*To*: Christian Sternagel <c.sternagel at gmail.com>*Subject*: Re: [isabelle] extending well-founded partial orders to total well-founded orders*From*: Andrei Popescu <uuomul at yahoo.com>*Date*: Tue, 19 Feb 2013 14:46:22 -0800 (PST)*Cc*: cl-isabelle-users at lists.cam.ac.uk*In-reply-to*: <5122F574.70306@gmail.com>

Hi Christian, No, this is not what I had in mind. Transfinite recursion does not need Zorn. But I refrain from any further explanations, since hopefully the other route works out for you. Andrei --- On Tue, 2/19/13, Christian Sternagel <c.sternagel at gmail.com> wrote: From: Christian Sternagel <c.sternagel at gmail.com> Subject: Re: [isabelle] extending well-founded partial orders to total well-founded orders To: "Andrei Popescu" <uuomul at yahoo.com> Cc: cl-isabelle-users at lists.cam.ac.uk Date: Tuesday, February 19, 2013, 5:45 AM Thanks Andrei, concerning your proof sketch, I have some problems applying your recipe (or the one from the blog post, for that matter) in a concrete proof. Lets first review my setting (please let me know if any of this is "strange" in any way). I use the following definitions (mostly from AFP/Well_Quasi_Orders): definition irreflp_on :: "('a ⇒ 'a ⇒ bool) ⇒ 'a set ⇒ bool" where "irreflp_on P A = (∀a∈A. ¬ P a a)" definition transp_on :: "('a ⇒ 'a ⇒ bool) ⇒ 'a set ⇒ bool" where "transp_on P A = (∀x∈A. ∀y∈A. ∀z∈A. P x y ∧ P y z ⟶ P x z)" definition po_on :: "('a ⇒ 'a ⇒ bool) ⇒ 'a set ⇒ bool" where "po_on P A = (irreflp_on P A ∧ transp_on P A)" definition total_on :: "('a ⇒ 'a ⇒ bool) ⇒ 'a set ⇒ bool" where "total_on P A = (∀x∈A. ∀y∈A. x = y ∨ P x y ∨ P y x)" definition wfp_on :: "('a ⇒ 'a ⇒ bool) ⇒ 'a set ⇒ bool" where "wfp_on P A = (¬ (∃f. ∀i. f i ∈ A ∧ P (f (Suc i)) (f i)))" definition wellorder_on where "wellorder_on P A = (po_on P A ∧ wfp_on P A ∧ total_on P A)" definition ext_on where "ext_on P Q A = (∀x∈A. ∀y∈A. Q x y ⟶ P x y)" For "wfp_on" I derived the following induction schema: wfp_on_induct: wfp_on ?P ?A ⟹ ?x ∈ ?A ⟹ (⋀y. y ∈ ?A ⟹ (⋀x. x ∈ ?A ⟹ ?P x y ⟹ ?Q x) ⟹ ?Q y) ⟹ ?Q ?x Moreover from Zorn.thy I derived the following variant of the well-order theorem: wellorder_on: "∃W. wellorder_on W A" Let P be the given well-founded partial order on A. Then, we obtain a well-order W on A by the well-order theorem. Using wfp_on_induct, I can start a proof { fix x assume "x ∈ A" with `wfp_on W A` have "wellorder_on N {y∈A. W^== x} ∧ ext_on N P {y∈A. W^== y x}" proof (induct rule: wfp_on_induct) for some appropriate definition of N (using worec (?)). Is that the transfinite induction you were referring to? Even if I would succeed with this proof, I don't see how I could derive "wellorder_on N A & ext_on N P A" from it. What am I doing wrong? cheers chris On 02/19/2013 09:33 AM, Andrei Popescu wrote: > Hi Christian, > > >> I guess since Isabelle2013 this is now "~~/src/HOL/Cardinals/", right? > > Right. > > >> Could you elaborate on the mentioned finite recursion combinator and > how it is used? > > The worec combinator, > > worec:: "(('a => 'b) => 'a => 'b) => 'a => 'b" > > (defined in the context of a fixed wellorder r on 'a) > > is just a slightly more convenient version of wfrec. > It is used to define a function f :: 'a => 'b by specifying, > for each x :: 'a, the value of f on x in terms of the values of f > on all elements less than x w.r.t. r, i.e., in my notation, all > elements ofunderS x. This would ideally employ an operator > of type > > Prod x : 'a. > (underS x => 'b) => 'b > > In HOL, the same is achieved by an > "admissible" operator of a less informative type. > > The only relevant facts are below: > > definition > adm_wo:: "(('a => 'b) => 'a => 'b) => bool" > where > "adm_wo H ≡ ∀f g x. (∀y ∈ underS x. f y = g y) --> H f x = H g x" > > lemma worec_fixpoint: > assumes "adm_wo H" > shows "worec H = H (worec H)" > > Cheers, > Andrei > > > > --- On *Mon, 2/18/13, Christian Sternagel /<c.sternagel at gmail.com>/* wrote: > > > From: Christian Sternagel <c.sternagel at gmail.com> > Subject: Re: [isabelle] extending well-founded partial orders to > total well-founded orders > To: "Andrei Popescu" <uuomul at yahoo.com> > Cc: cl-isabelle-users at lists.cam.ac.uk > Date: Monday, February 18, 2013, 8:32 AM > > Dear Andrei, > > finally deadlines are over for the time being and I found your email > again ;) > > On 01/19/2013 12:22 AM, Andrei Popescu wrote: > > My AFP formalization ordinals > > > > http://afp.sourceforge.net/entries/Ordinals_and_Cardinals.shtml > > I guess since Isabelle2013 this is now "~~/src/HOL/Cardinals/", right? > > > (hopefully) provides the necessary ingredients: Initial segments in > > Wellorder_Embedding, ordinal sum in theory > Constructions_on_Wellorders, > > and a transfinite recursion combinator (a small adaptation of the > > wellfounded combinator) in theory Wellorder_Relation. > > Could you elaborate on the mentioned finite recursion combinator and > how it is used? > > thanks in advance, > > chris > >

**References**:**Re: [isabelle] extending well-founded partial orders to total well-founded orders***From:*Christian Sternagel

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