(* Specifications: *)
Lemma clos_rt_step_trans :
forall (A : Set) (R : A -> A -> Prop) (x y z : A),
clos_refl_trans _ R x y -> clos_trans _ R y z -> clos_trans _ R x z.
Axiom A : Prop.
Axiom R : A->A->Prop.
Axiom Rstar: A->A->Prop.
Axiom unf_Rstar: (P:(A->A->Prop)->Prop)
(P [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))
->(P Rstar).
Axiom Rstar': A->A->Prop.
Axiom unf_Rstar': (P:(A->A->Prop)->Prop)
(P [x,y:A](P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y))
->(P Rstar').
Axiom coherence: A->A->Prop.
Axiom unf_coherence: (P:(A->A->Prop)->Prop)
(P [x:A][y:A](P:Prop)((z:A)(Rstar x z)->(Rstar y z)->P)->P)
->(P coherence).
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).
Theorem Rstar_reflexive:  (x:A)(Rstar x x).
Proof [x:A]
(unf_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)).
Theorem Rstar_R: (x:A)(y:A)(z:A)(R x y)->(Rstar y z)->(Rstar x z).
Proof [x,y:A]
[z:A]
[t1:(R x y)]
(unf_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))).
Theorem  Rstar_transitive:  (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z).
Proof [x,y:A]
[z:A]
(unf_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:A]
[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)))).
Theorem Rstar'_reflexive: (x:A)(Rstar' x x).
Proof [x:A]
(unf_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).
Theorem Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x y).
Proof [x,y:A]
[z:A]
[t1:(R x z)]
[t2:(Rstar z y)]
(unf_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)).
Theorem Rstar'_Rstar:  (x:A)(y:A)(Rstar' x y)->(Rstar x y).
Proof [x,y:A]
(unf_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))).
Theorem Rstar_Rstar':  (x:A)(y:A)(Rstar x y)->(Rstar' x y).
Proof  [x,y:A]
(unf_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:A]
[w:A]
[h1:(R u v)]
[h2:(Rstar' v w)](Rstar'_R u w v h1 (Rstar'_Rstar v w h2)))).
Theorem coherence_intro :  (x:A)(y:A)(z:A)(Rstar x z)->(Rstar y z)->(coherence x y).
Proof [x,y:A]
[z:A]
[H:(Rstar x z)]
[H0:(Rstar y z)]
(unf_coherence [P:A->A->Prop](P x y)
[P:Prop][H1:(z0:A)(Rstar x z0)->(Rstar y z0)->P](H1 z H H0)).
Lemma Rstar_coherence : (x:A)(y:A)(Rstar x y)->(coherence x y).
Proof  [x:A][y:A][h:(Rstar x y)](coherence_intro x y y h (Rstar_reflexive y)).
Lemma coherence_sym: (x:A)(y:A)(coherence x y)->(coherence y x).
Proof [x,y:A]
(unf_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))).
Theorem Diagram: (x:A)
((u:A)
(R x u)
->([x0:A]
(y:A)(z:A)(Rstar x0 y)->(Rstar x0 z)->(coherence y z)
u))
->(y:A)
(z:A)
(u:A)
(R x u)
->(Rstar u y)
->(v:A)(R x v)->(Rstar v z)->(coherence y z).
Axiom A : Type.
Axiom R : A->A->Prop.
(* Axiomes simulant les definitions *)
Axiom Rstar:A->A->Prop.
Axiom unfold_Rstar: (P:(A->A->Prop)->Prop)
(P [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))
->(P Rstar).
Axiom Rstar':A->A->Prop.
Axiom unfold_Rstar': (P:(A->A->Prop)->Prop)
(P [x,y:A](P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y))
->(P Rstar').
Axiom coherence: A->A->Prop.
Axiom unfold_coherence: (P:(A->A->Prop)->Prop)
(P [x:A][y:A](P:Prop)((z:A)(Rstar x z)->(Rstar y z)->P)->P)
->(P coherence).