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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