- full commit for the transtive closure of ltpss!

[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 c951ddac038e8d82b69c961a12d87fdc746385fc..3517eff98f8ae4187352097ffb812e1fd01d0bbf 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.
@@ -110,3 +117,16 @@ lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
 @SN_intro #a2 #HRa12 #HSa12
 elim (HSa12 ?) -HSa12 /2 width=1/
 qed.
+
+definition NF_sn: ∀A. relation A → relation A → predicate A ≝
+   λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
+
+inductive SN_sn (A) (R,S:relation A): predicate A ≝
+| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
+.
+
+lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
+#A #R #S #a2 #Ha2
+@SN_sn_intro #a1 #HRa12 #HSa12
+elim (HSa12 ?) -HSa12 /2 width=1/
+qed.