Variable A : Prop .
Variable R : A->A->Prop.
Definition Rstar:A->A->Prop :=
[x,y:A](P0:A->A->Prop)
((u:A)(P0 u u))
->((u:A)(v:A)(w:A)(R u v)->(P0 v w)->(P0 u w))->(P0 x y).
Definition Rstar':A->A->Prop :=
[x,y:A](P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y).
Definition coherence: A->A->Prop :=
[x:A][y:A](P:Prop)((z:A)(Rstar x z)->(Rstar y z)->P)->P.
(* Les 2 hypotheses du Lemme de Newman *)
(* Verification du lemme *)
Proof [x:A][P0:A->A->Prop]
[H:(u:A)(P0 u u)][_:(u:A)(v:A)(w:A)(R u v)->(P0 v w)->(P0 u w)](H x).
Proof [x,y,z:A][t1:(R x y)] [t2:(P0:A->A->Prop)
((u:A)(P0 u u))
->((u:A)(v:A)(w:A)(R u v)->(P0 v w)->(P0 u w))->(P0 y z)]
[P:A->A->Prop][h1:(u:A)(P u u)]
[h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)]
(h2 x y z t1 (t2 [a,a0:A](P a a0) h1 h2)).

Lemma Rstar_transitive: (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z).