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

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

module Equality where

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

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

record Reflexive-relation a : Set (lsuc a) where
  infix 4 _≡_
  field

    -- "Equality".

    _≡_ : {A : Set a}  A  A  Set a

    -- Reflexivity.

    refl :  {A} (x : A)  x  x

-- Some definitions.

module Reflexive-relation′
         (reflexive :    Reflexive-relation ) where

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

  -- Non-equality.

  infix 4 _≢_

  _≢_ :  {a} {A : Set a}  A  A  Set a
  x  y = ¬ (x  y)

  -- The property of having decidable equality.

  Decidable-equality :  {}  Set   Set 
  Decidable-equality A = Decidable (_≡_ {A = A})

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

  Contractible :  {}  Set   Set 
  Contractible A =  λ (x : A)   y  x  y

  -- Proof irrelevance (or maybe "data irrelevance", depending on what
  -- the type is used for).

  Proof-irrelevant :  {}  Set   Set 
  Proof-irrelevant A = (x y : A)  x  y

  -- Uniqueness of identity proofs (for a particular type).

  Uniqueness-of-identity-proofs :  {}  Set   Set 
  Uniqueness-of-identity-proofs A =
    {x y : A}  Proof-irrelevant (x  y)

  -- The K rule (without computational content).

  K-rule :  a p  Set (lsuc (a  p))
  K-rule a p = {A : Set a} (P : {x : A}  x  x  Set 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}  {A : Set a}  A  Set a
  Singleton x =  λ y  y  x

  -- A variant of Singleton.

  Other-singleton :  {a} {A : Set a}  A  Set a
  Other-singleton x =  λ y  x  y

  -- The inspect idiom.

  inspect :  {a} {A : Set a} (x : A)  Other-singleton x
  inspect x = x , refl x

  -- Extensionality for functions of a certain type.

  Extensionality′ :  {a b} (A : Set a)  (A  Set b)  Set (a  b)
  Extensionality′ A B =
    {f g : (x : A)  B x}  (∀ x  f x  g x)  f  g

  -- Extensionality for functions at certain levels.

  Extensionality : (a b : Level)  Set (lsuc (a  b))
  Extensionality a b =
    {A : Set a}  {B : A  Set b}  Extensionality′ A B

  -- Proofs of extensionality which behave well when applied to
  -- reflexivity.

  Well-behaved-extensionality :
     {a b} (A : Set a)  (A  Set b)  Set (a  b)
  Well-behaved-extensionality A B =
     λ (ext : Extensionality′ A B) 
       f  ext  x  refl (f x))  refl f

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

-- Parametrised by a reflexive relation.

record Equality-with-J
         a p (reflexive :    Reflexive-relation ) :
         Set (lsuc (a  p)) where

  open Reflexive-relation′ reflexive

  field

    -- The J rule.

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

    -- The usual computational behaviour of the J rule.

    elim-refl :  {A : Set a} (P : {x y : A}  x  y  Set p)
                (r :  x  P (refl x)) {x} 
                elim P r (refl x)  r x

-- Some derived properties.

module Equality-with-J′
  {reflexive :    Reflexive-relation }
  (eq :  {a p}  Equality-with-J a p reflexive)
  where

  private
    open Reflexive-relation′ reflexive public
    open module E {a p} = Equality-with-J (eq {a} {p}) public

  -- Congruence.

  cong :  {a b} {A : Set a} {B : Set b}
         (f : A  B) {x y : A}  x  y  f x  f y
  cong f = elim  {u v} _  f u  f v)  x  refl (f x))

  abstract

    -- "Evaluation rule" for cong.

    cong-refl :  {a b} {A : Set a} {B : Set b} (f : A  B) {x : A} 
                cong f (refl x)  refl (f x)
    cong-refl f = elim-refl  {u v} _  f u  f v) (refl  f)

  -- Substitutivity.

  subst :  {a p} {A : Set a} (P : A  Set p) {x y : A} 
          x  y  P x  P y
  subst P = elim  {u v} _  P u  P v)  x p  p)

  -- "Evaluation rules" for subst.

  subst-refl≡id :  {a p} {A : Set a} (P : A  Set p) {x} 
                  subst P (refl x)  id
  subst-refl≡id P = elim-refl  {u v} _  P u  P v)  x p  p)

  subst-refl :  {a p} {A : Set a} (P : A  Set p) {x} (p : P x) 
               subst P (refl x) p  p
  subst-refl P p = cong (_$ p) (subst-refl≡id P)

  -- Singleton types are contractible.

  private

    irr :  {a} {A : Set a} {x : A}
          (p : Singleton x)  (x , refl x)  p
    irr p =
      elim  {u v} u≡v  _≡_ {A = Singleton v}
                              (v , refl v) (u , u≡v))
            _  refl _)
           (proj₂ p)

  singleton-contractible :
     {a} {A : Set a} (x : A)  Contractible (Singleton x)
  singleton-contractible x = ((x , refl x) , irr)

  abstract

    -- "Evaluation rule" for singleton-contractible.

    singleton-contractible-refl :
       {a} {A : Set a} (x : A) 
      proj₂ (singleton-contractible x) (x , refl x)  refl (x , refl x)
    singleton-contractible-refl x =
      elim-refl  {u v} u≡v  _≡_ {A = Singleton v}
                                   (v , refl v) (u , u≡v))
                _

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

record Equality-with-substitutivity-and-contractibility
         a p (reflexive :    Reflexive-relation ) :
         Set (lsuc (a  p)) where

  open Reflexive-relation′ reflexive

  field

    -- Substitutivity.

    subst : {A : Set a} (P : A  Set p) {x y : A}  x  y  P x  P y

    -- The usual computational behaviour of substitutivity.

    subst-refl : {A : Set a} (P : A  Set p) {x : A} (p : P x) 
                 subst P (refl x) p  p

    -- Singleton types are contractible.

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

-- Some derived properties.

module Equality-with-substitutivity-and-contractibility′
  {reflexive :    Reflexive-relation }
  (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 :  {a b} {A : Set a} {B : Set b}
           (f : A  B) {x y : A}  x  y  f x  f y
    cong f {x} x≡y =
      subst  y  x  y  f x  f y) x≡y  _  refl (f x)) x≡y

  -- Symmetry.

  sym :  {a} {A : Set a} {x y : A}  x  y  y  x
  sym {x = x} x≡y = subst  z  x  z  z  x) x≡y id x≡y

  abstract

    -- "Evaluation rule" for sym.

    sym-refl :  {a} {A : Set a} {x : A}  sym (refl x)  refl x
    sym-refl {x = x} =
      cong  f  f (refl x)) $
        subst-refl  z  x  z  z  x) id

  -- Transitivity.

  trans :  {a} {A : Set a} {x y z : A}  x  y  y  z  x  z
  trans {x = x} = flip (subst (_≡_ x))

  abstract

    -- "Evaluation rule" for trans.

    trans-refl-refl :  {a} {A : Set a} {x : A} 
                      trans (refl x) (refl x)  refl x
    trans-refl-refl {x = x} = subst-refl (_≡_ x) (refl x)

  -- Equational reasoning combinators.

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

  _∎ :  {a} {A : Set a} (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-≡ :  {a} {A : Set a} x {y z : A}  y  z  x  y  x  z
  step-≡ _ = flip trans

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

  _≡⟨⟩_ :  {a} {A : Set a} x {y : A}  x  y  x  y
  _ ≡⟨⟩ x≡y = x≡y

  finally :  {a} {A : Set a} (x y : A)  x  y  x  y
  finally _ _ x≡y = x≡y

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

  abstract

    -- The J rule.

    elim :  {a p} {A : Set a} (P : {x y : A}  x  y  Set p) 
           (∀ x  P (refl x)) 
            {x y} (x≡y : x  y)  P x≡y
    elim P p {x} {y} x≡y =
      let lemma = proj₂ (singleton-contractible y) in
      subst {A = Singleton y}
            (P  proj₂)
            ((y , refl y)                      ≡⟨ sym (lemma (y , refl y)) 
             proj₁ (singleton-contractible y)  ≡⟨ lemma (x , x≡y) ⟩∎
             (x , x≡y)                         )
            (p y)

    -- Transitivity and symmetry sometimes cancel each other out.

    trans-sym :  {a} {A : Set a} {x y : A} (x≡y : x  y) 
                trans (sym x≡y) x≡y  refl y
    trans-sym =
      elim  {x y} (x≡y : x  y)  trans (sym x≡y) x≡y  refl y)
            x  trans (sym (refl x)) (refl x)  ≡⟨ cong  p  trans p (refl x)) sym-refl 
                  trans (refl x) (refl x)        ≡⟨ trans-refl-refl ⟩∎
                  refl x                         )

    -- "Evaluation rule" for elim.

    elim-refl :  {a p} {A : Set a} (P : {x y : A}  x  y  Set p)
                (p :  x  P (refl x)) {x} 
                elim P p (refl x)  p x
    elim-refl P p {x} =
      let lemma = proj₂ (singleton-contractible x) (x , refl x) in
      subst {A = Singleton x} (P  proj₂) (trans (sym lemma) lemma) (p x)  ≡⟨ cong  q  subst {A = Singleton x} (P  proj₂) q (p x))
                                                                                   (trans-sym lemma) 
      subst {A = Singleton x} (P  proj₂) (refl (x , refl x))       (p x)  ≡⟨ subst-refl {A = Singleton x} (P  proj₂) (p x) ⟩∎
      p x                                                                  

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

J⇒subst+contr :
   {reflexive} 
  (∀ {a p}  Equality-with-J 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′ eq

subst+contr⇒J :
   {reflexive} 
  (∀ {a p}  Equality-with-substitutivity-and-contractibility
               a p reflexive) 
   {a p}  Equality-with-J 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
  {reflexive}
  (eq :  {a p}  Equality-with-J a p reflexive)
  where

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

  open Equality-with-J′ eq public
  open Equality-with-substitutivity-and-contractibility′
         (J⇒subst+contr eq) public
    using ( sym; sym-refl
          ; trans; trans-refl-refl
          ; _∎; step-≡; _≡⟨⟩_; 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 :  {a p} {A : Set a} {y : A} (P :  {x}  x  y  Set p) 
          P (refl y) 
           {x} (x≡y : x  y)  P x≡y
  elim₁ {y = y} P p {x} x≡y =
    subst {A = Singleton y}
          (P  proj₂)
          (proj₂ (singleton-contractible y) (x , x≡y))
          p

  abstract

    -- "Evaluation rule" for elim₁.

    elim₁-refl :  {a p} {A : Set a} {y : A}
                 (P :  {x}  x  y  Set p) (p : P (refl y)) 
                 elim₁ P p (refl y)  p
    elim₁-refl {y = y} P p =
      subst {A = Singleton y} (P  proj₂)
            (proj₂ (singleton-contractible y) (y , refl y)) p    ≡⟨ cong  q  subst {A = Singleton y} (P  proj₂) q p)
                                                                         (singleton-contractible-refl y) 
      subst {A = Singleton y} (P  proj₂) (refl (y , refl y)) p  ≡⟨ subst-refl {A = Singleton y} (P  proj₂) p ⟩∎
      p                                                          

  -- A variant of singleton-contractible.

  private

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

  other-singleton-contractible :
     {a} {A : Set a} (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 :
       {a} {A : Set a} (x : A) 
      proj₂ (other-singleton-contractible x) (x , refl x) 
      refl (x , refl x)
    other-singleton-contractible-refl x =
      elim-refl  {u v} u≡v  _≡_ {A = Other-singleton u}
                                   (u , refl u) (v , u≡v))
                _

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

  elim¹ :  {a p} {A : Set a} {x : A} (P :  {y}  x  y  Set p) 
          P (refl x) 
           {y} (x≡y : x  y)  P x≡y
  elim¹ {x = x} P p {y} x≡y =
    subst {A = Other-singleton x}
          (P  proj₂)
          (proj₂ (other-singleton-contractible x) (y , x≡y))
          p

  abstract

    -- "Evaluation rule" for elim¹.

    elim¹-refl :  {a p} {A : Set a} {x : A}
                 (P :  {y}  x  y  Set p) (p : P (refl x)) 
                 elim¹ P p (refl x)  p
    elim¹-refl {x = x} P p =
      subst {A = Other-singleton x} (P  proj₂)
            (proj₂ (other-singleton-contractible x) (x , refl x)) p    ≡⟨ cong  q  subst {A = Other-singleton x} (P  proj₂) q p)
                                                                               (other-singleton-contractible-refl x) 
      subst {A = Other-singleton x} (P  proj₂) (refl (x , refl x)) p  ≡⟨ subst-refl {A = Other-singleton x} (P  proj₂) p ⟩∎
      p                                                                

  -- A generalisation of dependent-cong (which is defined below).

  dependent-cong′ :
     {a b} {A : Set a} {B : A  Set b} {x y}
    (f : (x : A)  x  y  B x) (x≡y : x  y) 
    subst B x≡y (f x x≡y)  f y (refl y)
  dependent-cong′ {B = B} {y = y} f x≡y = elim₁
     {x} (x≡y : x  y) 
       (f :  x  x  y  B x) 
       subst B x≡y (f x x≡y)  f y (refl y))
     f  subst B (refl y) (f y (refl y))  ≡⟨ subst-refl _ _ ⟩∎
           f y (refl y)                     )
    x≡y f

  abstract

    -- "Evaluation rule" for dependent-cong′.

    dependent-cong′-refl :
       {a b} {A : Set a} {B : A  Set b} {y}
      (f : (x : A)  x  y  B x) 
      dependent-cong′ f (refl y)  subst-refl B (f y (refl y))
    dependent-cong′-refl f = cong (_$ f) $ elim₁-refl _ _

  -- A dependent variant of cong.

  dependent-cong :
     {a b} {A : Set a} {B : A  Set b}
    (f : (x : A)  B x) {x y} (x≡y : x  y) 
    subst B x≡y (f x)  f y
  dependent-cong f = dependent-cong′ (const  f)

  abstract

    -- "Evaluation rule" for dependent-cong.

    dependent-cong-refl :
       {a b} {A : Set a} {B : A  Set b} (f : (x : A)  B x) {x} 
      dependent-cong f (refl x)  subst-refl B (f x)
    dependent-cong-refl _ = dependent-cong′-refl _

  -- Binary congruence.

  cong₂ :  {a b c} {A : Set a} {B : Set b} {C : Set c}
          (f : A  B  C) {x y : A} {u v : B} 
          x  y  u  v  f x u  f y v
  cong₂ f {x} {y} {u} {v} 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 :  {a b c} {A : Set a} {B : Set b} {C : Set c}
                 (f : A  B  C) {x : A} {y : B} 
                 cong₂ f (refl x) (refl y)  refl (f x y)
    cong₂-refl f {x} {y} =
      trans (cong (flip f y) (refl x)) (cong (f x) (refl y))  ≡⟨ cong₂ trans (cong-refl (flip f y)) (cong-refl (f x)) 
      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    ({A : Set }  Uniqueness-of-identity-proofs A)
  K⇔UIP = record
    { from = λ UIP P r {x} x≡x  subst P (UIP (refl x) x≡x) (r x)
    ; to   = λ K {_} 
        elim  p   q  p  q)
              x  K  {x} p  refl x  p)  x  refl (refl x)))
    }

  abstract

    -- Extensionality at given levels works at lower levels as well.

    lower-extensionality :
       {a}  {b}  
      Extensionality (a  ) (b  )  Extensionality a b
    lower-extensionality   ext f≡g =
      cong  h  lower  h  lift) $
        ext {A =   _} {B =    _} (cong lift  f≡g  lower)

    lower-extensionality₂ :
       {a} {A : Set a} {b}  
      ({B : A  Set (b  )}  Extensionality′ A B) 
      ({B : A  Set  b     }  Extensionality′ A B)
    lower-extensionality₂  ext f≡g =
      cong  h  lower  h) $
        ext {B =    _} (cong lift  f≡g)

  -- Extensionality for explicit function types works for implicit
  -- function types as well.

  implicit-extensionality :
     {a b} {A : Set a} {B : A  Set b} 
    Extensionality′ A B 
    {f g : {x : A}  B x} 
    (∀ x  f {x}  g {x})   {x}  f {x})  g
  implicit-extensionality ext f≡g =
    cong  f {x}  f x) $ ext f≡g

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

  abstract

    trans-reflʳ :  {a} {A : Set a} {x y : A} (x≡y : x  y) 
                  trans x≡y (refl y)  x≡y
    trans-reflʳ =
      elim  {u v} u≡v  trans u≡v (refl v)  u≡v)
            _  trans-refl-refl)

    trans-reflˡ :  {a} {A : Set a} {x y : A} (x≡y : x  y) 
                  trans (refl x) x≡y  x≡y
    trans-reflˡ =
      elim  {u v} u≡v  trans (refl u) u≡v  u≡v)
            _  trans-refl-refl)

    trans-assoc :  {a} {A : Set a} {x y z u : A}
                  (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  x≡y   y≡z z≡u  trans (trans x≡y y≡z) z≡u 
                                trans x≡y (trans y≡z z≡u))
            y y≡z z≡u 
              trans (trans (refl y) y≡z) z≡u ≡⟨ cong₂ trans (trans-reflˡ y≡z) (refl z≡u) 
              trans y≡z z≡u                  ≡⟨ sym $ trans-reflˡ (trans y≡z z≡u) ⟩∎
              trans (refl y) (trans y≡z z≡u) )

    sym-sym :  {a} {A : Set a} {x y : A} (x≡y : x  y) 
              sym (sym x≡y)  x≡y
    sym-sym = elim  {u v} u≡v  sym (sym u≡v)  u≡v)
                    x  sym (sym (refl x))  ≡⟨ cong sym (sym-refl {x = x}) 
                          sym (refl x)        ≡⟨ sym-refl ⟩∎
                          refl x              )

    sym-trans :  {a} {A : Set a} {x y z : A}
                (x≡y : x  y) (y≡z : y  z) 
                sym (trans x≡y y≡z)  trans (sym y≡z) (sym x≡y)
    sym-trans {a} =
      elim  x≡y   y≡z  sym (trans x≡y y≡z)  trans (sym y≡z) (sym x≡y))
            y y≡z  sym (trans (refl y) y≡z)        ≡⟨ cong sym (trans-reflˡ y≡z) 
                      sym y≡z                         ≡⟨ sym $ trans-reflʳ (sym y≡z) 
                      trans (sym y≡z) (refl y)        ≡⟨ cong {a = a} {b = a} (trans (sym y≡z)) (sym sym-refl)  ⟩∎
                      trans (sym y≡z) (sym (refl y))  )

    trans-symˡ :  {a} {A : Set a} {x y : A} (p : x  y) 
                 trans (sym p) p  refl y
    trans-symˡ =
      elim  p  trans (sym p) p  refl _)
            x  trans (sym (refl x)) (refl x)  ≡⟨ trans-reflʳ _ 
                  sym (refl x)                   ≡⟨ sym-refl ⟩∎
                  refl x                         )

    trans-symʳ :  {a} {A : Set a} {x y : A} (p : x  y) 
                 trans p (sym p)  refl _
    trans-symʳ =
      elim  p  trans p (sym p)  refl _)
            x  trans (refl x) (sym (refl x))  ≡⟨ trans-reflˡ _ 
                  sym (refl x)                   ≡⟨ sym-refl ⟩∎
                  refl x                         )

    cong-trans :  {a b} {A : Set a} {B : Set b} {x y z : A}
                 (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  x≡y   y≡z  cong f (trans x≡y y≡z) 
                            trans (cong f x≡y) (cong f y≡z))
            y y≡z  cong f (trans (refl y) y≡z)           ≡⟨ cong (cong f) (trans-reflˡ _) 
                      cong f y≡z                            ≡⟨ sym $ trans-reflˡ (cong f y≡z) 
                      trans (refl (f y)) (cong f y≡z)       ≡⟨ cong₂ trans (sym (cong-refl f {x = y})) (refl (cong f y≡z)) ⟩∎
                      trans (cong f (refl y)) (cong f y≡z)  )

    cong-id :  {a} {A : Set a} {x y : A} (x≡y : x  y) 
              x≡y  cong id x≡y
    cong-id = elim  u≡v  u≡v  cong id u≡v)
                    x  refl x            ≡⟨ sym (cong-refl id {x = x}) ⟩∎
                          cong id (refl x)  )

    cong-const :  {a b} {A : Set a} {B : Set b} {x y : A} {z : B}
                 (x≡y : x  y) 
                 cong (const z) x≡y  refl z
    cong-const {z = z} =
      elim  u≡v  cong (const z) u≡v  refl z)
            x  cong (const z) (refl x)  ≡⟨ cong-refl (const z) ⟩∎
                  refl z                   )

    cong-∘ :  {a b c} {A : Set a} {B : Set b} {C : Set c} {x y : A}
             (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  x≡y  cong f (cong g x≡y)  cong (f  g) x≡y)
                       x  cong f (cong g (refl x))  ≡⟨ cong (cong f) (cong-refl g) 
                             cong f (refl (g x))       ≡⟨ cong-refl f 
                             refl (f (g x))            ≡⟨ sym (cong-refl (f  g)) ⟩∎
                             cong (f  g) (refl x)     )

    cong-proj₁-cong₂-, :
       {a b} {A : Set a} {B : Set b} {x y : A} {u v : B}
      (x≡y : x  y) (u≡v : u  v) 
      cong proj₁ (cong₂ _,_ x≡y u≡v)  x≡y
    cong-proj₁-cong₂-, {x = x} {y} {u} {v} x≡y u≡v =
      cong proj₁ (trans (cong (flip _,_ u) x≡y) (cong (_,_ y) u≡v))  ≡⟨ cong-trans proj₁ _ _ 

      trans (cong proj₁ (cong (flip _,_ u) x≡y))
            (cong proj₁ (cong (_,_ y) u≡v))                          ≡⟨ cong₂ trans (cong-∘ proj₁ (flip _,_ u) x≡y) (cong-∘ proj₁ (_,_ y) u≡v) 

      trans (cong id x≡y) (cong (const y) u≡v)                       ≡⟨ cong₂ trans (sym $ cong-id x≡y) (cong-const u≡v) 

      trans x≡y (refl y)                                             ≡⟨ trans-reflʳ x≡y ⟩∎

      x≡y                                                            

    cong-proj₂-cong₂-, :
       {a b} {A : Set a} {B : Set b} {x y : A} {u v : B}
      (x≡y : x  y) (u≡v : u  v) 
      cong proj₂ (cong₂ _,_ x≡y u≡v)  u≡v
    cong-proj₂-cong₂-, {x = x} {y} {u} {v} x≡y u≡v =
      cong proj₂ (trans (cong (flip _,_ u) x≡y) (cong (_,_ y) u≡v))  ≡⟨ cong-trans proj₂ _ _ 

      trans (cong proj₂ (cong (flip _,_ u) x≡y))
            (cong proj₂ (cong (_,_ y) u≡v))                          ≡⟨ cong₂ trans (cong-∘ proj₂ (flip _,_ u) x≡y) (cong-∘ proj₂ (_,_ y) u≡v) 

      trans (cong (const u) x≡y) (cong id u≡v)                       ≡⟨ cong₂ trans (cong-const x≡y) (sym $ cong-id u≡v) 

      trans (refl u) u≡v                                             ≡⟨ trans-reflˡ u≡v ⟩∎

      u≡v                                                            

    cong-sym :  {a b} {A : Set a} {B : Set b} {x y : A}
               (f : A  B) (x≡y : x  y) 
               cong f (sym x≡y)  sym (cong f x≡y)
    cong-sym f = elim  x≡y  cong f (sym x≡y)  sym (cong f x≡y))
                       x  cong f (sym (refl x))  ≡⟨ cong (cong f) sym-refl 
                             cong f (refl x)        ≡⟨ cong-refl f 
                             refl (f x)             ≡⟨ sym sym-refl 
                             sym (refl (f x))       ≡⟨ cong sym $ sym (cong-refl f {x = x}) ⟩∎
                             sym (cong f (refl x))  )

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

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

    cong₂-cong-cong :
       {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}
        {x₁ x₂} {eq : x₁  x₂}
      (f : A  B) (g : A  C) (h : B  C  D) 
      cong₂ h (cong f eq) (cong g eq) 
      cong  x  h (f x) (g x)) eq
    cong₂-cong-cong f g h = elim¹
       eq  cong₂ h (cong f eq) (cong g eq) 
              cong  x  h (f x) (g x)) eq)
      (cong₂ h (cong f (refl _)) (cong g (refl _))  ≡⟨ cong₂ (cong₂ h) (cong-refl f) (cong-refl g) 
       cong₂ h (refl _) (refl _)                    ≡⟨ cong₂-refl h 
       refl _                                       ≡⟨ sym $ cong-refl  x  h (f x) (g x)) ⟩∎
       cong  x  h (f x) (g x)) (refl _)          )
      _

    cong-≡id :
       {a b} {A : Set a} (B : A  Set b) {x} {y : B x} {f : B x  B x}
      (f≡id : f  id) 
      cong  g  g (f y)) f≡id 
      cong  g  f (g y)) f≡id
    cong-≡id B = elim₁
       {f} p  cong  g  g (f _)) p  cong  g  f (g _)) p)
      (refl _)

    elim-∘ :
       {a p} {A : Set a} {x y : A}
      (P Q :  {x y}  x  y  Set 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
       x≡y  f (elim P r x≡y) 
               elim Q (f  r) x≡y)
      (f (elim P r (refl x))    ≡⟨ cong f $ elim-refl P _ 
       f (r x)                  ≡⟨ sym $ elim-refl Q _ ⟩∎
       elim Q (f  r) (refl x)  )
      x≡y

    elim-cong :
       {a b p} {A : Set a} {B : Set b} {x y : A}
      (P : B  B  Set p) (f : A  B)
      (r :  x  P x x) {x≡y : x  y} 
      elim  {x y} _  P x y) r (cong f x≡y) 
      elim  {x y} _  P (f x) (f y)) (r  f) x≡y
    elim-cong {x = x} P f r {x≡y} = elim¹
       x≡y  elim  {x y} _  P x y) r (cong f x≡y) 
               elim  {x y} _  P (f x) (f y)) (r  f) x≡y)
      (elim  {x y} _  P x y) r (cong f (refl x))       ≡⟨ cong (elim  {x y} _  P x y) _) $ cong-refl f 
       elim  {x y} _  P x y) r (refl (f x))            ≡⟨ elim-refl  {x y} _  P x y) _ 
       r (f x)                                            ≡⟨ sym $ elim-refl  {x y} _  P (f x) (f y)) _ ⟩∎
       elim  {x y} _  P (f x) (f y)) (r  f) (refl x)  )
      x≡y

  subst-const :  {a p} {A : Set a} {x₁ x₂ : A} (x₁≡x₂ : x₁  x₂)
                  {P : Set p} {p} 
                subst (const P) x₁≡x₂ p  p
  subst-const x₁≡x₂ {P = P} {p} =
    elim¹  x₁≡x₂  subst (const P) x₁≡x₂ p  p)
          (subst-refl (const P) _)
          x₁≡x₂

  abstract

    subst-∘ :  {a b p} {A : Set a} {B : Set b} {x y : A}
              (P : B  Set 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 _ =
      sym $ cong  g  g _) $ elim-cong  u v  P u  P v) f _

    subst-↑ :  {a p } {A : Set a} {x y}
              (P : A  Set p) {p :   (P x)} {x≡y : x  y} 
              subst (   P) x≡y p  lift (subst P x≡y (lower p))
    subst-↑ { = } P {p} = elim¹
       x≡y  subst (   P) x≡y p  lift (subst P x≡y (lower p)))
      (subst (   P) (refl _) p         ≡⟨ subst-refl (   P) _ 
       p                                  ≡⟨ cong lift $ sym $ subst-refl P _ ⟩∎
       lift (subst P (refl _) (lower p))  )
      _

    -- A fusion law for subst.

    subst-subst :
       {a p} {A : Set a} (P : A  Set p)
      {x y z : A} (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  {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)
            x y≡z p 
              subst P y≡z (subst P (refl x) p)  ≡⟨ cong (subst P y≡z) $ subst-refl P p 
              subst P y≡z p                     ≡⟨ cong  q  subst P q p) (sym $ trans-reflˡ _) ⟩∎
              subst P (trans (refl x) y≡z) p    )
           x≡y y≡z p

    -- "Computation rules" for subst-subst.

    subst-subst-reflˡ :
       {a p} {A : Set a} (P : A  Set p) {x y p} {x≡y : x  y} 
      subst-subst P (refl x) x≡y p 
      cong₂ (flip (subst P)) (subst-refl P p) (sym $ trans-reflˡ x≡y)
    subst-subst-reflˡ P =
      cong  f  f _ _) $
        elim-refl  {x y} x≡y   {z} (y≡z : y  z) p 
                     subst P y≡z (subst P x≡y p)  _)
                  _

    subst-subst-refl-refl :
       {a p} {A : Set a} (P : A  Set p) {x p} 
      subst-subst P (refl x) (refl x) p 
      cong₂ (flip (subst P)) (subst-refl P p) (sym trans-refl-refl)
    subst-subst-refl-refl P {x} {p} =
      subst-subst P (refl x) (refl x) p                              ≡⟨ subst-subst-reflˡ _ 

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

    -- The combinator trans is defined in terms of subst. It could
    -- have been defined in another way.

    subst-trans :
       {a} {A : Set a} {x y z : A} (x≡y : x  y) {y≡z : y  z} 
      subst  x  x  z) (sym x≡y) y≡z  trans x≡y y≡z
    subst-trans {y = y} {z} x≡y {y≡z} =
      elim₁  x≡y  subst  x  x  z) (sym x≡y) y≡z 
                     trans x≡y y≡z)
            (subst  x  x  z) (sym (refl y)) y≡z  ≡⟨ cong  eq  subst  x  x  z) eq y≡z) sym-refl 
             subst  x  x  z) (refl y) y≡z        ≡⟨ subst-refl  x  x  z) y≡z 
             y≡z                                     ≡⟨ sym $ trans-reflˡ y≡z ⟩∎
             trans (refl y) y≡z                      )
            x≡y

    -- Substitutivity and symmetry sometimes cancel each other out.

    subst-subst-sym :
       {a p} {A : Set a} (P : A  Set p) {x y : A}
      (x≡y : x  y) (p : P y) 
      subst P x≡y (subst P (sym x≡y) p)  p
    subst-subst-sym {A = A} P {y = y} x≡y p =
      subst P x≡y (subst P (sym x≡y) p)  ≡⟨ subst-subst P _ _ _ 
      subst P (trans (sym x≡y) x≡y) p    ≡⟨ cong  q  subst P q p) (trans-symˡ x≡y) 
      subst P (refl y) p                 ≡⟨ subst-refl P p ⟩∎
      p                                  

    -- Some corollaries and variants.

    trans-[trans-sym]- :  {a} {A : Set a} {a b c : A} 
                         (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 = subst-subst-sym (_≡_ _) c≡b a≡b

    trans-[trans]-sym :  {a} {A : Set a} {a b c : A} 
                        (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  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] :  {a} {A : Set a} {a b c : A} 
                         (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} {A : Set a} {a b c : A} 
                        (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  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 :
       {a p} {A : Set a} {x : A} {P : Set p} {p} 
      trans (sym $ subst-refl  _  P) p) (subst-const (refl x)) 
      refl p
    subst-refl-subst-const {x = x} {P} {p} =
      trans (sym $ subst-refl  _  P) p)
            (elim¹  eq  subst  _  P) eq p  p)
                   (subst-refl  _  P) _) _)        ≡⟨ cong (trans _) (elim¹-refl  eq  subst  _  P) eq p  p) _) 
      trans (sym $ subst-refl  _  P) p)
            (subst-refl  _  P) _)                  ≡⟨ trans-symˡ _ ⟩∎
      refl _                                          

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

    dependent-cong-subst-const-cong :
       {a b} {A : Set a} {B : Set b} (f : A  B) {x y} (x≡y : x  y) 
      dependent-cong f x≡y 
      (subst (const B) x≡y (f x)  ≡⟨ subst-const x≡y 
       f x                        ≡⟨ cong f x≡y ⟩∎
       f y                        )
    dependent-cong-subst-const-cong f = elim
       {x y} x≡y  dependent-cong f x≡y 
                     trans (subst-const x≡y) (cong f x≡y))
       x 
        dependent-cong f (refl x)                        ≡⟨ dependent-cong-refl f 
        subst-refl (const _) (f x)                       ≡⟨ sym $ trans-reflʳ _ 
        trans (subst-refl (const _) (f x)) (refl (f x))  ≡⟨ cong₂ trans
                                                                  (sym $ elim¹-refl _ _)
                                                                  (sym $ cong-refl f) ⟩∎
        trans (subst-const (refl x)) (cong f (refl x))   )

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

  Σ-≡,≡→≡ :  {a b} {A : Set a} {B : A  Set 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
     {x₁ y₁} (p : x₁  y₁)   {x₂ y₂} 
       subst B p x₂  y₂  (x₁ , x₂)  (y₁ , y₂))
     z₁ {x₂} {y₂} x₂≡y₂  cong (_,_ z₁) (
       x₂                    ≡⟨ sym $ subst-refl B x₂ 
       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-≡,≡←≡ :  {a b} {A : Set a} {B : A  Set 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} = elim
     {p₁ p₂ : Σ A B} _ 
        λ (p : proj₁ p₁  proj₁ p₂)  subst B p (proj₂ p₁)  proj₂ p₂)
     p  refl _ , subst-refl B _)

  abstract

    -- "Evaluation rules" for Σ-≡,≡→≡.

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

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

    Σ-≡,≡→≡-refl-subst-refl :
       {a b} {A : Set a} {B : A  Set b} {x y} 
      Σ-≡,≡→≡ (refl x) (subst-refl B y)  refl (x , y)
    Σ-≡,≡→≡-refl-subst-refl {B = B} {x} {y} =
      Σ-≡,≡→≡ (refl x) (subst-refl B y)                             ≡⟨ Σ-≡,≡→≡-reflˡ _ 
      cong (_,_ x) (trans (sym $ subst-refl B y) (subst-refl B y))  ≡⟨ cong (cong _) (trans-symˡ _) 
      cong (_,_ x) (refl y)                                         ≡⟨ cong-refl _ ⟩∎
      refl (x , y)                                                  

    -- "Evaluation rule" for Σ-≡,≡←≡.

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

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

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

    Σ-≡,≡→≡-cong :
       {a b} {A : Set a} {B : A  Set 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
       {q₁ q₂} q₁≡q₂ 
          {r₁ r₂}
         (r₁≡r₂ : trans (cong (flip (subst B) _) (sym q₁≡q₂)) r₁  r₂) 
         Σ-≡,≡→≡ q₁ r₁  Σ-≡,≡→≡ q₂ r₂)
       q {r₁ r₂} r₁≡r₂  cong (Σ-≡,≡→≡ q) (
         r₁                                                 ≡⟨ sym $ trans-reflˡ _ 
         trans (refl (subst B q _)) r₁                      ≡⟨ cong (flip trans _) $ sym $ cong-refl (flip (subst B) _) 
         trans (cong (flip (subst B) _) (refl q)) r₁        ≡⟨ cong  e  trans (cong (flip (subst B) _) e) _) $ sym sym-refl 
         trans (cong (flip (subst B) _) (sym (refl q))) r₁  ≡⟨ r₁≡r₂ ⟩∎
         r₂                                                 ))

    trans-Σ-≡,≡→≡ :
       {a b} {A : Set a} {B : A  Set 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
       {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)))
       x  elim₁
          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)))
          {y}  elim¹
             {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
               r₂₃ 
                 trans (Σ-≡,≡→≡ (refl _) (refl _))
                       (Σ-≡,≡→≡ (refl _) r₂₃) 
                 Σ-≡,≡→≡ (trans (refl _) (refl _))
                         (trans (sym $ subst-subst _ _ _ _)
                                (trans (cong (subst _ _) (refl _))
                                       r₂₃)))
              (let lemma₁ =
                     sym (subst-refl B (subst B (refl x) y))          ≡⟨⟩

                     sym (cong  f  f (subst B (refl x) y))
                               (subst-refl≡id B))                     ≡⟨ cong sym $ cong-≡id B _ 

                     sym (cong  f  subst B (refl x) (f y))
                               (subst-refl≡id B))                     ≡⟨ cong sym $ sym $ cong-∘ _ _ _ ⟩∎

                     sym (cong (subst B (refl x)) (subst-refl B y))   

                   lemma₂ =
                     sym (cong (subst B _) (subst-refl B _))          ≡⟨ 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 B _))))      ≡⟨ cong (flip trans _) $ sym $ cong-sym (flip (subst B) _) _ ⟩∎

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

                   lemma₃ =
                     trans (cong (flip (subst B) _) trans-refl-refl)
                           (sym (cong (subst B _) (subst-refl B _)))     ≡⟨ cong  e  trans (cong (flip (subst B) _) e)
                                                                                              (sym $ cong (subst B _) (subst-refl B _))) $
                                                                                 sym $ sym-sym _ 
                     trans (cong (flip (subst B) _)
                                 (sym $ sym trans-refl-refl))
                           (sym (cong (subst B _) (subst-refl B _)))     ≡⟨ cong (flip trans _) $ cong-sym (flip (subst B) _) _ 

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

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

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

               cong (_ ,_) (trans (sym (subst-refl B _))
                                  (sym (subst-refl B _)))                 ≡⟨ cong (cong (_ ,_)  trans _) lemma₁ 

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

               Σ-≡,≡→≡ (refl _)
                       (sym (cong (subst B _) (subst-refl B _)))          ≡⟨ cong (Σ-≡,≡→≡ _) lemma₂ 

               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 B _)))))        ≡⟨ sym $ Σ-≡,≡→≡-cong _ (refl _) 

               Σ-≡,≡→≡ (trans (refl _) (refl _))
                       (trans (cong (flip (subst B) _) trans-refl-refl)
                              (sym (cong (subst B _) (subst-refl B _))))  ≡⟨ cong (Σ-≡,≡→≡ (trans (refl _) (refl _))) lemma₃ ⟩∎

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

    Σ-≡,≡→≡-subst-const :
       {a b} {A : Set a} {B : Set b} {p₁ p₂ : A × B} 
      (p : proj₁ p₁  proj₁ p₂) (q : proj₂ p₁  proj₂ p₂) 
      Σ-≡,≡→≡ p (trans (subst-const p) q)  cong₂ _,_ p q
    Σ-≡,≡→≡-subst-const {B = B} {_ , y₁} {_ , y₂} p q = elim
       {x₁ y₁} (p : x₁  y₁) 
         Σ-≡,≡→≡ p (trans (subst-const _) q)  cong₂ _,_ p q)
       x 
         let lemma =
               trans (sym $ subst-refl  _  B) y₁)
                     (trans (subst-const _) q)               ≡⟨ sym $ trans-assoc _ _ _ 
               trans (trans (sym $ subst-refl  _  B) y₁)
                            (subst-const _))
                     q                                       ≡⟨ cong₂ trans subst-refl-subst-const (refl _) 
               trans (refl y₁) q                             ≡⟨ trans-reflˡ _ ⟩∎
               q                                              in

         Σ-≡,≡→≡ (refl x) (trans (subst-const _) q)           ≡⟨ Σ-≡,≡→≡-reflˡ _ 
         cong (_,_ x) (trans (sym $ subst-refl  _  B) y₁)
                             (trans (subst-const _) q))       ≡⟨ cong (cong (_,_ x)) lemma 
         cong (_,_ x) q                                       ≡⟨ sym $ cong₂-reflˡ _,_ ⟩∎
         cong₂ _,_ (refl x) q                                 )
      p

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

    proj₁-Σ-≡,≡←≡ :
       {a b} {A : Set a} {B : A  Set b} {p₁ p₂ : Σ A B}
      (p₁≡p₂ : p₁  p₂) 
      proj₁ (Σ-≡,≡←≡ p₁≡p₂)  cong proj₁ p₁≡p₂
    proj₁-Σ-≡,≡←≡ = elim
       p₁≡p₂  proj₁ (Σ-≡,≡←≡ p₁≡p₂)  cong proj₁ p₁≡p₂)
       p 
         proj₁ (Σ-≡,≡←≡ (refl p))  ≡⟨ cong proj₁ $ Σ-≡,≡←≡-refl 
         refl (proj₁ p)            ≡⟨ sym $ cong-refl proj₁ ⟩∎
         cong proj₁ (refl p)       )

  -- A binary variant of subst.

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

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

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

    -- A corollary of push-subst-pair.

    push-subst-pair′ :
       {a b c} {A : Set a} {y z : A} {y≡z : y  z}
      (B : A  Set b) (C : Σ A B  Set c) {p p₁} 
      (p₁≡p₁ : subst B y≡z (proj₁ p)  p₁) 
      subst  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  {p₁} p₁≡p₁ 
               subst  x  Σ (B x) (curry C x)) y≡z p 
               (p₁ , subst₂ C y≡z p₁≡p₁ (proj₂ p)))
            (push-subst-pair B C)

    -- A proof simplification rule for subst₂.

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

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

    push-subst-, :
       {a b c} {A : Set a} {y z : A} {y≡z : y  z}
      (B : A  Set b) (C : A  Set c) {p} 
      subst  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  x  B x × C x) y≡z (x , y)                           ≡⟨ push-subst-pair B (C  proj₁) 
      (subst B y≡z x , subst (C  proj₁) (Σ-≡,≡→≡ y≡z (refl _)) y)  ≡⟨ cong (_,_ _) $ subst₂-proj₁ C ⟩∎
      (subst B y≡z x , subst C y≡z y)                               

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

    push-subst-inj₁ :
       {a b c} {A : Set a} {y z : A} {y≡z : y  z}
      (B : A  Set b) (C : A  Set c) {x} 
      subst  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
       y≡z  subst  x  B x  C x) y≡z (inj₁ x) 
               inj₁ (subst B y≡z x))
      (subst  x  B x  C x) (refl _) (inj₁ x)  ≡⟨ subst-refl  x  B x  C x) _ 
       inj₁ x                                     ≡⟨ cong inj₁ $ sym $ subst-refl B _ ⟩∎
       inj₁ (subst B (refl _) x)                  )
      y≡z

    push-subst-inj₂ :
       {a b c} {A : Set a} {y z : A} {y≡z : y  z}
      (B : A  Set b) (C : A  Set c) {x} 
      subst  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
       y≡z  subst  x  B x  C x) y≡z (inj₂ x) 
               inj₂ (subst C y≡z x))
      (subst  x  B x  C x) (refl _) (inj₂ x)  ≡⟨ subst-refl  x  B x  C x) _ 
       inj₂ x                                     ≡⟨ cong inj₂ $ sym $ subst-refl C _ ⟩∎
       inj₂ (subst C (refl _) x)                  )
      y≡z

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

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

    push-subst-implicit-application :
       {a b c} {A : Set a} {B : A  Set b} {x₁ x₂ : A}
      (x₁≡x₂ : x₁  x₂) (C : (x : A)  B x  Set c)
      {f : (x : A)  B x} {g : {y : B x₁}  C x₁ y} 
      subst  x  C x (f x)) x₁≡x₂ (g {y = f x₁}) 
      subst  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
       {x₂} x₁≡x₂ 
         subst  x  C x (f x)) x₁≡x₂ (g {y = f x₁}) 
         subst  x   {y}  C x y) x₁≡x₂ g {y = f x₂})
      (subst  x  C x (f x)) (refl _) (g {y = f x₁})    ≡⟨ subst-refl  x  C x (f x)) _ 
       g {y = f x₁}                                       ≡⟨ cong  g  g {y = f x₁}) $ sym $ subst-refl  x   {y}  C x y) _ ⟩∎
       subst  x   {y}  C x y) (refl _) g {y = f x₁}  )
      x₁≡x₂

    subst-∀ :
       {a b c} {A : Set a} {B : A  Set b} {x₁ x₂ : A} {y : B x₁}
        {C : (x : A)  B x  Set c} {f : (y : B x₂)  C x₂ y}
        {x₁≡x₂ : x₁  x₂} 
      subst  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-∀ {B = B} {C = C} {x₁≡x₂ = x₁≡x₂} = elim
       {x₁ x₂} x₁≡x₂ 
         {y : B x₁} (f : (y : B x₂)  C x₂ y) 
         subst  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)))
       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  x  (y : B x)  C x y) (sym (refl x)) f y     ≡⟨ cong  eq  subst  x  (y : B x)  C x y) eq _ _) sym-refl 

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

         f y                                                    ≡⟨ sym $ dependent-cong 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  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-→ :
       {a b c} {A : Set a} {B : A  Set b} {x₁ x₂ : A} {y : B x₂}
      {C : A  Set c} {f : B x₁  C x₁}
      {x₁≡x₂ : x₁  x₂} 
      subst  x  B x  C x) x₁≡x₂ f y 
      subst C x₁≡x₂ (f (subst B (sym x₁≡x₂) y))
    subst-→ {B = B} {y = y} {C} {f} {x₁≡x₂} =
      subst  x  B x  C x) x₁≡x₂ f y                          ≡⟨ cong  eq  subst  x  B x  C x) eq f y) $ sym $
                                                                      sym-sym _ 
      subst  x  B x  C x) (sym $ sym x₁≡x₂) f y              ≡⟨ subst-∀ 

      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  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  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  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 :
       {a b c} {A : Set a} {x y : A}
      (B : A  Set b) {C : Set c} {f : B x  C}
      (x≡y : x  y) 
      subst  x  B x  C) x≡y f  f  subst B (sym x≡y)
    subst-→-domain B {C} x≡y = elim
       {x y} x≡y  (f : B x  C) 
                     subst  x  B x  C) x≡y f 
                     f  subst B (sym x≡y))
       x f 
         subst  x  B x  C) (refl x) f  ≡⟨ subst-refl  x  B x  _) _ 
         f                                 ≡⟨ cong (f ∘_) $ sym $ subst-refl≡id B 
         f  subst B (refl x)              ≡⟨ cong  p  f  subst B p) $ sym sym-refl ⟩∎
         f  subst B (sym (refl x))        )
      x≡y _

    -- 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 :
       {a b} {A : Set a} {B : Set b} {f g : A  B} {x y}
        {x≡y : x  y} {fx≡gx : f x  g x} 
      subst  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} = elim
       {x y} x≡y 
           (fx≡gx : f x  g x) 
           subst  z  f z  g z) x≡y fx≡gx 
           trans (sym (cong f x≡y)) (trans fx≡gx (cong g x≡y)))
       x fx≡gx 
           subst  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  p q  trans (sym p) (trans _ q))
                                                                                         (cong-refl f) (cong-refl g) ⟩∎
           trans (sym (cong f (refl x))) (trans fx≡gx (cong g (refl x)))   )
      _
      _

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

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

    -- Some rearrangement lemmas for equalities between equalities.

    [trans≡]≡[≡trans-symʳ] :
       {a} {A : Set a} {a₁ a₂ a₃ : A}
      (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
       {a₂ a₃} p₂₃ 
          {a₁} (p₁₂ : a₁  a₂) (p₁₃ : a₁  a₃) 
         (trans p₁₂ p₂₃  p₁₃)
           
         (p₁₂  trans p₁₃ (sym p₂₃)))
       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] :
       {a} {A : Set a} {a₁ a₂ a₃ : A}
      (p₁₂ : a₁  a₂) (p₁₃ : a₁  a₃) (p₂₃ : a₂  a₃) 
      (trans p₁₂ p₂₃  p₁₃)
        
      (p₂₃  trans (sym p₁₂) p₁₃)
    [trans≡]≡[≡trans-symˡ] p₁₂ = elim
       {a₁ a₂} p₁₂ 
          {a₃} (p₁₃ : a₁  a₃) (p₂₃ : a₂  a₃) 
         (trans p₁₂ p₂₃  p₁₃)
           
         (p₂₃  trans (sym p₁₂) p₁₃))
       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] :
       {a} {A : Set a} {x y : A} {p : x  y} {q : x  x} {r : y  y} 
      (subst  z  z  z) p q  r)
        
      (trans q p  trans p r)
    [subst≡]≡[trans≡trans] {p = p} {q} {r} = elim
       {x y} p  {q : x  x} {r : y  y} 
                   (subst  z  z  z) p q  r)
                     
                   (trans q p  trans p r))
       x {q r} 
         subst  z  z  z) (refl x) q  r   ≡⟨ cong (_≡ _) (subst-refl  z  z  z) _) 
         q  r                                ≡⟨ sym $ cong₂ _≡_ (trans-reflʳ _) (trans-reflˡ _) ⟩∎
         trans q (refl x)  trans (refl x) r  )
      p

    -- "Evaluation rule" for [subst≡]≡[trans≡trans].

    [subst≡]≡[trans≡trans]-refl :
       {a} {A : Set a} {x : A} {q : x  x} {r : x  x} 
      [subst≡]≡[trans≡trans] {p = refl x} {q = q} {r = r} 
      trans (cong (_≡ r) (subst-refl  z  z  z) q))
            (sym $ cong₂ _≡_ (trans-reflʳ q) (trans-reflˡ r))
    [subst≡]≡[trans≡trans]-refl {q = q} {r = r} =
      cong  f  f {q = q} {r = r}) $
        elim-refl  {x y} _  {q : x  x} {r : y  y}  _) _

    -- Sometimes one can turn two ("modified") copies of a proof into
    -- one.

    trans-cong-cong :
       {a b} {A : Set a} {B : Set b} {x y : A}
      (f : A  A  B) (p : x  y) 
      trans (cong  z  f z x) p)
            (cong  z  f y z) p) 
             cong  z  f z z) p
    trans-cong-cong f = elim
       {x y} p  trans (cong  z  f z x) p)
                         (cong  z  f y z) p) 
                          cong  z  f z z) p)
       x  trans (cong  z  f z x) (refl x))
                   (cong  z  f x z) (refl x))  ≡⟨ cong₂ trans (cong-refl  z  f z x)) (cong-refl  z  f x z)) 

             trans (refl (f x x)) (refl (f x x))  ≡⟨ trans-refl-refl 

             refl (f x x)                         ≡⟨ sym $ cong-refl  z  f z z) ⟩∎

             cong  z  f z z) (refl x)          )

    -- If f and g agree on a decidable subset of their common domain, then
    -- cong f eq is equal to (modulo some uses of transitivity) cong g eq
    -- for proofs eq between elements in this subset.

    cong-respects-relevant-equality :
       {a b} {A : Set a} {B : Set b} {x y} {x≡y : x  y} {f g : A  B}
      (p : A  Bool) (f≡g :  x  T (p x)  f x  g x)
      {px : T (p x)} {py : T (p y)} 
      trans (cong f x≡y) (f≡g y py)  trans (f≡g x px) (cong g x≡y)
    cong-respects-relevant-equality {f = f} {g} p f≡g = elim
       {x y} x≡y 
         {px : T (p x)} {py : T (p y)} 
         trans (cong f x≡y) (f≡g y py)  trans (f≡g x px) (cong g x≡y))
       x {px px′} 
         trans (cong f (refl x)) (f≡g x px′)  ≡⟨ cong (flip trans _) (cong-refl f) 
         trans (refl (f x)) (f≡g x px′)       ≡⟨ trans-reflˡ _ 
         f≡g x px′                            ≡⟨ cong (f≡g x) (T-irr (p x) px′ px) 
         f≡g x px                             ≡⟨ sym $ trans-reflʳ _ 
         trans (f≡g x px) (refl (g x))        ≡⟨ cong (trans _) (sym $ cong-refl _) ⟩∎
         trans (f≡g x px) (cong g (refl x))   )
      _
      where
      T-irr :  b  Proof-irrelevant (T b)
      T-irr true  _ _ = refl _
      T-irr false ()

    -- If f z evaluates to z for a decidable set of values which
    -- includes x and y, do we have
    --
    --   cong f x≡y ≡ x≡y
    --
    -- for any x≡y : x ≡ y? The equation above is not well-typed if f
    -- is a variable, but the approximation below can be proved.

    cong-roughly-id :
       {a} {A : Set a} (f : A  A) (p : A  Bool) {x y : A}
      (x≡y : x  y) (px : T (p x)) (py : T (p y))
      (f≡id :  z  T (p z)  f z  z) 
      cong f x≡y 
      trans (f≡id x px) (trans x≡y $ sym (f≡id y py))
    cong-roughly-id f p {x} {y} x≡y px py f≡id =
      let lemma =
            trans (cong id x≡y) (sym (f≡id y py))  ≡⟨ cong-respects-relevant-equality p  x  sym  f≡id x) ⟩∎
            trans (sym (f≡id x px)) (cong f x≡y)   
      in
      cong f x≡y                                                 ≡⟨ sym $ subst  eq  eq  trans (f≡id x px)
                                                                                                   (trans (cong id x≡y) (sym (f≡id y py))) 
                                                                                             cong f x≡y)
                                                                                ([trans≡]≡[≡trans-symˡ] _ _ _) id lemma 
      trans (f≡id x px) (trans (cong id x≡y) $ sym (f≡id y py))  ≡⟨ cong  eq  trans _ (trans eq _)) (sym $ cong-id _) ⟩∎
      trans (f≡id x px) (trans x≡y $ sym (f≡id y py))