------------------------------------------------------------------------
-- The Agda standard library
--
-- Induction over Subset
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Data.Fin.Subset.Induction where
open import Data.Nat.Base using (ℕ)
open import Data.Nat.Induction using (<-wellFounded)
open import Data.Fin.Subset using (Subset; ∁; _⊂_; _⊃_; ∣_∣)
open import Data.Fin.Subset.Properties using (p⊂q⇒∣p∣<∣q∣; p⊂q⇒∁p⊃∁q)
open import Induction using (RecStruct)
open import Induction.WellFounded as WF
open import Level using (Level)
import Relation.Binary.Construct.On as On using (wellFounded)
private
variable
ℓ : Level
n : ℕ
------------------------------------------------------------------------
-- Re-export accessability
open WF public using (Acc; acc)
------------------------------------------------------------------------
-- Complete induction based on _⊂_
⊂-Rec : RecStruct (Subset n) ℓ ℓ
⊂-Rec = WfRec _⊂_
⊂-wellFounded : WellFounded {A = Subset n} _⊂_
⊂-wellFounded = Subrelation.wellFounded p⊂q⇒∣p∣<∣q∣
(On.wellFounded ∣_∣ <-wellFounded)
------------------------------------------------------------------------
-- Complete induction based on _⊃_
⊃-Rec : RecStruct (Subset n) ℓ ℓ
⊃-Rec = WfRec _⊃_
⊃-wellFounded : WellFounded {A = Subset n} _⊃_
⊃-wellFounded = Subrelation.wellFounded p⊂q⇒∁p⊃∁q
(On.wellFounded ∁ ⊂-wellFounded)
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