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

  private
    open module R {} = Reflexive (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

  -- 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

  -- 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 ) :
         Set (lsuc (a  p)) where

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

  private
    open Reflexive′ 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 rule" for subst.

  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  h  h p) $
      elim-refl  {u v} _  P u  P v)  x p  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 ) :
         Set (lsuc (a  p)) where

  open Reflexive′ 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 }
  (eq :   {a p}  Equality-with-substitutivity-and-contractibility
                     a p reflexive)
  where

  private
    open Reflexive′ 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.

  infixr 0 _≡⟨_⟩_ _≡⟨⟩_
  infix  0 finally

  _≡⟨_⟩_ :  {a} {A : Set a} x {y z : A}  x  y  y  z  x  z
  _ ≡⟨ x≡y  y≡z = trans 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
          ; _≡⟨_⟩_; _≡⟨⟩_; 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.

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

  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                                                                

  -- 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 inspect idiom.

  data Inspect {a} {A : Set a} (x : A) : Set a where
    _with-≡_ : (y : A) (eq : x  y)  Inspect x

  inspect :  {a} {A : Set a} (x : A)  Inspect x
  inspect x = x with-≡ refl x

  -- 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

  -- Proof irrelevance (or maybe "data irrelevance", depending on what
  -- the set 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 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 _)          )
      _

  abstract

    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

    -- 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 (used in
    -- Equivalence.equality-equivalence-lemma).

    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                                          

    -- 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 _                                          

  -- 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))                   

    -- "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 simplification 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-≡,≡→≡-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} {x : A} {B : Set b} {y₁ y₂ : B}
      (y₁≡y₂ : y₁  y₂) (C : A  B  Set c) {f : (x : A)  C x y₁} 
      subst (C x) y₁≡y₂ (f x) 
      subst  y  (x : A)  C x y) y₁≡y₂ f x
    push-subst-application {x = x} y₁≡y₂ C {f} = elim¹
       y₁≡y₂  subst (C x) y₁≡y₂ (f x) 
                 subst  y   x  C x y) y₁≡y₂ f x)
      (subst (C x) (refl _) (f x)              ≡⟨ subst-refl (C x) _ 
       f x                                     ≡⟨ cong  g  g x) $ sym $ subst-refl  y   x  C x y) _ ⟩∎
       subst  y   x  C x y) (refl _) f x  )
      y₁≡y₂

    push-subst-implicit-application :
       {a b c} {A : Set a} {x : A} {B : Set b} {y₁ y₂ : B}
      (y₁≡y₂ : y₁  y₂) (C : A  B  Set c) {f : {x : A}  C x y₁} 
      subst (C x) y₁≡y₂ (f {x = x}) 
      subst  y  {x : A}  C x y) y₁≡y₂ f {x = x}
    push-subst-implicit-application {x = x} y₁≡y₂ C {f} =
      elim¹
         y₁≡y₂  subst (C x) y₁≡y₂ f 
                   subst  y   {x}  C x y) y₁≡y₂ f)
        (subst (C x) (refl _) f                  ≡⟨ subst-refl (C x) _ 
         f                                       ≡⟨ cong  g  g {x = x}) $ sym $ subst-refl  y   {x}  C x y) _ ⟩∎
         subst  y   {x}  C x y) (refl _) f  )
        y₁≡y₂

    -- 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 {A = A} f p =
      elim  {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)))
            x px px′ f≡id  helper x (p x) px px′ (f≡id x))
      where
      helper :
        (x : A) (b : Bool) (px px′ : T b)
        (f≡id : T b  f x  x) 
        cong f (refl x) 
        trans (f≡id px) (trans (refl x) $ sym (f≡id px′))
      helper x false px _ f≡id = ⊥-elim px
      helper x true  _  _ f≡id =
        cong f (refl x)                                 ≡⟨ cong-refl f 
        refl (f x)                                      ≡⟨ sym $ trans-symʳ _ 
        trans (f≡id _) (sym (f≡id _))                   ≡⟨ cong (trans (f≡id _)) $ sym $ trans-reflˡ _ ⟩∎
        trans (f≡id _) (trans (refl x) $ sym (f≡id _))