(* Les 2 hypotheses du Lemme de Newman *)

(* Specifications: *)
Axiom Hyp1:(x:A)(P:A->Prop)((y:A)((z:A)(R y z)->(P z))->(P y))->(P x).
Axiom Hyp2:(x:A)(y:A)(z:A)(R x y)->(R x z)->(coherence y z).
Check
([Rstar_reflexive: (x:A)(Rstar x x)]
([Rstar_R: (x:A)(y:A)(z:A)(R x y)->(Rstar y z)->(Rstar x z)]
([Rstar_transitive:  (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z)]
([Rstar'_reflexive: (x:A)(Rstar' x x)]
([Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x y)]
([Rstar'_Rstar:  (x:A)(y:A)(Rstar' x y)->(Rstar x y)]
([Rstar_Rstar':  (x:A)(y:A)(Rstar x y)->(Rstar' x y)]
([coherence_intro :  (x:A)(y:A)(z:A)(Rstar x z)->(Rstar y z)
->(coherence x y)]
([Rstar_coherence : (x:A)(y:A)(Rstar x y)->(coherence x y)]
([coherence_sym: (x:A)(y:A)(coherence x y)->(coherence y x)]
([Diagram:
(x:A)((u:A)(R x u)->(y:A)(z:A)(Rstar u y)->(Rstar u z)->(coherence y z))
->(y,z,u:A)(R x u)->(Rstar u y)
->(v:A)(R x v)->(Rstar v z)->(coherence y z)]
([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)]
([Ind_proof :
(x:A)((u:A)(R x u)->(y:A)(z:A)(Rstar u y)->(Rstar u z)->(coherence y z))
->(y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z)]
[x:A](Hyp1 x
[x:A](y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z) Ind_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)]
(unfold_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))
)
[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)][u:A][t1:(R x u)][t2:(Rstar u y)]
(unfold_Rstar' [P:A->A->Prop](P x z)->(coherence y z)
[hyp_:(P:A->A->Prop)(P x x)
->((u0:A)(R x u0)->(Rstar u0 z)->(P x z))->(P x z)]
(hyp_ [_:A][a:A](coherence y a)
(coherence_sym x y (Rstar_coherence x y h1))
(Diagram x hyp_ind y z u t1 t2))
(Rstar_Rstar' x z h2))
)
[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)]
(unfold_coherence
[P:A->A->Prop]
((x0,y0,z0:A)(R x0 y0)->(R x0 z0)->(P y0 z0))
->((u0:A)(R x u0)
->(y0,z0:A)(Rstar u0 y0)->(Rstar u0 z0)->(P y0 z0))
->(coherence y 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)]
(unfold_coherence [P:A->A->Prop](P y z)
[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)
)
[x,y:A](unfold_coherence [P:A->A->Prop](P x y)->(P y x)
[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)))
)
[x,y:A][h:(Rstar x y)](coherence_intro x y y h (Rstar_reflexive y))
)
[x,y,z:A][H:(Rstar x z)][H0:(Rstar y z)]
(unfold_coherence [P:A->A->Prop](P x y)
[P:Prop][H1:(z0:A)(Rstar x z0)->(Rstar y z0)->P](H1 z H H0))
)
[x,y:A](unfold_Rstar [P:A->A->Prop](P x y)->(Rstar' x y)
[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))))
)
[x,y:A](unfold_Rstar' [P:A->A->Prop](P x y)->(Rstar x y)
[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)))
)
[x,y,z:A][t1:(R x z)][t2:(Rstar z y)]
(unfold_Rstar' [P:A->A->Prop](P x y)
[P:A->A->Prop]
[_:(P x x)][h2:(u:A)(R x u)->(Rstar u y)->(P x y)](h2 z t1 t2))
)
[x:A](unfold_Rstar' [P:A->A->Prop](P x x)
[P:A->A->Prop][H:(P x x)][_:(u:A)(R x u)->(Rstar u x)->(P x x)]H)
)
[x,y,z:A](unfold_Rstar [P:A->A->Prop](P x y)->(Rstar y z)->(Rstar x z)
[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))))
)
[x,y,z:A][t1:(R x y)]
(unfold_Rstar [P:A->A->Prop](P y z)->(P x z)
[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)))
)
[x:A](unfold_Rstar [P:A->A->Prop](P x x)
[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))
)
:: (x:A)(y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z).