# [isabelle-dev] rtrancl lemma proposal

Peter Lammich lammich at in.tum.de
Wed Oct 1 14:41:52 CEST 2014

```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
>
> Caveat: It seems to be necessary to strongly instantiate it before usage.
>
> cheers
>
> chris
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```