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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basics/star.ma".
+include "Ground-2/xoa_props.ma".
+
+(* PROPERTIES of RELATIONS **************************************************)
+
+definition confluent: ∀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.
+
+definition transitive: ∀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.
+
+lemma TC_strip1: ∀A,R1,R2. confluent A R1 R2 →
+                 ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
+                 ∃∃a. R2 a1 a & TC … R1 a2 a.
+#A #R1 #R2 #HR12 #a0 #a1 #H elim H -H a1
+[ #a1 #Ha01 #a2 #Ha02
+  elim (HR12 … Ha01 … Ha02) -HR12 a0 /3/
+| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+  elim (IHa0 … Ha02) -IHa0 Ha02 a0 #a0 #Ha0 #Ha20
+  elim (HR12 … Ha1 … Ha0) -HR12 a /4/
+]
+qed.
+
+lemma TC_strip2: ∀A,R1,R2. confluent A R1 R2 →
+                 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
+                 ∃∃a. TC … R2 a1 a & R1 a2 a.
+#A #R1 #R2 #HR12 #a0 #a2 #H elim H -H a2
+[ #a2 #Ha02 #a1 #Ha01
+  elim (HR12 … Ha01 … Ha02) -HR12 a0 /3/
+| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
+  elim (IHa0 … Ha01) -IHa0 Ha01 a0 #a0 #Ha10 #Ha0
+  elim (HR12 … Ha0 … Ha2) -HR12 a /4/
+]
+qed.
+
+lemma TC_confluent: ∀A,R1,R2.
+                    confluent A R1 R2 → confluent A (TC … R1) (TC … R2).
+#A #R1 #R2 #HR12 #a0 #a1 #H elim H -H a1
+[ #a1 #Ha01 #a2 #Ha02
+  elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 a0 /3/
+| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+  elim (IHa0 … Ha02) -IHa0 Ha02 a0 #a0 #Ha0 #Ha20
+  elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 a /4/
+]
+qed.
+
+lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
+                R a1 a → TC … R a a2 → TC … R a1 a2.
+/3/ qed.
+
+lemma TC_strap1: ∀A,R1,R2. transitive A R1 R2 →
+                 ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
+                 ∃∃a. R2 a1 a & TC … R1 a a2.
+#A #R1 #R2 #HR12 #a1 #a0 #H elim H -H a0
+[ #a0 #Ha10 #a2 #Ha02
+  elim (HR12 … Ha10 … Ha02) -HR12 a0 /3/
+| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
+  elim (HR12 … Ha0 … Ha02) -HR12 a0 #a0 #Ha0 #Ha02
+  elim (IHa … Ha0) -a /4/
+]
+qed.
+
+lemma TC_strap2: ∀A,R1,R2. transitive A R1 R2 →
+                 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
+                 ∃∃a. TC … R2 a1 a & R1 a a2.
+#A #R1 #R2 #HR12 #a0 #a2 #H elim H -H a2
+[ #a2 #Ha02 #a1 #Ha10
+  elim (HR12 … Ha10 … Ha02) -HR12 a0 /3/
+| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
+  elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
+  elim (HR12 … Ha0 … Ha02) -HR12 a /4/
+]
+qed.
+
+lemma TC_transitive: ∀A,R1,R2.
+                     transitive A R1 R2 → transitive A (TC … R1) (TC … R2).
+#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
+[ #a0 #Ha10 #a2 #Ha02
+  elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 a0 /3/
+| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
+  elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 a0 #a0 #Ha0 #Ha02
+  elim (IHa … Ha0) -a /4/
+]
+qed.
+
+lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
+/2/ qed.
+
+lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:A→Prop.
+                   P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
+                   ∀a2. TC … R a1 a2 → P a2.
+#A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -Ha12 a2 /3/
+qed.