------------------------------------------------------------------------
-- The Agda standard library
--
-- Various forms of induction for natural numbers
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Nat.Induction where

open import Data.Nat.Base
open import Data.Nat.Properties using (<⇒<′)
open import Data.Product.Base using (_×_; _,_)
open import Data.Unit.Polymorphic.Base
open import Induction
open import Induction.WellFounded as WF
open import Level using (Level)

private
  variable
     : Level

------------------------------------------------------------------------
-- Re-export accessability

open WF public using (Acc; acc)

------------------------------------------------------------------------
-- Ordinary induction

Rec :    RecStruct   
Rec  P zero    = 
Rec  P (suc n) = P n

recBuilder : RecursorBuilder (Rec )
recBuilder P f zero    = _
recBuilder P f (suc n) = f n (recBuilder P f n)

rec : Recursor (Rec )
rec = build recBuilder

------------------------------------------------------------------------
-- Complete induction

CRec :    RecStruct   
CRec  P zero    = 
CRec  P (suc n) = P n × CRec  P n

cRecBuilder : RecursorBuilder (CRec )
cRecBuilder P f zero    = _
cRecBuilder P f (suc n) = f n ih , ih
  where ih = cRecBuilder P f n

cRec : Recursor (CRec )
cRec = build cRecBuilder

------------------------------------------------------------------------
-- Complete induction based on _<′_

<′-Rec : RecStruct   
<′-Rec = WfRec _<′_

-- mutual definition

<′-wellFounded : WellFounded _<′_
<′-wellFounded′ :  n  <′-Rec (Acc _<′_) n

<′-wellFounded n = acc (<′-wellFounded′ n)

<′-wellFounded′ (suc n) n <′-base       = <′-wellFounded n
<′-wellFounded′ (suc n) m (<′-step m<n) = <′-wellFounded′ n m m<n

module _ { : Level} where
  open WF.All <′-wellFounded  public
    renaming ( wfRecBuilder to <′-recBuilder
             ; wfRec        to <′-rec
             )
    hiding (wfRec-builder)

------------------------------------------------------------------------
-- Complete induction based on _<_

<-Rec : RecStruct   
<-Rec = WfRec _<_

<-wellFounded : WellFounded _<_
<-wellFounded = Subrelation.wellFounded <⇒<′ <′-wellFounded

-- A version of `<-wellFounded` that cheats by skipping building
-- the first billion proofs. Use this when you require the function
-- using the proof of well-foundedness to evaluate fast.
--
-- IMPORTANT: You have to be a little bit careful when using this to always
-- make the function be strict in some other argument than the accessibility
-- proof, otherwise you will have neutral terms unfolding a billion times
-- before getting stuck.
<-wellFounded-fast : WellFounded _<_
<-wellFounded-fast = <-wellFounded-skip 1000000000
  where
  <-wellFounded-skip :  (k : )  WellFounded _<_
  <-wellFounded-skip zero    n       = <-wellFounded n
  <-wellFounded-skip (suc k) zero    = <-wellFounded 0
  <-wellFounded-skip (suc k) (suc n) = acc  m _  <-wellFounded-skip k m)

module _ { : Level} where
  open WF.All <-wellFounded  public
    renaming ( wfRecBuilder to <-recBuilder
             ; wfRec        to <-rec
             )
    hiding (wfRec-builder)