Re: [isabelle] Finite_Set comp_fun_commute



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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