context-free parallel reduction on closures is confluent!

[helm.git] / matita / matita / contribs / lambda_delta / ground_2 / star.ma

diff --git a/matita/matita/contribs/lambda_delta/ground_2/star.ma b/matita/matita/contribs/lambda_delta/ground_2/star.ma
index 18f028acc6eda865266c5937b79e1a6b393d7d72..2779a0940c35c8e8dcc49983d25e6b612da4c24b 100644 (file)
--- a/matita/matita/contribs/lambda_delta/ground_2/star.ma
+++ b/matita/matita/contribs/lambda_delta/ground_2/star.ma
@@ -20,6 +20,13 @@ include "ground_2/notation.ma".
 
 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
 
+definition Confluent: ∀A. ∀R: relation A. Prop ≝ λA,R.
+                      ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 →
+                      ∃∃a. R a1 a & R a2 a.
+
+definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
+                       ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
+
 definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
                        ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
                        ∃∃a. R2 a1 a & R1 a2 a.
@@ -28,6 +35,10 @@ definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
                         ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
                         ∃∃a. R2 a1 a & R1 a a2.
 
+definition bi_confluent:  ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
+                          ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
+                          ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
+
 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
                  ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
                  ∃∃a. R2 a1 a & TC … R1 a2 a.