X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fdynamic%2Fsnv_cpcs.ma;h=e62507f9b2458548de0e676c1ae9eafda82a4d7d;hb=e76eade57c0454a58b0d58e5484efe9af417847e;hp=7078c9f3e561a9c6c4bd0a739c0c478deebe0a4b;hpb=583c59b229ba770c9694c703b381542ff2e67f4e;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
index 7078c9f3e..e62507f9b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
@@ -12,156 +12,159 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/ssta_ssta.ma".
-include "basic_2/computation/ygt.ma".
-include "basic_2/dynamic/snv_ltpss_dx.ma".
-include "basic_2/dynamic/snv_sstas.ma".
+include "basic_2/unfold/lsstas_lsstas.ma".
+include "basic_2/computation/fpbs_lift.ma".
+include "basic_2/computation/fpbg_fpns.ma".
+include "basic_2/equivalence/cpes_cpds.ma".
+include "basic_2/dynamic/snv.ma".
 
 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
 
 (* Inductive premises for the preservation results **************************)
-(*
-definition IH_snv_dxprs: ∀h:sh. sd h → relation2 lenv term ≝
-                         λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
-                         ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-*)
-definition IH_ssta_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
-                             λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
-                             ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                             ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 →
-                             ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-
-definition IH_snv_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
-                            λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
-                            ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-
-definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
-                        λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
-                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] U1 → ⦃h, L1⦄ ⊩ U1 :[g].
-
-fact ssta_ltpr_cpr_aux: ∀h,g,L1,T1. IH_ssta_ltpr_tpr h g L1 T1 →
-                        ⦃h, L1⦄ ⊩ T1 :[g] →
-                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                        ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 →
-                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L1 #T1 #IH #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 * #T #HT1T #HTT2
-elim (IH … HTU1 … HL12 … HT1T) // -T1 #U #HTU #HU1
-elim (ssta_tpss_conf … HTU … HTT2) -T #U2 #HTU2 #HU2
-lapply (ltpr_cpr_trans … HL12 U U2 ?) -HL12 /2 width=3/ -HU2 #HU2
-lapply (cpcs_cprs_strap1 … HU1 … HU2) -U /2 width=3/
-qed-.
 
-fact snv_ltpr_cpr_aux: ∀h,g,L1,T1. IH_snv_ltpr_tpr h g L1 T1 →
-                       ⦃h, L1⦄ ⊩ T1 :[g] →
-                       ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-#h #g #L1 #T1 #IH #HT1 #L2 #HL12 #T2 * #T #HT1T #HTT2
-lapply (IH … HL12 … HT1T) -HL12 // -T1 #HT0
-lapply (snv_tpss_conf … HT0 … HTT2) -T //
+definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+                           λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                           ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+
+definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+                          λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                          ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
+                          ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                          ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
+
+definition IH_lsstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+                              λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                              ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+                              ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, g, l2] U1 →
+                              ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                              ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, g, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+
+definition IH_snv_lsstas: ∀h:sh. sd h → relation3 genv lenv term ≝
+                          λh,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+                          ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T ▪[h, g] l1 →
+                          ∀U. ⦃G, L⦄ ⊢ T •*[h, g, l2] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+
+(* Properties for the preservation results **********************************)
+
+fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                       ∀G,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                       ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
+@(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/
 qed-.
 
-fact snv_cprs_aux: ∀h,g,L0,T0.
-                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                   ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                   ∀T2. L1 ⊢ T1 ➡* T2 → ⦃h, L1⦄ ⊩ T2 :[g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #T2 #H
-@(cprs_ind … H) -T2 // -HT1
-/4 width=6 by snv_ltpr_cpr_aux, ygt_cprs_trans/
+fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                      ∀G,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                      ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
+                      ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H
+@(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
 qed-.
 
-fact ssta_cprs_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                    ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                    ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 → ∀T2. L1 ⊢ T1 ➡* T2 →
-                    ∃∃U2. ⦃h, L1⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
-@(cprs_ind … H) -T2 [ /2 width=3/ ]
-#T #T2 #HT1T #HTT2 * #U #HTU #HU1 
-elim (ssta_ltpr_cpr_aux … HTU … HTT2) //
-[2: /3 width=7 by snv_cprs_aux, ygt_cprs_trans/
-|3: /3 width=3 by ygt_cprs_trans/
-] -L0 -T0 -T1 -T #U2 #HTU2 #HU2
-lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
+fact da_cpcs_aux: ∀h,g,G0,L0,T0.
+                  (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                  (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                  ∀G,L,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+                  ∀T2. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] →
+                  ∀l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ∀l2. ⦃G, L⦄ ⊢ T2 ▪[h, g] l2 →
+                  ⦃G, L⦄ ⊢ T1 ⬌* T2 → l1 = l2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #l1 #Hl1 #l2 #Hl2 #H
+elim (cpcs_inv_cprs … H) -H /4 width=18 by da_cprs_lpr_aux, da_mono/
 qed-.
 
-fact ssta_cpcs_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                    ∀L,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T2⦄ →
-                    ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
-                    ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
-                    ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
-                    L ⊢ T1 ⬌* T2 →
-                    l1 = l2 ∧ L ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (ssta_cprs_aux … HLT01 HT1 … HTU1 … H1) -T1 /2 width=1/ #W1 #H1 #HUW1
-elim (ssta_cprs_aux … HLT02 HT2 … HTU2 … H2) -T2 /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
-elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
-lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
+fact ssta_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+                       ∀G,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                       ∀l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l+1 →
+                       ∀U1. ⦃G, L1⦄ ⊢ T1 •[h, g] U1 →
+                       ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                       ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #H01 #HT1 #l #Hl #U1 #HTU1 #T2 #HT12 #L2 #HL12
+elim (IH … H01 … 1 … Hl U1 … HT12 … HL12)
+/3 width=3 by lsstas_inv_SO, ssta_lsstas, ex2_intro/
 qed-.
 
-fact snv_sstas_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                    ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                    ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ⦃h, L1⦄ ⊩ U1 :[g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #U1 #H
-@(sstas_ind … H) -U1 // -HT1 /3 width=5 by ygt_sstas_trans/
+fact lsstas_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+                          ∀G,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                          ∀l1,l2. l2 ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 →
+                          ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, g, l2] U1 →
+                          ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                          ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, g, l2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=10 by/ ]
+#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
+elim (IHT1 L1) // -IHT1 #U #HTU #HU1
+elim (IH1 … Hl21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
+[2: /3 width=12 by da_cprs_lpr_aux/
+|3: /3 width=10 by snv_cprs_lpr_aux/
+|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
+] -G0 -L0 -T0 -T1 -T -l1 #U2 #HTU2 #HU2
+/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
 qed-.
 
-fact sstas_cprs_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                     ∀T2. L1 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
-                     ∃∃U2. ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #T2 #HT12 #U1 #H
-@(sstas_ind … H) -U1 [ /3 width=3/ ]
-#U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
-lapply (snv_cprs_aux … IH2 … HT1 … HT12) // #HT2
-elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
-elim (ssta_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12
-[2,3: /3 width=7 by snv_sstas_aux, ygt_cprs_trans/
-|4,5: /3 width=3 by ygt_sstas_trans, ygt_cprs_trans/
-] -L0 -T0 -T1 -HT2 #H #HW12 destruct /3 width=4/
+fact lsstas_cpcs_lpr_aux: ∀h,g,G0,L0,T0.
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                          (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+                          ∀G,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                          ∀l,l1. l ≤ l1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] l1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, g, l] U1 →
+                          ∀T2. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+                          ∀l2. l ≤ l2 → ⦃G, L1⦄ ⊢ T2 ▪[h, g] l2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, g, l] U2 →
+                          ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l #l1 #Hl1 #HTl1 #U1 #HTU1 #T2 #H02 #HT2 #l2 #Hl2 #HTl2 #U2 #HTU2 #H #L2 #HL12
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (lsstas_cprs_lpr_aux … H01 HT1 … Hl1 HTl1 … HTU1 … H1 … HL12) -T1 /2 width=1 by/ #W1 #H1 #HUW1
+elim (lsstas_cprs_lpr_aux … H02 HT2 … Hl2 HTl2 … HTU2 … H2 … HL12) -T2 /2 width=1 by/ #W2 #H2 #HUW2 -L0 -T0
+lapply (lsstas_mono … H1 … H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/
 qed-.
 
-fact dxprs_cprs_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                     ∀T2. L1 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
-                     ∃∃U2. ⦃h, L1⦄ ⊢ T2 •*➡*[g] U2 & L1 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #T2 #HT12 #U1 * #W1 #HTW1 #HWU1
-elim (sstas_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
-lapply (cpcs_cprs_conf … HWU1 … HW12) -W1 #H
-elim (cpcs_inv_cprs … H) -H /3 width=3/
+fact snv_ssta_aux: ∀h,g,G0,L0,T0.
+                   (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
+                   ∀G,L,T. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+                   ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 →
+                   ∀U. ⦃G, L⦄ ⊢ T •[h, g] U → ⦃G, L⦄ ⊢ U ¡[h, g].
+/3 width=8 by lsstas_inv_SO, ssta_lsstas/ qed-.
+
+fact lsstas_cpds_aux: ∀h,g,G0,L0,T0.
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                      (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+                      ∀G,L,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+                      ∀l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
+                      ∀U1. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] U1 → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
+                      ∃∃U2,l. l ≤ l2 & ⦃G, L⦄ ⊢ T2 •*[h, g, l] U2 & ⦃G, L⦄ ⊢ U1 •*⬌*[h, g] U2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 * #T #l0 #l #Hl0 #H #HT1T #HTT2
+lapply (da_mono … H … Hl1) -H #H destruct
+lapply (lsstas_da_conf … HTU1 … Hl1) #Hl12
+elim (le_or_ge l2 l) #Hl2
+[ lapply (lsstas_conf_le … HTU1 … HT1T) -HT1T // #HU1T
+  /5 width=11 by cpds_cpes_dx, monotonic_le_minus_l, ex3_2_intro, ex4_3_intro/
+| lapply (lsstas_da_conf … HT1T … Hl1) #Hl1l
+  lapply (lsstas_conf_le … HT1T … HTU1) -HTU1 // #HTU1
+  elim (lsstas_cprs_lpr_aux … IH3 IH2 IH1 … Hl1l … HTU1 … HTT2 L)
+  /3 width=8 by fpbg_fpbs_trans, lsstas_fpbs, monotonic_le_minus_l/ -T #U2 #HTU2 #HU12
+  /3 width=5 by cpcs_cpes, ex3_2_intro/
+]
 qed-.
 
-fact sstas_ltpr_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊩ T :[g] →
-                     ∀L2. L1 ➡ L2 → ∀U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
-                     ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T #H01 #HT #L2 #HL12 #U1 #H
-@(sstas_ind … H) -U1 [ /4 width=3/ ]
-#W1 #U1 #l1 #HTW1 #HWU1 * #W2 #HTW2 #HW12
-lapply (snv_sstas_aux … IH2 … HTW1) // -IH2 -HT #HW1
-
-(*
-elim (IH1 … HWU1 … HL12 W1) // -IH1 -HW1 -HWU1 -HL12
-[2: /2 width=3 by ygt_sstas_trans/ ] #U2 
-*)
-(*
-fact sstas_dxprs_aux: ∀h,g,L0,T0.
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                      ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                      ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀T2. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T2 →
-                      ∃∃U2. ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #HTU1 #T2 * #T #HT1T #HTT2
-*)
\ No newline at end of file
+fact cpds_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+                       (∀G1,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+                       ∀G,L1,T1. ⦃G0, L0, T0⦄ >⋕[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+                       ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
+                       ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                       ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 * #W1 #l1 #l2 #Hl21 #Hl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12
+lapply (IH2 … H01 … Hl1 … HT12 … HL12) -L0 -T0 // -T1 #Hl1
+lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
+elim (cpcs_inv_cprs … H) -H /3 width=7 by ex4_3_intro, ex2_intro/
+qed-.