(* Title: HOL/Induct/Ordinals.thy
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
*)
section \<open>Ordinals\<close>
theory Ordinals
imports Main
begin
text \<open>
Some basic definitions of ordinal numbers. Draws an Agda
development (in Martin-L\"of type theory) by Peter Hancock (see
\<^url>\<open>http://www.dcs.ed.ac.uk/home/pgh/chat.html\<close>).
\<close>
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 (\n. 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) = \(\x. cantor x b) a"
| "cantor a (Limit f) = Limit (\n. 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 (\n. \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 (\n. veblen (f n)))"
definition "veb a = veblen a Zero"
definition "\\<^sub>0 = veb Zero"
definition "\\<^sub>0 = Limit (\n. iter veb n Zero)"
end
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