End REC_COP3_ALT.
Section REC_COP3_RELAT.
(* relation parent *)
Variable c0 : Config.
Variable s1 s2 : Site.
Hypothesis diff_site : s1 <> s2.
Hypothesis rt_true : rt c0 s2 = false.
Hypothesis
ancestor_rt : forall s1 s2 : Site, ancestor c0 s2 s1 -> rt c0 s2 = true.
Hypothesis le_dt_time : forall s0 : Site, dt c0 s0 <= time c0.
Remark no_ancestor_s2 : forall s : Site, ~ ancestor c0 s2 s.
Remark rcop3_trans_parent :
forall s3 s4 : Site,
s1 <> s4 \/ s2 <> s3 ->
parent (rec_copy3_trans c0 s1 s2) s4 s3 -> parent c0 s4 s3.
Remark rcop3_trans_no_parent :
forall s3 s4 : Site,
s1 <> s4 \/ s2 <> s3 ->
~ parent c0 s4 s3 -> ~ parent (rec_copy3_trans c0 s1 s2) s4 s3.

Lemma rcop3_trans_parent_cr :
forall s3 s4 : Site,
(parent c0 s3 s4 -> dt c0 s3 < dt c0 s4) ->
parent (rec_copy3_trans c0 s1 s2) s3 s4 ->
dt (rec_copy3_trans c0 s1 s2) s3 < dt (rec_copy3_trans c0 s1 s2) s4.