Proof [x:A]
[hyp_ind:(u:A)
(R x u)->(y:A)(z:A)(Rstar u y)->(Rstar u z)->(coherence y z)]
[y,z:A]
[h1:(Rstar x y)]
[h2:(Rstar x z)]
(unf_Rstar' [P:A->A->Prop](P x y)->(coherence y z)
[hyp_:(P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y)]
(hyp_ [_:A][a:A](coherence a z) (Rstar_coherence x z h2)
(caseRxy x hyp_ind y z h1 h2))
(Rstar_Rstar' x y h1)).
Syntactic Definition local_confluent :=
(x:A)(y:A)(z:A)(R x y)->(R x z)->(coherence y z).
Syntactic Definition noetherian :=
(x:A)(P:A->Prop)((y:A)((z:A)(R y z)->(P z))->(P y))->(P x).

Theorem Newman : (x:A)(y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z).
Proof [x:A](Hyp1 x [x:A](y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z)
Ind_proof).