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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".
-include "Ground_2/notation.ma".
-
-(* PROPERTIES OF RELATIONS **************************************************)
-
-definition Decidable: Prop → Prop ≝ λR. R ∨ (R → False).
-
-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.
-
-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.
-
-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.
-#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
-[ #a1 #Ha01 #a2 #Ha02
-  elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
-  elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-  elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_strip2: ∀A,R1,R2. confluent2 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 -a2
-[ #a2 #Ha02 #a1 #Ha01
-  elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
-  elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
-  elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_confluent2: ∀A,R1,R2.
-                     confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
-#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
-[ #a1 #Ha01 #a2 #Ha02
-  elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
-| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
-  elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-  elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_strap1: ∀A,R1,R2. transitive2 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 -a0
-[ #a0 #Ha10 #a2 #Ha02
-  elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
-  elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
-  elim (IHa … Ha0) -a /4 width=3/
-]
-qed.
-
-lemma TC_strap2: ∀A,R1,R2. transitive2 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 -a2
-[ #a2 #Ha02 #a1 #Ha10
-  elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
-  elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
-  elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_transitive2: ∀A,R1,R2.
-                      transitive2 A R1 R2 → transitive2 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 width=3/
-| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
-  elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
-  elim (IHa … Ha0) -a /4 width=3/
-]
-qed.
-
-definition NF: ∀A. relation A → relation A → predicate A ≝
-   λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
-
-inductive SN (A) (R,S:relation A): predicate A ≝
-| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → False) → SN A R S a2) → SN A R S a1
-.
-
-lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
-#A #R #S #a1 #Ha1
-@SN_intro #a2 #HRa12 #HSa12
-elim (HSa12 ?) -HSa12 /2 width=1/
-qed.