--- /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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_append.ma".
+include "basic_2/reduction/crr.ma".
+
+(* REDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION **************************)
+
+(* Advanved properties ******************************************************)
+
+lemma crr_append_sn: ∀G,L,K,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, K @@ L⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L #K0 #T #H elim H -L -T /2 width=1/
+#L #K #V #i #HLK
+lapply (ldrop_fwd_length_lt2 … HLK) #Hi
+lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=2/ -Hi /2 width=3/
+qed.
+
+lemma trr_crr: ∀G,L,T. ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L #T #H lapply (crr_append_sn … H) //
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact crr_inv_labst_last_aux: ∀G,L1,T,W. ⦃G, L1⦄ ⊢ ➡ 𝐑⦃T⦄ →
+ ∀L2. L1 = ⋆.ⓛW @@ L2 → ⦃G, L2⦄ ⊢ ➡ 𝐑⦃T⦄.
+#G #L1 #T #W #H elim H -L1 -T /2 width=1/ /3 width=1/
+[ #L1 #K1 #V1 #i #HLK1 #L2 #H destruct
+ lapply (ldrop_fwd_length_lt2 … HLK1)
+ >append_length >commutative_plus normalize in ⊢ (??% → ?); #H
+ elim (le_to_or_lt_eq i (|L2|)) /2 width=1/ -H #Hi destruct
+ [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2
+ lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi
+ normalize #H destruct /2 width=3/
+ | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // <minus_n_n #H
+ lapply (ldrop_inv_O2 … H) -H #H destruct
+ ]
+| #a #I #L1 #V #T #HI #_ #IHT #L2 #H destruct /3 width=1/
+]
+qed.
+
+lemma crr_inv_labst_last: ∀G,L,T,W. ⦃G, ⋆.ⓛW @@ L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄.
+/2 width=4/ qed-.
+
+lemma crr_inv_trr: ∀G,T,W. ⦃G, ⋆.ⓛW⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, ⋆⦄ ⊢ ➡ 𝐑⦃T⦄.
+/2 width=4/ qed-.