[CakeML] Set theory axioms

[Ramana Kumar]

Hi Peter,

This indeed looks very nice.

One suggestion I would make: would it be possible to extend the existing
theory, rather than making an extended copy?
It seems to me that your additional definitions and proofs could go into
the existing setSpecScript.sml without any problem.
The only place of conflict is where you want to define is_set_theory with
the additional axioms. Perhaps you could simply define a new constant
is_full_set_theory or similar for that purpose?

I am hoping to avoid diverging forks in the future, in case we make any
updates to setSpecScript.sml (unlikely, but there may be a few little
cleanups).

Cheers,
Ramana

On 30 June 2016 at 12:48, Konrad Slind <konrad.slind at gmail.com> wrote:

> Looks nice! Note that Michael has done an extensive development of the
> ordinals in
>
>   <holdir>/examples/set-theory/hol_sets
>
> and transfinite induction is in there somewhere.
>
> Konrad.
>
>
> On Wed, Jun 29, 2016 at 5:09 PM, Peter Vincent Homeier <
> palantir at trustworthytools.com> wrote:
>
>> I've just made my first contribution to CakeML. This consists of two
>> files within the subdirectory "flame/set-theory", which are modified
>> versions of the corresponding files under "candle/set-theory".
>>
>> My primary reference for this work was "Set Theory, The Third Millennium
>> Edition" by Thomas Jech, Springer, 2006.
>>
>> These files extend the existing specification of set theory to full ZFC.
>> This includes two axioms that were previously omitted, the Axiom of
>> Regularity and the Axiom Schema of Replacement:
>>
>> val regular_def = Define`
>>   regular ^mem ⇔ ∀x. (∃y. mem y x) ⇒ ∃y. mem y x ∧ ∀z. ~(mem z x ∧ mem z
>> y)`
>>
>> val is_functional_def = Define`
>>   is_functional (R:'a -> 'b -> bool) ⇔ ∀x y z. R x y ∧ R x z ⇒ y = z`
>>
>> val replacement_def = Define`
>>   replacement ^mem ⇔
>>       ∀R. is_functional R ⇒
>>           ∀d. ∃r. ∀y. mem y r ⇔ ∃x. mem x d ∧ R x y`
>>
>> val is_set_theory_def = Define`
>>   is_set_theory ^mem ⇔
>>     extensional mem ∧
>>     (∃sub. is_separation mem sub) ∧
>>     (∃power. is_power mem power) ∧
>>     (∃union. is_union mem union) ∧
>>     (∃upair. is_upair mem upair) ∧
>>     regular mem ∧
>>     replacement mem`
>>
>> In addition, I have modified the Axiom of Infinity,
>>
>> val is_infinite_def = Define`
>>   is_infinite ^mem s = INFINITE {a | a <: s}`
>>
>> not really deleting it, but replacing it by the inductive property, taken
>> from page 12 of "Set Theory, The Third Millennium Edition" by Thomas Jech,
>> Springer, 2006, using the successor operator as defined (Definition 1.20)
>> on page 19 of "Introduction to Set Theory" by J. Donald Monk, McGraw Hill,
>> 1969.
>>
>> val suc_def = Define`
>>   suc ^mem x = x ∪ Unit x`
>>
>> val _ = Parse.overload_on("Suc",``suc ^mem``)
>>
>> val is_inductive_def = Define`
>>   is_inductive ^mem s ⇔
>>       ∅ <: s ∧ ∀x. x <: s ⇒ Suc x <: s`
>>
>> val is_model_def = Define`
>>   is_model ^s ⇔
>>     is_set_theory mem ∧
>>     is_inductive mem indset ∧
>>     is_choice mem ch`
>>
>> There is included a proof that the inductive axiom implies the prior
>> infinity axiom; this proof depends on the Axiom of Regularity.
>>
>>     inductive_set_infinite
>>       |- is_set_theory mem ∧ is_inductive mem indset ⇒
>>          is_infinite mem indset
>>
>> I also added several new set theory constants, such as binary
>> intersection, inverse of a function, image of a function on a set,
>> dependent function space, and dependent product space. The last three
>> required the Axiom Schema of Replacement to define.
>>
>> I am hoping to stimulate some discussion and debate. It is necessary to
>> extend the existing partial specification of set theory to support the
>> construction of the set-theory model of HOL-Omega. Besides that, there will
>> probably be other uses for ZFC set theory, and I felt we needed a more
>> complete axiomatic basis.
>>
>> The next steps would presumably be the definition of the ordinals, with
>> transfinite induction, and the cardinals.
>>
>> Enjoy, and please let me know what you think.
>>
>> Peter
>>
>>
>> "In Your majesty ride prosperously
>> because of truth, humility, and righteousness;
>> and Your right hand shall teach You awesome things." (Psalm 45:4)
>>
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>>
>>
>
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