Theory Ordinals

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theory Ordinals
imports Main

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


header {* Ordinals *}

theory Ordinals imports Main begin

text {*
Some basic definitions of ordinal numbers. Draws an Agda
development (in Martin-L\"of type theory) by Peter Hancock (see
\url{http://www.dcs.ed.ac.uk/home/pgh/chat.html}).
*}


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


consts
pred :: "ordinal => nat => ordinal option"

primrec
"pred Zero n = None"
"pred (Succ a) n = Some a"
"pred (Limit f) n = Some (f n)"


consts
iter :: "('a => 'a) => nat => ('a => 'a)"

primrec
"iter f 0 = id"
"iter f (Suc n) = f o (iter f n)"


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


definition
OpItw :: "(ordinal => ordinal) => (ordinal => ordinal)" ("\<Squnion>") where
"\<Squnion>f = OpLim (iter f)"


consts
cantor :: "ordinal => ordinal => ordinal"

primrec
"cantor a Zero = Succ a"
"cantor a (Succ b) = \<Squnion>(λx. cantor x b) a"
"cantor a (Limit f) = Limit (λn. cantor a (f n))"


consts
Nabla :: "(ordinal => ordinal) => (ordinal => ordinal)" ("∇")

primrec
"∇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 = ∇(\<Squnion>f)"


consts
veblen :: "ordinal => ordinal => ordinal"

primrec
"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 0 = veb Zero"
definition 0 = Limit (λn. iter veb n Zero)"

end