--- /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/notation/relations/sn_5.ma".
+include "basic_2/reduction/cnx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
+
+definition csx: ∀h. sd h → relation3 genv lenv term ≝
+ λh,o,G,L. SN … (cpx h o G L) (eq …).
+
+interpretation
+ "context-sensitive extended strong normalization (term)"
+ 'SN h o G L T = (csx h o G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) →
+ R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H elim H -T1
+/5 width=1 by SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: sn3_pr2_intro *)
+lemma csx_intro: ∀h,o,G,L,T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] T2) →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] T1.
+/4 width=1 by SN_intro/ qed.
+
+lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+elim (eq_term_dec T1 T2) #HT12 destruct /3 width=4 by/
+qed-.
+
+(* Basic_1: was just: sn3_nf2 *)
+lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+/2 width=1 by NF_to_SN/ qed.
+
+lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬊*[h, o] ⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
+#Hkd @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
+[ #H destruct elim HX //
+| -HX * #d0 #_ #H destruct -d0 /2 width=1 by/
+]
+qed.
+
+(* Basic_1: was just: sn3_cast *)
+lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, o] W →
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓝW.T.
+#h #o #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+@csx_intro #X #H1 #H2
+elim (cpx_inv_cast1 … H1) -H1
+[ * #W0 #T0 #HLW0 #HLT0 #H destruct
+ elim (eq_false_inv_tpair_sn … H2) -H2
+ [ /3 width=3 by csx_cpx_trans/
+ | -HLW0 * #H destruct /3 width=1 by/
+ ]
+|2,3: /3 width=3 by csx_cpx_trans/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+ ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csx_intro #V2 #HLV2 #HV2
+@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
+#H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: was just: sn3_gen_head *)
+lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
+/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
+
+fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+ ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
+@csx_intro #T2 #HLT2 #HT2
+@(IH (ⓑ{a,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+#H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: was just: sn3_gen_bind *)
+lemma csx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
+
+fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csx_intro #T2 #HLT2 #HT2
+@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
+#H destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: was just: sn3_gen_flat *)
+lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
+
+lemma csx_fwd_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
+
+lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
+
+(* Basic_1: removed theorems 14:
+ sn3_cdelta
+ sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
+ sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
+ sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
+*)