[isabelle-dev] cardinality primitives in Isabelle/HOL?

Lawrence Paulson lp15 at cam.ac.uk
Tue Jan 22 15:58:44 CET 2019

I’m trying to install some of my new material and I’m wondering what to do with equipollence and related concepts:

definition eqpoll :: "'a set ⇒ 'b set ⇒ bool" (infixl "≈" 50)
  where "eqpoll A B ≡ ∃f. bij_betw f A B"

definition lepoll :: "'a set ⇒ 'b set ⇒ bool" (infixl "≲" 50)
  where "lepoll A B ≡ ∃f. inj_on f A ∧ f ` A ⊆ B"

definition lesspoll :: "'a set ⇒ 'b set ⇒ bool" (infixl ‹≺› 50) 
  where "A ≺ B == A ≲ B ∧ ~(A ≈ B)"

The raw definitions are extremely simple and could even be installed in the main Isabelle/HOL distribution. The basic properties of these concepts require few requisites. However, more advanced material requires the Cardinals development. 

Where do people think these definitions and proofs should be installed?


> On 27 Dec 2018, at 20:29, Makarius <makarius at sketis.net> wrote:
> On 27/12/2018 17:45, Traytel  Dmitriy wrote:
>> Hi Larry,
>> the HOL-Cardinals library might be just right for the purpose:
>> theory Scratch
>>  imports "HOL-Cardinals.Cardinals"
>> begin
>> lemma "|A| ≤o |B| ⟷ (∃f. inj_on f A ∧ f ` A ⊆ B)"
>>  by (rule card_of_ordLeq[symmetric])
>> lemma "|A| =o |B| ⟷ (∃f. bij_betw f A B)"
>>  by (rule card_of_ordIso[symmetric])
>> lemma
>>  assumes "|A| ≤o |B|" "|B| ≤o |A|"
>>  shows "|A| =o |B|"
>>  by (simp only: assms ordIso_iff_ordLeq)
>> end
>> The canonical entry point for the library is the above HOL-Cardinals.Cardinals. Many of the theorems and definitions are already in Main, because the (co)datatype package is based on them. The syntax is hidden in main—one gets it by importing HOL-Library.Cardinal_Notations (which HOL-Cardinals.Cardinals does transitively).
> It would be great to see this all consolidated and somehow unified, i.e.
> some standard notation in Main as proposed by Larry (potentially as
> bundles as proposed by Florian), and avoidance of too much no_syntax
> remove non-standard notation from Main.
> 	Makarius

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