-(**************************************************************************)
-(* ___ *)
-(* ||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/reduction/cnx_lift.ma".
-include "basic_2/computation/gcp.ma".
-include "basic_2/computation/csx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was just: sn3_lift *)
-lemma csx_lift: ∀h,o,G,L2,L1,T1,b,l,k. ⦃G, L1⦄ ⊢ ⬊*[h, o] T1 →
- ∀T2. ⬇[b, l, k] L2 ≡ L1 → ⬆[l, k] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G #L2 #L1 #T1 #b #l #k #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
-@csx_intro #T #HLT2 #HT2
-elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
-@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_mono … HT0 … HT12) in HT2; -T1 /2 width=1 by/
-qed.
-
-(* Basic_1: was just: sn3_gen_lift *)
-lemma csx_inv_lift: ∀h,o,G,L2,L1,T1,b,l,k. ⦃G, L1⦄ ⊢ ⬊*[h, o] T1 →
- ∀T2. ⬇[b, l, k] L1 ≡ L2 → ⬆[l, k] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G #L2 #L1 #T1 #b #l #k #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
-@csx_intro #T #HLT2 #HT2
-elim (lift_total T l k) #T0 #HT0
-lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
-@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1 by/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was: sn3_gen_def *)
-lemma csx_inv_lref_bind: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V →
- ⦃G, L⦄ ⊢ ⬊*[h, o] #i → ⦃G, K⦄ ⊢ ⬊*[h, o] V.
-#h #o #I #G #L #K #V #i #HLK #Hi
-elim (lift_total V 0 (i+1))
-/4 width=9 by csx_inv_lift, csx_cpx_trans, cpx_delta, drop_fwd_drop2/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was just: sn3_abbr *)
-lemma csx_lref_bind: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, o] V → ⦃G, L⦄ ⊢ ⬊*[h, o] #i.
-#h #o #I #G #L #K #V #i #HLK #HV
-@csx_intro #X #H #Hi
-elim (cpx_inv_lref1 … H) -H
-[ #H destruct elim Hi //
-| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
- lapply (drop_mono … HLK0 … HLK) -HLK #H destruct
- /3 width=8 by csx_lift, csx_cpx_trans, drop_fwd_drop2/
-]
-qed.
-
-lemma csx_appl_simple: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, o] V → ∀T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.T2) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.T1.
-#h #o #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
-@csx_intro #X #H1 #H2
-elim (cpx_inv_appl1_simple … H1) // -H1
-#V0 #T0 #HLV0 #HLT10 #H destruct
-elim (eq_false_inv_tpair_dx … H2) -H2
-[ -IHV -HT1 /4 width=3 by csx_cpx_trans, cpx_pair_sn/
-| -HLT10 * #H #HV0 destruct
- @IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/ (**) (* full auto 17s *)
-]
-qed.
-
-lemma csx_fqu_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/2 width=8 by csx_inv_lref_bind, csx_inv_lift, csx_fwd_flat_dx, csx_fwd_bind_dx, csx_fwd_pair_sn/
-qed-.
-
-lemma csx_fquq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
-[ /2 width=5 by csx_fqu_conf/
-| * #HG #HL #HT destruct //
-]
-qed-.
-
-lemma csx_fqup_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=5 by csx_fqu_conf/
-qed-.
-
-lemma csx_fqus_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fqus_inv_gen … H12) -H12
-[ /2 width=5 by csx_fqup_conf/
-| * #HG #HL #HT destruct //
-]
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem csx_gcp: ∀h,o. gcp (cpx h o) (eq …) (csx h o).
-#h #o @mk_gcp
-[ normalize /3 width=13 by cnx_lift/
-| #G #L elim (deg_total h o 0) /3 width=8 by cnx_sort_iter, ex_intro/
-| /2 width=8 by csx_lift/
-| /2 width=3 by csx_fwd_flat_dx/
-]
-qed.