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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 "pointer_order.ma".
-
-(* POINTER SEQUENCE *********************************************************)
-
-(* Policy: pointer sequence metavariables: r, s *)
-definition pseq: Type[0] ≝ list ptr.
-
-(* Note: a "head" computation contracts just redexes in the head *)
-definition is_head: predicate pseq ≝ All … in_head.
-
-lemma is_head_dx: ∀s. is_head s → is_head (dx:::s).
-#s elim s -s //
-#p #s #IHs * /3 width=1/ 
-qed.
-
-lemma is_head_append: ∀r. is_head r → ∀s. is_head s → is_head (r@s).
-#r elim r -r //
-#q #r #IHr * /3 width=1/
-qed.
-
-(* Note: to us, a "normal" computation contracts redexes in non-decreasing positions *)
-definition is_le: predicate pseq ≝ Allr … ple.
-
-lemma is_le_compatible: ∀c,s. is_le s → is_le (c:::s).
-#c #s elim s -s // #p * //
-#q #s #IHs * /3 width=1/
-qed.
-
-lemma is_le_cons: ∀p,s. is_le s → is_le ((dx::p)::sn:::s).
-#p #s elim s -s // #q1 * /2 width=1/
-#q2 #s #IHs * /4 width=1/
-qed.
-
-lemma is_le_append: ∀r. is_le r → ∀s. is_le s → is_le ((dx:::r)@sn:::s).
-#r elim r -r /3 width=1/ #p * /2 width=1/
-#q #r #IHr * /3 width=1/
-qed.
-
-theorem is_head_is_le: ∀s. is_head s → is_le s.
-#s elim s -s // #p * //
-#q #s #IHs * /3 width=1/
-qed.
-
-lemma is_le_in_head: ∀p. in_head p → ∀s. is_le s → is_le (p::s).
-#p #Hp * // /3 width=1/
-qed.
-
-theorem is_head_is_le_trans: ∀r. is_head r → ∀s. is_le s → is_le (r@s).
-#r elim r -r // #p *
-[ #_ * /2 width=1/
-| #q #r #IHr * /3 width=1/
-]
-qed.
-
-definition ho_compatible_rc: predicate (pseq→relation term) ≝ λR.
-                             ∀s,A1,A2. R s A1 A2 → R (sn:::s) (𝛌.A1) (𝛌.A2).
-
-definition ho_compatible_sn: predicate (pseq→relation term) ≝ λR.
-                             ∀s,B1,B2,A. R s B1 B2 → R (sn:::s) (@B1.A) (@B2.A).
-
-definition ho_compatible_dx: predicate (pseq→relation term) ≝ λR.
-                             ∀s,B,A1,A2. R s A1 A2 → R (dx:::s) (@B.A1) (@B.A2).
-
-lemma lstar_compatible_rc: ∀R. compatible_rc R → ho_compatible_rc (lstar … R).
-#R #HR #s #A1 #A2 #H @(lstar_ind_l ????????? H) -s -A1 // /3 width=3/
-qed.
-
-lemma lstar_compatible_sn: ∀R. compatible_sn R → ho_compatible_sn (lstar … R).
-#R #HR #s #B1 #B2 #A #H @(lstar_ind_l ????????? H) -s -B1 // /3 width=3/
-qed.
-
-lemma lstar_compatible_dx: ∀R. compatible_dx R → ho_compatible_dx (lstar … R).
-#R #HR #s #B #A1 #A2 #H @(lstar_ind_l ????????? H) -s -A1 // /3 width=3/
-qed.