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

open import Relation.Binary

module Induction.WellFounded where

open import Data.Product
open import Function
open import Induction
open import Level
open import Relation.Unary

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

WfRec :  {a r} {A : Set a}  Rel A r   {}  RecStruct A  _
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 } {A : Set a} (_<_ : Rel A ) (x : A) : Set (a  ) 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.

Well-founded :  {a } {A : Set a}  Rel A   Set _
Well-founded _<_ =  x  Acc _<_ x

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

module Some {a lt} {A : Set a} {_<_ : Rel A lt} {} where

  wfRec-builder : SubsetRecursorBuilder (Acc _<_) (WfRec _<_ {ℓ = })
  wfRec-builder P f x (acc rs) = λ y y<x 
    f y (wfRec-builder P f y (rs y y<x))

  wfRec : SubsetRecursor (Acc _<_) (WfRec _<_)
  wfRec = subsetBuild wfRec-builder

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

module All {a lt} {A : Set a} {_<_ : Rel A lt}
           (wf : Well-founded _<_)  where

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

  wfRec : Recursor (WfRec _<_)
  wfRec = build wfRec-builder

-- 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 {a ℓ₁ ℓ₂} {A : Set a}
                   {_<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  y y<x  accessible (rs y (<₁⇒<₂ y<x)))

  well-founded : Well-founded _<₂_  Well-founded _<₁_
  well-founded wf = λ x  accessible (wf x)

module Inverse-image {a b } {A : Set a} {B : Set b} {_<_ : Rel B }
                     (f : A  B) where

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

  well-founded : Well-founded _<_  Well-founded (_<_ on f)
  well-founded wf = λ x  accessible (wf (f x))

module Transitive-closure {a } {A : Set a} (_<_ : Rel A ) where

  infix 4 _<⁺_

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

  downwards-closed :  {x y}  Acc _<⁺_ y  x <⁺ y  Acc _<⁺_ x
  downwards-closed (acc rs) x<y = acc  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) =
      downwards-closed (accessible′ acc-x _ z<x) y<z

  well-founded : Well-founded _<_  Well-founded _<⁺_
  well-founded wf = λ x  accessible (wf x)

module Lexicographic {a b ℓ₁ ℓ₂} {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}  Well-founded (RelB x)) 
                 Acc _<_ (x , y)
    accessible accA wfB = acc (accessible′ accA (wfB _) wfB)

    accessible′ :
       {x y} 
      Acc RelA x  Acc (RelB x) y  (∀ {x}  Well-founded (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)

  well-founded : Well-founded RelA  (∀ {x}  Well-founded (RelB x)) 
                 Well-founded _<_
  well-founded wfA wfB p = accessible (wfA (proj₁ p)) wfB