-(**************************************************************************)
-(* ___ *)
-(* ||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/multiple/lleq_drop.ma".
-include "basic_2/reduction/lpx_drop.ma".
-include "basic_2/computation/lsx.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* Advanced properties ******************************************************)
-
-lemma lsx_lref_free: ∀h,o,G,L,l,i. |L| ≤ i → G ⊢ ⬊*[h, o, #i, l] L.
-#h #o #G #L1 #l #i #HL1 @lsx_intro
-#L2 #HL12 #H elim H -H
-/4 width=6 by lpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
-qed.
-
-lemma lsx_lref_skip: ∀h,o,G,L,l,i. yinj i < l → G ⊢ ⬊*[h, o, #i, l] L.
-#h #o #G #L1 #l #i #HL1 @lsx_intro
-#L2 #HL12 #H elim H -H
-/3 width=4 by lpx_fwd_length, lleq_skip/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lsx_fwd_lref_be: ∀h,o,I,G,L,l,i. l ≤ yinj i → G ⊢ ⬊*[h, o, #i, l] L →
- ∀K,V. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, o, V, 0] K.
-#h #o #I #G #L #l #i #Hli #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 #K1 #V #HLK1 @lsx_intro
-#K2 #HK12 #HnK12 lapply (drop_fwd_drop2 … HLK1)
-#H2LK1 elim (drop_lpx_trans … H2LK1 … HK12) -H2LK1 -HK12
-#L2 #HL12 #H2LK2 #H elim (lreq_drop_conf_be … H … HLK1) -H /2 width=1 by ylt_inj/
-#Y #_ #HLK2 lapply (drop_fwd_drop2 … HLK2)
-#HY lapply (drop_mono … HY … H2LK2) -HY -H2LK2 #H destruct
-/4 width=10 by lleq_inv_lref_ge/
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma lsx_lift_le: ∀h,o,G,K,T,U,lt,l,k. lt ≤ yinj l →
- ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, T, lt] K →
- ∀L. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, U, lt] L.
-#h #o #G #K #T #U #lt #l #k #Hltl #HTU #H @(lsx_ind … H) -K
-#K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
-#L2 #HL12 #HnU elim (lpx_drop_conf … HLK1 … HL12) -HL12
-/4 width=10 by lleq_lift_le/
-qed-.
-
-lemma lsx_lift_ge: ∀h,o,G,K,T,U,lt,l,k. yinj l ≤ lt →
- ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, T, lt] K →
- ∀L. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, U, lt + k] L.
-#h #o #G #K #T #U #lt #l #k #Hllt #HTU #H @(lsx_ind … H) -K
-#K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
-#L2 #HL12 #HnU elim (lpx_drop_conf … HLK1 … HL12) -HL12
-/4 width=9 by lleq_lift_ge/
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma lsx_inv_lift_le: ∀h,o,G,L,T,U,lt,l,k. lt ≤ yinj l →
- ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L →
- ∀K. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, T, lt] K.
-#h #o #G #L #T #U #lt #l #k #Hltl #HTU #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
-#K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
-/4 width=10 by lleq_inv_lift_le/
-qed-.
-
-lemma lsx_inv_lift_be: ∀h,o,G,L,T,U,lt,l,k. yinj l ≤ lt → lt ≤ l + k →
- ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L →
- ∀K. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, T, l] K.
-#h #o #G #L #T #U #lt #l #k #Hllt #Hltlm #HTU #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
-#K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
-/4 width=11 by lleq_inv_lift_be/
-qed-.
-
-lemma lsx_inv_lift_ge: ∀h,o,G,L,T,U,lt,l,k. yinj l + yinj k ≤ lt →
- ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L →
- ∀K. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, T, lt-k] K.
-#h #o #G #L #T #U #lt #l #k #Hlmlt #HTU #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
-#K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
-/4 width=9 by lleq_inv_lift_ge/
-qed-.