--- /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/computation/fpbs_fpb.ma".
+include "basic_2/computation/fpbs_fpbs.ma".
+include "basic_2/computation/csx_fpbs.ma".
+include "basic_2/computation/lsx_csx.ma".
+include "basic_2/computation/fsb_alt.ma".
+
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
+
+(* Advanced propreties on context-sensitive extended normalizing terms ******)
+
+lemma csx_fsb_fpbs: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #L1 #T1 #H @(csx_ind … H) -T1
+#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind … G2 L2 T2) -G2 -L2 -T2
+#G0 #L0 #T0 #IHu #H10 lapply (csx_fpbs_conf … H10) // -HT1
+#HT0 generalize in match IHu; -IHu generalize in match H10; -H10
+@(lsx_ind … (csx_lsx … HT0 0)) -L0
+#L0 #_ #IHd #H10 #IHu @fsb_intro
+#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
+[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
+| #T2 #HT02 #HnT02 elim (fpbs_cpx_trans_neq … H10 … HT02 HnT02) -T0
+ /3 width=4 by/
+| #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
+ [ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
+ | #G3 #L3 #T3 #H03 #_ elim (lpx_fqup_trans … H03 … HL02) -L2
+ #L4 #T4 elim (eq_term_dec T0 T4) [ -IHc | -IHu ]
+ [ #H destruct /5 width=5 by fsb_fpbs_trans, lpxs_fpbs, fpbs_fqup_trans, lpx_lpxs/
+ | #HnT04 #HT04 #H04 #HL43 elim (cpxs_neq_inv_step_sn … HT04 HnT04) -HT04 -HnT04
+ #T2 #HT02 #HnT02 #HT24 elim (fpbs_cpx_trans_neq … H10 … HT02 HnT02) -T0
+ lapply (fpbs_intro_alt … G3 … L4 … L3 L3 … T3 … HT24 ? ? ?) -HT24
+ /3 width=8 by fpbs_trans, lpx_lpxs, fqup_fqus/ (**) (* full auto too slow *)
+ ]
+ ]
+]
+qed.
+
+lemma csx_fsb: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦥[h, o] ⦃G, L, T⦄.
+/2 width=5 by csx_fsb_fpbs/ qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma csx_ind_fpb: ∀h,o. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R G L T.
+/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
+
+lemma csx_ind_fpbg: ∀h,o. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R G L T.
+/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.