Looks like a special case of simulation:
lemma rtrancl_sim:
assumes "⋀x y a. ⟦(x, y) ∈ r; (x,a)∈S⟧ ⟹ ∃b. (y,b)∈S ∧ (a, b) ∈ s"
and "(x, y) ∈ r⇧*" and "(x,a)∈S"
shows "∃b. (y,b)∈S ∧ (a, b) ∈ s⇧*"
using assms(2, 1,3)
by (induct) (fastforce intro: rtrancl.rtrancl_into_rtrancl)+
lemma rtrancl_map:
assumes "⋀x y. (x, y) ∈ r ⟹ (f x, f y) ∈ s"
and "(x, y) ∈ r⇧*"
shows "(f x, f y) ∈ s⇧*"
using rtrancl_sim[where S="{(x,f x) | x. True}"] assms
by simp blast
--
Peter
On Mi, 2014-10-01 at 14:27 +0200, Christian Sternagel wrote:
> Dear developers,
>
> the following lemma seems like a basic fact about rtrancl:
>
> lemma rtrancl_map:
> assumes "⋀x y. (x, y) ∈ r ⟹ (f x, f y) ∈ s"
> and "(x, y) ∈ r⇧*"
> shows "(f x, f y) ∈ s⇧*"
> using assms(2, 1)
> by (induct) (auto intro: rtrancl.rtrancl_into_rtrancl)
>
> I didn't find it in Main. Did anybody else ever use/need it? Is it
> already part of some theory I did overlook? How about adding it?
>
> Caveat: It seems to be necessary to strongly instantiate it before usage.
>
> cheers
>
> chris
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