(*  Title:      HOL/Induct/Ordinals.thy
    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
*)

section ‹Ordinals›

theory Ordinals
imports Main
begin

text ‹
 Some basic definitions of ordinal numbers. Draws an Agda
 development (in Martin-Löf type theory) by Peter Hancock (see
 🌐‹http://www.dcs.ed.ac.uk/home/pgh/chat.html›).
 ›

datatype ordinal =
    Zero
  | Succ ordinal
  | Limit "nat → ordinal"

primrec pred :: "ordinal → nat → ordinal option"
where
  "pred Zero n = None"
| "pred (Succ a) n = Some a"
| "pred (Limit f) n = Some (f n)"

abbreviation (input) iter :: "('a → 'a) → nat → ('a → 'a)"
  where "iter f n ≡ f ^^ n"

definition OpLim :: "(nat → (ordinal → ordinal)) → (ordinal → ordinal)"
  where "OpLim F a = Limit (λn. F n a)"

definition OpItw :: "(ordinal → ordinal) → (ordinal → ordinal)"  (‹⊔›)
  where "⊔f = OpLim (iter f)"

primrec cantor :: "ordinal → ordinal → ordinal"
where
  "cantor a Zero = Succ a"
| "cantor a (Succ b) = ⊔(λx. cantor x b) a"
| "cantor a (Limit f) = Limit (λn. cantor a (f n))"

primrec Nabla :: "(ordinal → ordinal) → (ordinal → ordinal)"  (‹∇›)
where
  "∇f Zero = f Zero"
| "∇f (Succ a) = f (Succ (∇f a))"
| "∇f (Limit h) = Limit (λn. ∇f (h n))"

definition deriv :: "(ordinal → ordinal) → (ordinal → ordinal)"
  where "deriv f = ∇(⊔f)"

primrec veblen :: "ordinal → ordinal → ordinal"
where
  "veblen Zero = ∇(OpLim (iter (cantor Zero)))"
| "veblen (Succ a) = ∇(OpLim (iter (veblen a)))"
| "veblen (Limit f) = ∇(OpLim (λn. veblen (f n)))"

definition "veb a = veblen a Zero"
definition "ε₀ = veb Zero"
definition "Γ₀ = Limit (λn. iter veb n Zero)"

end