--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⧁ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'RestSupTerm $L1 $T1 $L2 $T2 }.
+
+include "basic_2/grammar/cl_weight.ma".
+include "basic_2/substitution/lift.ma".
+
+(* RESTRICTED SUPCLOSURE ****************************************************)
+
+inductive frsup: bi_relation lenv term ≝
+| frsup_bind_sn: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) L V
+| frsup_bind_dx: ∀a,I,L,V,T. frsup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T
+| frsup_flat_sn: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L V
+| frsup_flat_dx: ∀I,L,V,T. frsup L (ⓕ{I}V.T) L T
+.
+
+interpretation
+ "restricted structural predecessor (closure)"
+ 'RestSupTerm L1 T1 L2 T2 = (frsup L1 T1 L2 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact frsup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
+ ∀J. T1 = ⓪{J} → ⊥.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a #I #L #V #T #J #H destruct
+| #a #I #L #V #T #J #H destruct
+| #I #L #V #T #J #H destruct
+| #I #L #V #T #J #H destruct
+]
+qed-.
+
+lemma frsup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ ⧁ ⦃L2, T2⦄ → ⊥.
+/2 width=7 by frsup_inv_atom1_aux/ qed-.
+
+fact frsup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
+ ∀b,J,W,U. T1 = ⓑ{b,J}W.U →
+ (L2 = L1 ∧ T2 = W) ∨
+ (L2 = L1.ⓑ{J}W ∧ T2 = U).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
+| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
+| #I #L #V #T #b #J #W #U #H destruct
+| #I #L #V #T #b #J #W #U #H destruct
+]
+qed-.
+
+lemma frsup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⧁ ⦃L2, T2⦄ →
+ (L2 = L1 ∧ T2 = W) ∨
+ (L2 = L1.ⓑ{J}W ∧ T2 = U).
+/2 width=4 by frsup_inv_bind1_aux/ qed-.
+
+fact frsup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ →
+ ∀J,W,U. T1 = ⓕ{J}W.U →
+ L2 = L1 ∧ (T2 = W ∨ T2 = U).
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a #I #L #V #T #J #W #U #H destruct
+| #a #I #L #V #T #J #W #U #H destruct
+| #I #L #V #T #J #W #U #H destruct /3 width=1/
+| #I #L #V #T #J #W #U #H destruct /3 width=1/
+]
+qed-.
+
+lemma frsup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⧁ ⦃L2, T2⦄ →
+ L2 = L1 ∧ (T2 = W ∨ T2 = U).
+/2 width=4 by frsup_inv_flat1_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma frsup_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/
+qed-.
+
+lemma frsup_fwd_lw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{L1} ≤ ♯{L2}.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/
+qed-.
+
+lemma frsup_fwd_tw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ♯{T2} < ♯{T1}.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=1 by le_minus_to_plus/
+qed-.
+
+lemma frsup_fwd_append: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⧁ ⦃L2, T2⦄ → ∃L. L2 = L1 @@ L.
+#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
+[ #a
+| #a #I #L #V #_ @(ex_intro … (⋆.ⓑ{I}V)) //
+]
+#I #L #V #T @(ex_intro … (⋆)) //
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lift_frsup_trans: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀L,K,U2. ⦃L, U1⦄ ⧁ ⦃L @@ K, U2⦄ →
+ ∃T2. ⇧[d + |K|, e] T2 ≡ U2.
+#T1 #U1 #d #e * -T1 -U1 -d -e
+[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HTU1 #L #K #X #H
+ elim (frsup_inv_bind1 … H) -H *
+ [ -HTU1 #H1 #H2 destruct
+ >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/
+ | -HVW1 #H1 #H2 destruct
+ >(append_inv_pair_dx … H1) -L -K normalize /2 width=2/
+ ]
+|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #K #X #H
+ elim (frsup_inv_flat1 … H) -H #H1 * #H2 destruct
+ >(append_inv_refl_dx … H1) -L -K normalize /2 width=2/
+]
+#i #d #e [2,3: #_ ] #L #K #X #H
+elim (frsup_inv_atom1 … H)
+qed-.