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{-# OPTIONS --cubical-compatible #-}
module Padova2025.ProvingBasics.Termination.BoveCapretta.Digits where

Digits, revisited

open import Padova2025.ProgrammingBasics.Naturals.Base
open import Padova2025.ProgrammingBasics.Naturals.Arithmetic
open import Padova2025.ProvingBasics.Termination.Ordering
open import Padova2025.ProvingBasics.Termination.WellFounded.Base

Representing the function

data Def : ℕ → Set
f₀ : (n : ℕ) → Def n → ℕ

data Def where
  base : Def zero
  step : {x : ℕ} → Def (half (succ x)) → Def (succ x)

f₀ zero       base     = zero
f₀ x@(succ _) (step p) = succ (f₀ (half x) p)

Verifying totality by well-founded induction

total : (x : ℕ) → Def x
total x = go x (ℕ-wf x)
  where
  go : (x : ℕ) (gas : Acc _<'_ x) → Def x
  go zero       gas     = {!!}
  go x@(succ _) (acc f) = {!!}

digits₅ : ℕ → ℕ
digits₅ x = f₀ x (total x)

Verifying the defining equation

open import Padova2025.ProvingBasics.Equality.Base
open import Padova2025.ProvingBasics.Equality.General

Def-prop : {x : ℕ} (p q : Def x) → p ≡ q
Def-prop = {!!}

f₀-extensional : (x : ℕ) (p q : Def x) → f₀ x p ≡ f₀ x q
f₀-extensional = {!!}

digits₅-eq : (x : ℕ) → digits₅ (succ x) ≡ succ (digits₅ (half (succ x)))
digits₅-eq = {!!}