Re: [isabelle] Finite_Set comp_fun_commute



Is your desired theorem true?

I would find it easier to believe if it also assumed "x : A" and "g ` A <= A".

Larry Paulson


On 19 Feb 2013, at 16:43, John Wickerson <jpw48 at cam.ac.uk> wrote:

> Hi Peter, thanks very much for this. Forgive me if I'm mistaken, but I don't understand how either of these approaches would help. I think I would still need to reason about terms like 
> 
>> fold f s (insert a A)
> 
> 
> in order to complete the induction, and I can't reason about such terms without knowing that f satisfies the "comp_fun_commute" property.
> 
> Let me state my problem more concretely... 
> 
> Finite_Set provides the following lemma (the first assumption comes from the context "comp_fun_commute"):
> 
>> lemma fold_image:
>>  assumes "⋀x y. f x ∘ f y = f y ∘ f x"
>>  assumes "finite A" and "inj_on g A"
>>  shows "fold f x (g ` A) = fold (f ∘ g) x A"
> 
> But I want the following lemma:
> 
>> lemma fold_image_stronger:
>>  assumes "⋀x y. ⟦ x ∈ A ; y ∈ A ⟧ ⟹ f x ∘ f y = f y ∘ f x"
>>  assumes "finite A" and "inj_on g A"
>>  shows "fold f x (g ` A) = fold (f ∘ g) x A"
> 
> 
> How might I prove it? It's tricky because all the other lemmas about Finite_Set.fold are in the "comp_fun_commute" context where 
> 
>> ⋀x y. f x ∘ f y = f y ∘ f x
> 
> holds, whereas I only have the weaker property
> 
>> ⋀x y. ⟦ x ∈ A ; y ∈ A ⟧ ⟹ f x ∘ f y = f y ∘ f x
> 
> available to me.
> 
> Thanks very much,
> 
> john
> 
> 
> 
> 
> On 19 Feb 2013, at 16:38, Peter Lammich wrote:
> 
>> Hi.
>> 
>> An alternative is to use an invariant rule, i.e., something like:
>> 
>> 
>> I s a0   !!x s a. [| I s a; x\in s |] ==> I (s-{x}) (f x a)
>> ------------------------------------------------------------ if finite s
>> I {} (fold f s a0)
>> 
>> 
>> or, alternatively, show that your proposition holds for folding over any
>> distinct list representing the set:
>> 
>> 
>> !!l. [| distinct l; set l = s |] ==> P (foldl f l a0)
>> --------------------------------------------------------  if finite s
>> P (fold f s a0)
>> 
>> 
>> Both rules (modulo my typos) should be provable by induction over the
>> finite set s.
>> 
>> --
>> Peter
>> 
>> 
>> 
>> On Di, 2013-02-19 at 16:01 +0100, John Wickerson wrote:
>>> Dear Isabelle,
>>> 
>>> This question is directed at anybody familiar with the Finite_Set theory...
>>> 
>>> http://isabelle.in.tum.de/library/HOL/Finite_Set.html
>>> 
>>> ... in particular, the Finite_Set.fold functional. Consider the term
>>> 
>>> Finite_Set.fold f s A
>>> 
>>> Various lemmas (e.g. Finite_Set.comp_fun_commute.fold_image) require me to show that f satisfies the "comp_fun_commute" property, i.e.
>>> 
>>> (1)    f x o f y = f y o f x
>>> 
>>> for all x and y. This is too strong a requirement for me. I can show that (1) holds for all x and y in A, but not for all x and y in general. Morally, I *should* only have to show that f commutes when given inputs drawn from A.
>>> 
>>> It would be quite a bit of hassle for me to convert these lemmas to stronger versions. So I was wondering if anybody has come across this problem before, or knows how to easily strengthen these lemmas, or has any other advice on this topic?
>>> 
>>> Thanks,
>>> john
>> 
>> 
>> 
> 
> 






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