[isabelle-dev] Simplification theorems with more general typeclasses

[Mathias Fleury]

Hi all,

after some more experiments, I found out that there is another difference between explicit typeclass annotations and lemmas in the context: the former theorems are included in instantiations but are not included in interpretations. This usually does not make a difference, since there is usually a single order on a type.

Instead, I introduced an additional typeclass to the hierarchy. The change was successfully tested on testboard (mercurial diff <http://isabelle.in.tum.de/repos/testboard/rev/18f26b6779ad>, status <https://ci.isabelle.systems/jenkins/job/testboard/117/>), and does not need any AFP change.


Does someone have an opinion on this change?


Mathias
 
> On 04 Jul 2016, at 14:20, Mathias Fleury <Mathias.Fleury at ens-rennes.fr> wrote:
> 
> Hi Johannes,
> 
> 
> the multiset ordering (contrary to the subset ordering) does not have this property:
> 
> 	lemma "{#0#} <= {#Suc 0#}”
> 	  unfolding Multiset_Order.le_multiset⇩H⇩O by auto
> 
> (the actual notation is #⊆# and not <=).
> 
> 
> I tried locally to apply the changes of my previous email this week-end. Except some proofs inside the typeclass definitions (i.e. in the files Groups.thy, Rings.thy, and Missing_Ring.thy), no other changes were needed in Isabelle or the AFP.
> 
> 
> Thanks for your answer,
> Mathias
> 
> 
>> On 04 Jul 2016, at 13:22, Johannes Hölzl <hoelzl at in.tum.de <mailto:hoelzl at in.tum.de>> wrote:
>> 
>> Hi Mathias,
>> 
>> there is at least the type class 'canonically_ordered_monoid' which has
>> the property a <= b <--> ?c. a + c = b which implies 0 <= a for all a.
>> Are the multisets already in this typeclass?
>> 
>>  - Johannes
>> 
>> 
>> Am Dienstag, den 28.06.2016, 10:04 +0100 schrieb Mathias Fleury:
>>> Dear type-classes and simplifier experts,
>>> 
>>> in the plan of instantiating multisets with the multiset ordering, I
>>> am trying to instantiate the multisets with additional typeclasses to
>>> get specific simplification theorems. The aim is to mimic the
>>> simplifier’s behaviour of other types like natural numbers. One of my
>>> problems can be nicely illustrated by the following lemma: “M <= M +
>>> N <-> 0 <= N”. 
>>> 
>>> 
>>> Analog simplification rules already exist for rings (e.g., natural
>>> numbers*) and ordered groups too:
>>> 	thm
>>> Rings.linordered_semiring_class.less_eq_add_cancel_left_greater_eq_ze
>>> ro
>>> 	thm Groups.ordered_ab_group_add_class.le_add_same_cancel1
>>> Both rules are stating that “M <= M + N <—> 0 <= N” and are marked as
>>> [simp].
>>> 
>>> 
>>> However, the multisets are neither a group (no inverse for the law
>>> “+”) nor a ring (no multiplication). I could duplicate the theorems,
>>> but I noticed that the proofs of the theorems do only rely on the
>>> fact it is a monoid_add (for the zero element) and an
>>> ordered_ab_semigroup_add_imp_le (for the order). The following
>>> theorem would work too and is general enough to include the multiset
>>> case:
>>> 
>>> lemma le_add_same_cancel1 [simp]:
>>>   “(a :: 'a :: {monoid_add, ordered_ab_semigroup_add_imp_le}) ≤ a + b
>>> ⟷ 0 ≤ b”
>>>   using add_le_cancel_left [of a 0] by simp
>>> 
>>> 
>>> Are there any obvious differences between this more general version
>>> with explicit type class annotations
>>> and Groups.ordered_ab_group_add_class.le_add_same_cancel1? If no,
>>> would it make sense to use this version in Isabelle?
>>> 
>>> 
>>> 
>>> Thanks in advance,
>>> Mathias Fleury
>>> 
>>> 
>>> 
>>> 
>>> * for natural numbers, the simproc
>>> Numeral_Simprocs.natle_cancel_numerals is able to do it too.
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