```------------------------------------------------------------------------
-- The Agda standard library
--
-- Some properties imply others
------------------------------------------------------------------------

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

module Relation.Binary.Consequences where

open import Data.Maybe.Base using (just; nothing; decToMaybe)
open import Data.Sum.Base as Sum using (inj₁; inj₂)
open import Data.Product using (_,_)
open import Data.Empty.Irrelevant using (⊥-elim)
open import Function.Base using (_∘_; _\$_; flip)
open import Level using (Level)
open import Relation.Binary.Core
open import Relation.Binary.Definitions
open import Relation.Nullary using (yes; no; recompute)
open import Relation.Nullary.Decidable.Core using (map′)
open import Relation.Unary using (∁; Pred)

private
variable
a ℓ ℓ₁ ℓ₂ ℓ₃ ℓ₄ p : Level
A B : Set a

------------------------------------------------------------------------
-- Substitutive properties

module _ {_∼_ : Rel A ℓ} (R : Rel A p) where

subst⇒respˡ : Substitutive _∼_ p → R Respectsˡ _∼_
subst⇒respˡ subst {y} x′∼x Px′y = subst (flip R y) x′∼x Px′y

subst⇒respʳ : Substitutive _∼_ p → R Respectsʳ _∼_
subst⇒respʳ subst {x} y′∼y Pxy′ = subst (R x) y′∼y Pxy′

subst⇒resp₂ : Substitutive _∼_ p → R Respects₂ _∼_
subst⇒resp₂ subst = subst⇒respʳ subst , subst⇒respˡ subst

module _ {_∼_ : Rel A ℓ} {P : Pred A p} where

resp⇒¬-resp : Symmetric _∼_ → P Respects _∼_ → (∁ P) Respects _∼_
resp⇒¬-resp sym resp x∼y ¬Px Py = ¬Px (resp (sym x∼y) Py)

------------------------------------------------------------------------
-- Proofs for non-strict orders

module _ {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} where

total⇒refl : _≤_ Respects₂ _≈_ → Symmetric _≈_ →
Total _≤_ → _≈_ ⇒ _≤_
total⇒refl (respʳ , respˡ) sym total {x} {y} x≈y with total x y
... | inj₁ x∼y = x∼y
... | inj₂ y∼x = respʳ x≈y (respˡ (sym x≈y) y∼x)

total∧dec⇒dec : _≈_ ⇒ _≤_ → Antisymmetric _≈_ _≤_ →
Total _≤_ → Decidable _≈_ → Decidable _≤_
total∧dec⇒dec refl antisym total _≟_ x y with total x y
... | inj₁ x≤y = yes x≤y
... | inj₂ y≤x = map′ refl (flip antisym y≤x) (x ≟ y)

module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {≤₁ : Rel A ℓ₃} {≤₂ : Rel B ℓ₄} where

mono⇒cong : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ →
∀ {f} → f Preserves ≤₁ ⟶ ≤₂ → f Preserves ≈₁ ⟶ ≈₂
mono⇒cong sym reflexive antisym mono x≈y = antisym
(mono (reflexive x≈y))
(mono (reflexive (sym x≈y)))

antimono⇒cong : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ →
∀ {f} → f Preserves ≤₁ ⟶ (flip ≤₂) → f Preserves ≈₁ ⟶ ≈₂
antimono⇒cong sym reflexive antisym antimono p≈q = antisym
(antimono (reflexive (sym p≈q)))
(antimono (reflexive p≈q))

------------------------------------------------------------------------
-- Proofs for strict orders

module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where

trans∧irr⇒asym : Reflexive _≈_ → Transitive _<_ →
Irreflexive _≈_ _<_ → Asymmetric _<_
trans∧irr⇒asym refl trans irrefl x<y y<x =
irrefl refl (trans x<y y<x)

irr∧antisym⇒asym : Irreflexive _≈_ _<_ → Antisymmetric _≈_ _<_ →
Asymmetric _<_
irr∧antisym⇒asym irrefl antisym x<y y<x =
irrefl (antisym x<y y<x) x<y

asym⇒antisym : Asymmetric _<_ → Antisymmetric _≈_ _<_
asym⇒antisym asym x<y y<x = ⊥-elim (asym x<y y<x)

asym⇒irr : _<_ Respects₂ _≈_ → Symmetric _≈_ →
Asymmetric _<_ → Irreflexive _≈_ _<_
asym⇒irr (respʳ , respˡ) sym asym {x} {y} x≈y x<y =
asym x<y (respʳ (sym x≈y) (respˡ x≈y x<y))

tri⇒asym : Trichotomous _≈_ _<_ → Asymmetric _<_
tri⇒asym tri {x} {y} x<y x>y with tri x y
... | tri< _   _ x≯y = x≯y x>y
... | tri≈ _   _ x≯y = x≯y x>y
... | tri> x≮y _ _   = x≮y x<y

tri⇒irr : Trichotomous _≈_ _<_ → Irreflexive _≈_ _<_
tri⇒irr compare {x} {y} x≈y x<y with compare x y
... | tri< _   x≉y y≮x = x≉y x≈y
... | tri> x≮y x≉y y<x = x≉y x≈y
... | tri≈ x≮y _   y≮x = x≮y x<y

tri⇒dec≈ : Trichotomous _≈_ _<_ → Decidable _≈_
tri⇒dec≈ compare x y with compare x y
... | tri< _ x≉y _ = no  x≉y
... | tri≈ _ x≈y _ = yes x≈y
... | tri> _ x≉y _ = no  x≉y

tri⇒dec< : Trichotomous _≈_ _<_ → Decidable _<_
tri⇒dec< compare x y with compare x y
... | tri< x<y _ _ = yes x<y
... | tri≈ x≮y _ _ = no  x≮y
... | tri> x≮y _ _ = no  x≮y

trans∧tri⇒respʳ : Symmetric _≈_ → Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respectsʳ _≈_
trans∧tri⇒respʳ sym ≈-tr <-tr tri {x} {y} {z} y≈z x<y with tri x z
... | tri< x<z _ _ = x<z
... | tri≈ _ x≈z _ = ⊥-elim (tri⇒irr tri (≈-tr x≈z (sym y≈z)) x<y)
... | tri> _ _ z<x = ⊥-elim (tri⇒irr tri (sym y≈z) (<-tr z<x x<y))

trans∧tri⇒respˡ : Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respectsˡ _≈_
trans∧tri⇒respˡ ≈-tr <-tr tri {z} {_} {y} x≈y x<z with tri y z
... | tri< y<z _ _ = y<z
... | tri≈ _ y≈z _ = ⊥-elim (tri⇒irr tri (≈-tr x≈y y≈z) x<z)
... | tri> _ _ z<y = ⊥-elim (tri⇒irr tri x≈y (<-tr x<z z<y))

trans∧tri⇒resp : Symmetric _≈_ → Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respects₂ _≈_
trans∧tri⇒resp sym ≈-tr <-tr tri =
trans∧tri⇒respʳ sym ≈-tr <-tr tri ,
trans∧tri⇒respˡ ≈-tr <-tr tri

------------------------------------------------------------------------
-- Without Loss of Generality

module _  {_R_ : Rel A ℓ₁} {Q : Rel A ℓ₂} where

wlog : Total _R_ → Symmetric Q →
(∀ a b → a R b → Q a b) →
∀ a b → Q a b
wlog r-total q-sym prf a b with r-total a b
... | inj₁ aRb = prf a b aRb
... | inj₂ bRa = q-sym (prf b a bRa)

------------------------------------------------------------------------
-- Other proofs

module _ {R : REL A B p} where

dec⇒weaklyDec : Decidable R → WeaklyDecidable R
dec⇒weaklyDec dec x y = decToMaybe (dec x y)

dec⇒recomputable : Decidable R → Recomputable R
dec⇒recomputable dec {a} {b} = recompute \$ dec a b

module _ {R : REL A B ℓ₁} {S : REL A B ℓ₂} where

map-NonEmpty : R ⇒ S → NonEmpty R → NonEmpty S
map-NonEmpty f x = nonEmpty (f (NonEmpty.proof x))

module _ {R : REL A B ℓ₁} {S : REL B A ℓ₂} where

flip-Connex : Connex R S → Connex S R
flip-Connex f x y = Sum.swap (f y x)

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.6

subst⟶respˡ = subst⇒respˡ
{-# WARNING_ON_USAGE subst⟶respˡ
"Warning: subst⟶respˡ was deprecated in v1.6.
#-}
subst⟶respʳ = subst⇒respʳ
{-# WARNING_ON_USAGE subst⟶respʳ
"Warning: subst⟶respʳ was deprecated in v1.6.
#-}
subst⟶resp₂ = subst⇒resp₂
{-# WARNING_ON_USAGE subst⟶resp₂
"Warning: subst⟶resp₂ was deprecated in v1.6.
#-}
P-resp⟶¬P-resp = resp⇒¬-resp
{-# WARNING_ON_USAGE P-resp⟶¬P-resp
"Warning: P-resp⟶¬P-resp was deprecated in v1.6.
#-}
total⟶refl = total⇒refl
{-# WARNING_ON_USAGE total⟶refl
"Warning: total⟶refl was deprecated in v1.6.
#-}
total+dec⟶dec = total∧dec⇒dec
{-# WARNING_ON_USAGE total+dec⟶dec
"Warning: total+dec⟶dec was deprecated in v1.6.
#-}
trans∧irr⟶asym = trans∧irr⇒asym
{-# WARNING_ON_USAGE trans∧irr⟶asym
"Warning: trans∧irr⟶asym was deprecated in v1.6.
#-}
irr∧antisym⟶asym = irr∧antisym⇒asym
{-# WARNING_ON_USAGE irr∧antisym⟶asym
"Warning: irr∧antisym⟶asym was deprecated in v1.6.
#-}
asym⟶antisym = asym⇒antisym
{-# WARNING_ON_USAGE asym⟶antisym
"Warning: asym⟶antisym was deprecated in v1.6.
#-}
asym⟶irr = asym⇒irr
{-# WARNING_ON_USAGE asym⟶irr
"Warning: asym⟶irr was deprecated in v1.6.
#-}
tri⟶asym = tri⇒asym
{-# WARNING_ON_USAGE tri⟶asym
"Warning: tri⟶asym was deprecated in v1.6.
#-}
tri⟶irr = tri⇒irr
{-# WARNING_ON_USAGE tri⟶irr
"Warning: tri⟶irr was deprecated in v1.6.
#-}
tri⟶dec≈ = tri⇒dec≈
{-# WARNING_ON_USAGE tri⟶dec≈
"Warning: tri⟶dec≈ was deprecated in v1.6.
#-}
tri⟶dec< = tri⇒dec<
{-# WARNING_ON_USAGE tri⟶dec<
"Warning: tri⟶dec< was deprecated in v1.6.
#-}
trans∧tri⟶respʳ≈ = trans∧tri⇒respʳ
{-# WARNING_ON_USAGE trans∧tri⟶respʳ≈
"Warning: trans∧tri⟶respʳ≈ was deprecated in v1.6.
#-}
trans∧tri⟶respˡ≈ = trans∧tri⇒respˡ
{-# WARNING_ON_USAGE trans∧tri⟶respˡ≈
"Warning: trans∧tri⟶respˡ≈ was deprecated in v1.6.
#-}
trans∧tri⟶resp≈ = trans∧tri⇒resp
{-# WARNING_ON_USAGE trans∧tri⟶resp≈
"Warning: trans∧tri⟶resp≈ was deprecated in v1.6.