*To*: Andrei Popescu <uuomul at yahoo.com>*Subject*: Re: [isabelle] extending well-founded partial orders to total well-founded orders*From*: Christian Sternagel <c.sternagel at gmail.com>*Date*: Tue, 19 Feb 2013 12:45:56 +0900*Cc*: cl-isabelle-users at lists.cam.ac.uk*In-reply-to*: <1361234001.18881.YahooMailClassic@web120601.mail.ne1.yahoo.com>*References*: <1361234001.18881.YahooMailClassic@web120601.mail.ne1.yahoo.com>*User-agent*: Mozilla/5.0 (X11; Linux x86_64; rv:17.0) Gecko/20130110 Thunderbird/17.0.2

Thanks Andrei,

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

wellorder_on: "∃W. wellorder_on W A"

{ 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)

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

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