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)).
Proof [x,y,z:A][h:(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)]
(h [a,a0:A](Rstar a0 z)->(Rstar a z) [u:A][H:(Rstar u z)]H
[u,v,w:A][t1:(R u v)][t2:(Rstar w z)->(Rstar v z)]
[t3:(Rstar w z)](Rstar_R u v z t1 (t2 t3))).
Proof [x:A][P:A->A->Prop][H:(P x x)][_:(u:A)(R x u)->(Rstar u x)->(P x x)]H.
Proof [x,y,z:A][t1:(R x z)][t2:(Rstar z y)][P:A->A->Prop]
[_:(P x x)][h2:(u:A)(R x u)->(Rstar u y)->(P x y)](h2 z t1 t2).
Proof [x,y:A][h:(P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y)]
(h [a,a0:A](Rstar a a0) (Rstar_reflexive x) [u:A](Rstar_R x u y)).
Proof [x,y:A][h:(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)]
(h Rstar' [u:A](Rstar'_reflexive u)
[u,v,w:A][h1:(R u v)]
[h2:(Rstar' v w)](Rstar'_R u w v h1 (Rstar'_Rstar v w h2))).
Proof [x,y,z:A][H:(Rstar x z)][H0:(Rstar y z)][P:Prop]
[H1:(z0:A)(Rstar x z0)->(Rstar y z0)->P](H1 z H H0).
Proof [x,y:A][h:(Rstar x y)](coherence_intro x y y h (Rstar_reflexive y)).
Proof [x,y:A][H:(P:Prop)((z:A)(Rstar x z)->(Rstar y z)->P)->P][P:Prop]
[H0:(z:A)(Rstar y z)->(Rstar x z)->P]
(H P [z:A][H1:(Rstar x z)][H2:(Rstar y z)](H0 z H2 H1)).
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,u:A][t1:(R x u)][t2:(Rstar u y)][v:A][u1:(R x v)][u2:(Rstar v z)]
([Hyp0:(x0,y0,z0:A)(R x0 y0)->(R x0 z0)
->(P:Prop)((z1:A)(Rstar y0 z1)->(Rstar z0 z1)->P)->P]
[hyp_ind0:(u0:A)(R x u0)->(y0,z0:A)(Rstar u0 y0)->(Rstar u0 z0)
->(P:Prop)((z1:A)(Rstar y0 z1)->(Rstar z0 z1)->P)->P]
(Hyp0 x u v t1 u1 (coherence y z)
[z0:A][H:(Rstar u z0)][H0:(Rstar v z0)]
(hyp_ind0 u t1 y z0 t2 H (coherence y z)
[z1:A][H1:(Rstar y z1)][H2:(Rstar z0 z1)]
(hyp_ind0 v u1 z z1 u2
(Rstar_transitive v z0 z1 H0 H2)
(coherence y z)
[z2:A][H3:(Rstar z z2)][H4:(Rstar z1 z2)]
[P:Prop][H5:(z3:A)(Rstar y z3)->(Rstar z z3)->P]
(H5 z2
(Rstar_transitive y z1 z2 H1 H4) H3))))
Hyp2 hyp_ind).

Lemma caseRxy:
(x:A)((u:A)(R x u)->(y,z:A)(Rstar u y)->(Rstar u z)->(coherence y z))
->(y,z:A)(Rstar x y)->(Rstar x z)
->(u:A)(R x u)->(Rstar u y)->(coherence y z).