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.