```------------------------------------------------------------------------
-- The Agda standard library
--
-- Well-founded induction
------------------------------------------------------------------------

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

open import Relation.Binary

module Induction.WellFounded where

open import Data.Product
open import Function
open import Induction
open import Level
open import Relation.Binary.PropositionalEquality hiding (trans)
open import Relation.Unary

private
variable
a b ℓ ℓ₁ ℓ₂ r : Level
A : Set a
B : Set b

------------------------------------------------------------------------
-- Definitions

-- When using well-founded recursion you can recurse arbitrarily, as
-- long as the arguments become smaller, and "smaller" is
-- well-founded.

WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
WfRec _<_ P x = ∀ y → y < x → P y

-- The accessibility predicate: x is accessible if everything which is
-- smaller than x is also accessible (inductively).

data Acc {A : Set a} (_<_ : Rel A ℓ) (x : A) : Set (a ⊔ ℓ) where
acc : (rs : WfRec _<_ (Acc _<_) x) → Acc _<_ x

-- The accessibility predicate encodes what it means to be
-- well-founded; if all elements are accessible, then _<_ is
-- well-founded.

WellFounded : Rel A ℓ → Set _
WellFounded _<_ = ∀ x → Acc _<_ x

------------------------------------------------------------------------
-- Basic properties

acc-inverse : ∀ {_<_ : Rel A ℓ} {x : A} (q : Acc _<_ x) →
(y : A) → y < x → Acc _<_ y
acc-inverse (acc rs) y y<x = rs y y<x

Acc-resp-≈ : {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} → Symmetric _≈_ →
_<_ Respectsʳ _≈_ → (Acc _<_) Respects _≈_
Acc-resp-≈ sym respʳ x≈y (acc rec) =
acc (λ z z<y → rec z (respʳ (sym x≈y) z<y))

------------------------------------------------------------------------
-- Well-founded induction for the subset of accessible elements:

module Some {_<_ : Rel A r} {ℓ} where

wfRecBuilder : SubsetRecursorBuilder (Acc _<_) (WfRec _<_ {ℓ = ℓ})
wfRecBuilder P f x (acc rs) = λ y y<x →
f y (wfRecBuilder P f y (rs y y<x))

wfRec : SubsetRecursor (Acc _<_) (WfRec _<_)
wfRec = subsetBuild wfRecBuilder

unfold-wfRec : (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P) {x : A} (q : Acc _<_ x) →
wfRec P f x q ≡ f x (λ y y<x → wfRec P f y (acc-inverse q y y<x))
unfold-wfRec P f (acc rs) = refl

------------------------------------------------------------------------
-- Well-founded induction for all elements, assuming they are all
-- accessible:

module All {_<_ : Rel A r} (wf : WellFounded _<_) ℓ where

wfRecBuilder : RecursorBuilder (WfRec _<_ {ℓ = ℓ})
wfRecBuilder P f x = Some.wfRecBuilder P f x (wf x)

wfRec : Recursor (WfRec _<_)
wfRec = build wfRecBuilder

wfRec-builder = wfRecBuilder

module FixPoint
{_<_ : Rel A r} (wf : WellFounded _<_)
(P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P)
(f-ext : (x : A) {IH IH′ : WfRec _<_ P x} → (∀ {y} y<x → IH y y<x ≡ IH′ y y<x) → f x IH ≡ f x IH′)
where

some-wfRec-irrelevant : ∀ x → (q q′ : Acc _<_ x) → Some.wfRec P f x q ≡ Some.wfRec P f x q′
some-wfRec-irrelevant = All.wfRec wf _
(λ x → (q q′ : Acc _<_ x) → Some.wfRec P f x q ≡ Some.wfRec P f x q′)
(λ { x IH (acc rs) (acc rs′) → f-ext x (λ y<x → IH _ y<x (rs _ y<x) (rs′ _ y<x)) })

open All wf ℓ
wfRecBuilder-wfRec : ∀ {x y} y<x → wfRecBuilder P f x y y<x ≡ wfRec P f y
wfRecBuilder-wfRec {x} {y} y<x with wf x
... | acc rs = some-wfRec-irrelevant y (rs y y<x) (wf y)

unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x (λ y _ → wfRec P f y)
unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec

------------------------------------------------------------------------
-- It might be useful to establish proofs of Acc or Well-founded using
-- combinators such as the ones below (see, for instance,
-- "Constructing Recursion Operators in Intuitionistic Type Theory" by
-- Lawrence C Paulson).

module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂}
(<₁⇒<₂ : ∀ {x y} → x <₁ y → x <₂ y) where

accessible : Acc _<₂_ ⊆ Acc _<₁_
accessible (acc rs) = acc (λ y y<x → accessible (rs y (<₁⇒<₂ y<x)))

wellFounded : WellFounded _<₂_ → WellFounded _<₁_
wellFounded wf = λ x → accessible (wf x)

-- DEPRECATED in v1.4.
module InverseImage {_<_ : Rel B ℓ} (f : A → B) where

accessible : ∀ {x} → Acc _<_ (f x) → Acc (_<_ on f) x
accessible (acc rs) = acc (λ y fy<fx → accessible (rs (f y) fy<fx))

wellFounded : WellFounded _<_ → WellFounded (_<_ on f)
wellFounded wf = λ x → accessible (wf (f x))

well-founded = wellFounded
{-# WARNING_ON_USAGE accessible
"Warning: accessible was deprecated in v1.4.
#-}
{-# WARNING_ON_USAGE wellFounded
"Warning: wellFounded was deprecated in v1.4.
#-}

-- DEPRECATED in v1.5.
module TransitiveClosure {A : Set a} (_<_ : Rel A ℓ) where

infix 4 _<⁺_

data _<⁺_ : Rel A (a ⊔ ℓ) where
[_]   : ∀ {x y} (x<y : x < y) → x <⁺ y
trans : ∀ {x y z} (x<y : x <⁺ y) (y<z : y <⁺ z) → x <⁺ z

downwardsClosed : ∀ {x y} → Acc _<⁺_ y → x <⁺ y → Acc _<⁺_ x
downwardsClosed (acc rs) x<y = acc (λ z z<x → rs z (trans z<x x<y))

mutual

accessible : ∀ {x} → Acc _<_ x → Acc _<⁺_ x
accessible acc-x = acc (accessible′ acc-x)

accessible′ : ∀ {x} → Acc _<_ x → WfRec _<⁺_ (Acc _<⁺_) x
accessible′ (acc rs) y [ y<x ]         = accessible (rs y y<x)
accessible′ acc-x    y (trans y<z z<x) =
downwardsClosed (accessible′ acc-x _ z<x) y<z

wellFounded : WellFounded _<_ → WellFounded _<⁺_
wellFounded wf = λ x → accessible (wf x)

{-# WARNING_ON_USAGE _<⁺_
"Warning: _<⁺_ was deprecated in v1.5.
#-}

-- DEPRECATED in v1.3.
module Lexicographic {A : Set a} {B : A → Set b}
(RelA : Rel A ℓ₁)
(RelB : ∀ x → Rel (B x) ℓ₂) where

data _<_ : Rel (Σ A B) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
left  : ∀ {x₁ y₁ x₂ y₂} (x₁<x₂ : RelA   x₁ x₂) → (x₁ , y₁) < (x₂ , y₂)
right : ∀ {x y₁ y₂}     (y₁<y₂ : RelB x y₁ y₂) → (x  , y₁) < (x  , y₂)

mutual

accessible : ∀ {x y} →
Acc RelA x → (∀ {x} → WellFounded (RelB x)) →
Acc _<_ (x , y)
accessible accA wfB = acc (accessible′ accA (wfB _) wfB)

accessible′ :
∀ {x y} →
Acc RelA x → Acc (RelB x) y → (∀ {x} → WellFounded (RelB x)) →
WfRec _<_ (Acc _<_) (x , y)
accessible′ (acc rsA) _    wfB ._ (left  x′<x) = accessible (rsA _ x′<x) wfB
accessible′ accA (acc rsB) wfB ._ (right y′<y) =
acc (accessible′ accA (rsB _ y′<y) wfB)

wellFounded : WellFounded RelA → (∀ {x} → WellFounded (RelB x)) →
WellFounded _<_
wellFounded wfA wfB p = accessible (wfA (proj₁ p)) wfB

well-founded = wellFounded

{-# WARNING_ON_USAGE _<_
"Warning: _<_ was deprecated in v1.3.
#-}
{-# WARNING_ON_USAGE accessible
"Warning: accessible was deprecated in v1.3."
#-}
{-# WARNING_ON_USAGE accessible′
"Warning: accessible′ was deprecated in v1.3."
#-}
{-# WARNING_ON_USAGE wellFounded
"Warning: wellFounded was deprecated in v1.3.
#-}

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

-- Version 1.0

module Inverse-image = InverseImage
module Transitive-closure = TransitiveClosure
```