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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 "basic_2/notation/relations/suptermstar_6.ma".
+include "basic_2/relocation/fquq.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
+
+interpretation "star-iterated structural successor (closure)"
+   'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
+                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
+qed-.
+
+lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
+                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fqus_refl: tri_reflexive … fqus.
+/2 width=1 by tri_inj/ qed.
+
+lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+                   ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed.
+
+lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃* ⦃G2, L2, T2⦄ →
+                   ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed.
+
+lemma fqus_ldrop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ →
+                  ∀L1,U1,e. ⇩[0, e] L1 ≡ K1 → ⇧[0, e] T1 ≡ U1 →
+                  ⦃G1, L1, U1⦄ ⊃* ⦃G2, K2, T2⦄.
+#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
+/3 width=3 by fquq_fwd_fw, transitive_le/
+qed-.