl _ _ ⟩
g {y = f x₁}                                       ≡⟨ cong (λ g → g {y = f x₁}) \$ sym \$ subst-refl (λ x → ∀ {y} → C x y) _ ⟩∎
subst (λ x → ∀ {y} → C x y) (refl _) g {y = f x₁}  ∎)
x₁≡x₂

subst-∀-sym :
∀ {B : A → Type b} {y : B x₁}
{C : (x : A) → B x → Type c} {f : (y : B x₂) → C x₂ y}
{x₁≡x₂ : x₁ ≡ x₂} →
subst (λ x → (y : B x) → C x y) (sym x₁≡x₂) f y ≡
subst (uncurry C) (sym \$ Σ-≡,≡→≡ x₁≡x₂ (refl _))
(f (subst B x₁≡x₂ y))
subst-∀-sym {B = B} {C = C} {x₁≡x₂ = x₁≡x₂} = elim
(λ {x₁ x₂} x₁≡x₂ →
{y : B x₁} (f : (y : B x₂) → C x₂ y) →
subst (λ x → (y : B x) → C x y) (sym x₁≡x₂) f y ≡
subst (uncurry C) (sym \$ Σ-≡,≡→≡ x₁≡x₂ (refl _))
(f (subst B x₁≡x₂ y)))
(λ x {y} f →
let lemma =
cong (x ,_) (subst-refl B y)              ≡⟨ cong (cong (x ,_)) \$ sym \$ sym-sym _ ⟩
cong (x ,_) (sym \$ sym \$ subst-refl B y)  ≡⟨ cong-sym _ _ ⟩
sym \$ cong (x ,_) (sym \$ subst-refl B y)  ≡⟨ cong sym \$ sym Σ-≡,≡→≡-refl-refl ⟩∎
sym \$ Σ-≡,≡→≡ (refl x) (refl _)           ∎
in
subst (λ x → (y : B x) → C x y) (sym (refl x)) f y     ≡⟨ cong (λ eq → subst (λ x → (y : B x) → C x y) eq _ _) sym-refl ⟩

subst (λ x → (y : B x) → C x y) (refl x) f y           ≡⟨ cong (_\$ y) \$ subst-refl (λ x → (_ : B x) → _) _ ⟩

f y                                                    ≡⟨ sym \$ dcong f _ ⟩

subst (C x) (subst-refl B _) (f (subst B (refl x) y))  ≡⟨ subst-∘ _ _ _ ⟩

subst (uncurry C) (cong (x ,_) (subst-refl B y))
(f (subst B (refl x) y))                             ≡⟨ cong (λ eq → subst (uncurry C) eq (f (subst B (refl x) y))) lemma ⟩∎

subst (uncurry C) (sym \$ Σ-≡,≡→≡ (refl x) (refl _))
(f (subst B (refl x) y))                             ∎)
x₁≡x₂ _

subst-∀ :
∀ {B : A → Type b} {y : B x₂}
{C : (x : A) → B x → Type c} {f : (y : B x₁) → C x₁ y}
{x₁≡x₂ : x₁ ≡ x₂} →
subst (λ x → (y : B x) → C x y) x₁≡x₂ f y ≡
subst (uncurry C) (sym \$ Σ-≡,≡→≡ (sym x₁≡x₂) (refl _))
(f (subst B (sym x₁≡x₂) y))
subst-∀ {B = B} {y = y} {C = C} {f = f} {x₁≡x₂ = x₁≡x₂} =
subst (λ x → (y : B x) → C x y) x₁≡x₂ f y               ≡⟨ cong (λ eq → subst (λ x → (y : B x) → C x y) eq _ _) \$ sym \$ sym-sym _ ⟩
subst (λ x → (y : B x) → C x y) (sym (sym x₁≡x₂)) f y   ≡⟨ subst-∀-sym ⟩∎
subst (uncurry C) (sym \$ Σ-≡,≡→≡ (sym x₁≡x₂) (refl _))
(f (subst B (sym x₁≡x₂) y))                           ∎

subst-→ :
{B : A → Type b} {y : B x₂} {C : A → Type c} {f : B x₁ → C x₁} →
subst (λ x → B x → C x) x₁≡x₂ f y ≡
subst C x₁≡x₂ (f (subst B (sym x₁≡x₂) y))
subst-→ {x₁≡x₂ = x₁≡x₂} {B = B} {y = y} {C} {f} =
subst (λ x → B x → C x) x₁≡x₂ f y                          ≡⟨ cong (λ eq → subst (λ x → B x → C x) eq f y) \$ sym \$
sym-sym _ ⟩
subst (λ x → B x → C x) (sym \$ sym x₁≡x₂) f y              ≡⟨ subst-∀-sym ⟩

subst (C ∘ proj₁) (sym \$ Σ-≡,≡→≡ (sym x₁≡x₂) (refl _))
(f (subst B (sym x₁≡x₂) y))                              ≡⟨ subst-∘ _ _ _ ⟩

subst C (cong proj₁ \$ sym \$ Σ-≡,≡→≡ (sym x₁≡x₂) (refl _))
(f (subst B (sym x₁≡x₂) y))                              ≡⟨ cong (λ eq → subst C eq (f (subst B (sym x₁≡x₂) y))) \$
cong-sym _ _ ⟩
subst C (sym \$ cong proj₁ \$ Σ-≡,≡→≡ (sym x₁≡x₂) (refl _))
(f (subst B (sym x₁≡x₂) y))                              ≡⟨ cong (λ eq → subst C (sym eq) (f (subst B (sym x₁≡x₂) y))) \$
proj₁-Σ-≡,≡→≡ _ _ ⟩
subst C (sym \$ sym x₁≡x₂) (f (subst B (sym x₁≡x₂) y))      ≡⟨ cong (λ eq → subst C eq (f (subst B (sym x₁≡x₂) y))) \$
sym-sym _ ⟩∎
subst C x₁≡x₂ (f (subst B (sym x₁≡x₂) y))                  ∎

subst-→-domain :
(B : A → Type b) {f : B x → C} (x≡y : x ≡ y) {u : B y} →
subst (λ x → B x → C) x≡y f u ≡ f (subst B (sym x≡y) u)
subst-→-domain {C = C} B x≡y {u = u} = elim₁
(λ {x} x≡y → (f : B x → C) →
subst (λ x → B x → C) x≡y f u ≡
f (subst B (sym x≡y) u))
(λ f →
subst (λ x → B x → C) (refl _) f u  ≡⟨ cong (_\$ u) \$ subst-refl (λ x → B x → _) _ ⟩
f u                                 ≡⟨ cong f \$ sym \$ subst-refl _ _ ⟩
f (subst B (refl _) u)              ≡⟨ cong (λ p → f (subst B p u)) \$ sym sym-refl ⟩∎
f (subst B (sym (refl _)) u)        ∎)
x≡y _

-- A "computation rule".

subst-→-domain-refl :
{B : A → Type b} {f : B x → C} {u : B x} →
subst-→-domain B {f = f} (refl x) {u = u} ≡
trans (cong (_\$ u) (subst-refl (λ x → B x → _) _))
(trans (cong f (sym (subst-refl _ _)))
(cong (f ∘ flip (subst B) u) (sym sym-refl)))
subst-→-domain-refl {C = C} {B = B} {u = u} =
cong (_\$ _) \$
elim₁-refl
(λ {x} x≡y → (f : B x → C) →
subst (λ x → B x → C) x≡y f u ≡
f (subst B (sym x≡y) u))
_

-- The following lemma is Proposition 2 from "Generalizations of
-- Hedberg's Theorem" by Kraus, Escardó, Coquand and Altenkirch.

subst-in-terms-of-trans-and-cong :
{x≡y : x ≡ y} {fx≡gx : f x ≡ g x} →
subst (λ z → f z ≡ g z) x≡y fx≡gx ≡
trans (sym (cong f x≡y)) (trans fx≡gx (cong g x≡y))
subst-in-terms-of-trans-and-cong {f = f} {g = g} = elim
(λ {x y} x≡y →
(fx≡gx : f x ≡ g x) →
subst (λ z → f z ≡ g z) x≡y fx≡gx ≡
trans (sym (cong f x≡y)) (trans fx≡gx (cong g x≡y)))
(λ x fx≡gx →
subst (λ z → f z ≡ g z) (refl x) fx≡gx                         ≡⟨ subst-refl _ _ ⟩
fx≡gx                                                          ≡⟨ sym \$ trans-reflˡ _ ⟩
trans (refl (f x)) fx≡gx                                       ≡⟨ sym \$ cong₂ trans sym-refl (trans-reflʳ _)  ⟩
trans (sym (refl (f x))) (trans fx≡gx (refl (g x)))            ≡⟨ sym \$ cong₂ (λ p q → trans (sym p) (trans _ q))
(cong-refl _) (cong-refl _) ⟩∎
trans (sym (cong f (refl x))) (trans fx≡gx (cong g (refl x)))  ∎ )
_
_

-- A variant of subst-in-terms-of-trans-and-cong that works for
-- dependent functions.

subst-in-terms-of-trans-and-dcong :
{f g : (x : A) → P x} {x≡y : x ≡ y} {fx≡gx : f x ≡ g x} →
subst (λ z → f z ≡ g z) x≡y fx≡gx ≡
trans (sym (dcong f x≡y))
(trans (cong (subst P x≡y) fx≡gx) (dcong g x≡y))
subst-in-terms-of-trans-and-dcong {P = P} {f = f} {g = g} = elim
(λ {x y} x≡y →
(fx≡gx : f x ≡ g x) →
subst (λ z → f z ≡ g z) x≡y fx≡gx ≡
trans (sym (dcong f x≡y))
(trans (cong (subst P x≡y) fx≡gx) (dcong g x≡y)))
(λ x fx≡gx →
subst (λ z → f z ≡ g z) (refl x) fx≡gx               ≡⟨ subst-refl _ _ ⟩

fx≡gx                                                ≡⟨ elim¹
(λ {gx} eq →
eq ≡
trans (sym (subst-refl P (f x)))
(trans (cong (subst P (refl x)) eq)
(subst-refl P gx)))
(
refl (f x)                                             ≡⟨ sym \$ trans-symˡ _ ⟩

trans (sym (subst-refl P (f x)))
(subst-refl P (f x))                                 ≡⟨ cong (trans _) \$
trans (sym \$ trans-reflˡ _) \$
cong (flip trans _) \$
sym \$ cong-refl _ ⟩∎
trans (sym (subst-refl P (f x)))
(trans (cong (subst P (refl x)) (refl (f x)))
(subst-refl P (f x)))                             ∎)
fx≡gx ⟩
trans (sym (subst-refl P (f x)))
(trans (cong (subst P (refl x)) fx≡gx)
(subst-refl P (g x)))                           ≡⟨ sym \$
cong₂ (λ p q → trans (sym p) (trans (cong (subst P (refl x)) fx≡gx) q))
(dcong-refl _)
(dcong-refl _) ⟩∎
trans (sym (dcong f (refl x)))
(trans (cong (subst P (refl x)) fx≡gx)
(dcong g (refl x)))                             ∎)
_
_

-- Sometimes cong can be "pushed" inside subst. The following
-- lemma provides one example.

cong-subst :
{B : A → Type b} {C : A → Type c}
{f : ∀ {x} → B x → C x} {g h : (x : A) → B x}
(eq₁ : x ≡ y) (eq₂ : g x ≡ h x) →
cong f (subst (λ x → g x ≡ h x) eq₁ eq₂) ≡
subst (λ x → f (g x) ≡ f (h x)) eq₁ (cong f eq₂)
cong-subst {f = f} {g} {h} = elim₁
(λ eq₁ → ∀ eq₂ →
cong f (subst (λ x → g x ≡ h x) eq₁ eq₂) ≡
subst (λ x → f (g x) ≡ f (h x)) eq₁ (cong f eq₂))
(λ eq₂ →
cong f (subst (λ x → g x ≡ h x) (refl _) eq₂)          ≡⟨ cong (cong f) \$ subst-refl _ _ ⟩
cong f eq₂                                             ≡⟨ sym \$ subst-refl _ _ ⟩∎
subst (λ x → f (g x) ≡ f (h x)) (refl _) (cong f eq₂)  ∎)

-- Some rearrangement lemmas for equalities between equalities.

[trans≡]≡[≡trans-symʳ] :
(p₁₂ : a₁ ≡ a₂) (p₁₃ : a₁ ≡ a₃) (p₂₃ : a₂ ≡ a₃) →
(trans p₁₂ p₂₃ ≡ p₁₃)
≡
(p₁₂ ≡ trans p₁₃ (sym p₂₃))
[trans≡]≡[≡trans-symʳ] p₁₂ p₁₃ p₂₃ = elim
(λ {a₂ a₃} p₂₃ →
∀ {a₁} (p₁₂ : a₁ ≡ a₂) (p₁₃ : a₁ ≡ a₃) →
(trans p₁₂ p₂₃ ≡ p₁₃)
≡
(p₁₂ ≡ trans p₁₃ (sym p₂₃)))
(λ a₂₃ p₁₂ p₁₃ →
trans p₁₂ (refl a₂₃) ≡ p₁₃        ≡⟨ cong₂ _≡_ (trans-reflʳ _) (sym \$ trans-reflʳ _) ⟩
p₁₂ ≡ trans p₁₃ (refl a₂₃)        ≡⟨ cong ((_ ≡_) ∘ trans _) (sym sym-refl) ⟩∎
p₁₂ ≡ trans p₁₃ (sym (refl a₂₃))  ∎)
p₂₃ p₁₂ p₁₃

[trans≡]≡[≡trans-symˡ] :
(p₁₂ : a₁ ≡ a₂) (p₁₃ : a₁ ≡ a₃) (p₂₃ : a₂ ≡ a₃) →
(trans p₁₂ p₂₃ ≡ p₁₃)
≡
(p₂₃ ≡ trans (sym p₁₂) p₁₃)
[trans≡]≡[≡trans-symˡ] p₁₂ = elim
(λ {a₁ a₂} p₁₂ →
∀ {a₃} (p₁₃ : a₁ ≡ a₃) (p₂₃ : a₂ ≡ a₃) →
(trans p₁₂ p₂₃ ≡ p₁₃)
≡
(p₂₃ ≡ trans (sym p₁₂) p₁₃))
(λ a₁₂ p₁₃ p₂₃ →
trans (refl a₁₂) p₂₃ ≡ p₁₃        ≡⟨ cong₂ _≡_ (trans-reflˡ _) (sym \$ trans-reflˡ _) ⟩
p₂₃ ≡ trans (refl a₁₂) p₁₃        ≡⟨ cong ((_ ≡_) ∘ flip trans _) (sym sym-refl) ⟩∎
p₂₃ ≡ trans (sym (refl a₁₂)) p₁₃  ∎)
p₁₂

-- The following lemma is basically Theorem 2.11.5 from the HoTT
-- book (the book's lemma gives an equivalence between equality
-- types, rather than an equality between equality types).

[subst≡]≡[trans≡trans] :
{p : x ≡ y} {q : x ≡ x} {r : y ≡ y} →
(subst (λ z → z ≡ z) p q ≡ r)
≡
(trans q p ≡ trans p r)
[subst≡]≡[trans≡trans] {p = p} {q} {r} = elim
(λ {x y} p → {q : x ≡ x} {r : y ≡ y} →
(subst (λ z → z ≡ z) p q ≡ r)
≡
(trans q p ≡ trans p r))
(λ x {q r} →
subst (λ z → z ≡ z) (refl x```