Module Coq.Relations.Newman

Require Rstar.

Section Newman.

Variable A: Type.
Variable R: A->A->Prop.

Local Rstar := (Rstar A R).
Local Rstar_reflexive := (Rstar_reflexive A R).
Local Rstar_transitive := (Rstar_transitive A R).
Local Rstar_Rstar' := (Rstar_Rstar' A R).

Definition coherence := [x:A][y:A] (exT2 ? (Rstar x) (Rstar y)).

Theorem coherence_intro : (x:A)(y:A)(z:A)(Rstar x z)->(Rstar y z)->(coherence x y).
Proof [x:A][y:A][z:A][h1:(Rstar x z)][h2:(Rstar y z)]
      (exT_intro2 A (Rstar x) (Rstar y) z h1 h2).
  
A very simple case of coherence :

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)).
  
coherence is symmetric

Lemma coherence_sym: (x:A)(y:A)(coherence x y)->(coherence y x).
 Proof [x:A][y:A][h:(coherence x y)]
        (exT2_ind A (Rstar x) (Rstar y) (coherence y x)
                 [w:A][h1:(Rstar x w)][h2:(Rstar y w)]
                      (coherence_intro y x w h2 h1) h).

Definition confluence :=
     [x:A](y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z).
  
Definition local_confluence :=
     [x:A](y:A)(z:A)(R x y)->(R x z)->(coherence y z).
  
Definition noetherian :=
     (x:A)(P:A->Prop)((y:A)((z:A)(R y z)->(P z))->(P y))->(P x).
  
Section Newman_section.

The general hypotheses of the theorem

Hypothesis Hyp1:noetherian.
Hypothesis Hyp2:(x:A)(local_confluence x).
  
The induction hypothesis

Section Induct.
  Variable x:A.
  Hypothesis hyp_ind:(u:A)(R x u)->(confluence u).
  
Confluence in x

  Variables y,z:A.
  Hypothesis h1:(Rstar x y).
  Hypothesis h2:(Rstar x z).
  
particular case x->u and u->*y

Section Newman_.
  Variable u:A.
  Hypothesis t1:(R x u).
  Hypothesis t2:(Rstar u y).
  
In the usual diagram, we assume also x->v and v->*z

Theorem Diagram : (v:A)(u1:(R x v))(u2:(Rstar v z))(coherence y z).

Proof
   [v:A][u1:(R x v)][u2:(Rstar v z)]
   (exT2_ind A (Rstar u) (Rstar v)
      (coherence y z)
      ([w:A][s1:(Rstar u w)][s2:(Rstar v w)]
   (exT2_ind A (Rstar y) (Rstar w)
      (coherence y z)
      ([a:A][v1:(Rstar y a)][v2:(Rstar w a)]
   (exT2_ind A (Rstar a) (Rstar z)
      (coherence y z)
      ([b:A][w1:(Rstar a b)][w2:(Rstar z b)]
        (coherence_intro y z b (Rstar_transitive y a b v1 w1) w2))
      (hyp_ind v u1 a z (Rstar_transitive v w a s2 v2) u2)))
     (hyp_ind u t1 y w t2 s1)))
    (Hyp2 x u v t1 u1)).
  
Theorem caseRxy : (coherence y z).
Proof (Rstar_Rstar' x z h2
      ([v:A][w:A](coherence y w))
      (coherence_sym x y (Rstar_coherence x y h1))
      Diagram).
End Newman_.

Theorem Ind_proof : (coherence y z).
Proof (Rstar_Rstar' x y h1 ([u:A][v:A](coherence v z))
        (Rstar_coherence x z h2)
        caseRxy).
End Induct.

Theorem Newman : (x:A)(confluence x).
Proof [x:A](Hyp1 x confluence Ind_proof).

End Newman_section.

End Newman.


Index