X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fdynamic%2Fnta_preserve.ma;h=eb8d953a23c649d61bc13a90853971f41297036e;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=8d69b3ca21f7a2bc1246e88499586634eb6a363c;hpb=dc20d16b32940a94d29a04de0d4fe1f80e00a73f;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma
index 8d69b3ca2..eb8d953a2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma
@@ -12,97 +12,97 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_5_3.ma".
 include "basic_2/rt_equivalence/cpcs_cpcs.ma".
-include "basic_2/dynamic/cnv_cpcs.ma".
+include "basic_2/dynamic/cnv_preserve_cpcs.ma".
 include "basic_2/dynamic/nta.ma".
 
 (* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
 
 (* Properties based on preservation *****************************************)
 
-lemma cnv_cpms_nta (a) (h) (G) (L):
-      ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀U.⦃G,L⦄ ⊢ T ➡*[1,h] U → ⦃G,L⦄ ⊢ T :[a,h] U.
+lemma cnv_cpms_nta (h) (a) (G) (L):
+      ∀T. ❪G,L❫ ⊢ T ![h,a] → ∀U.❪G,L❫ ⊢ T ➡*[1,h] U → ❪G,L❫ ⊢ T :[h,a] U.
 /3 width=4 by cnv_cast, cnv_cpms_trans/ qed.
 
-lemma cnv_nta_sn (a) (h) (G) (L):
-      ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T :[a,h] U.
-#a #h #G #L #T #HT
+lemma cnv_nta_sn (h) (a) (G) (L):
+      ∀T. ❪G,L❫ ⊢ T ![h,a] → ∃U. ❪G,L❫ ⊢ T :[h,a] U.
+#h #a #G #L #T #HT
 elim (cnv_fwd_cpm_SO … HT) #U #HTU
 /4 width=2 by cnv_cpms_nta, cpm_cpms, ex_intro/
 qed-.
 
 (* Basic_1: was: ty3_typecheck *)
-lemma nta_typecheck (a) (h) (G) (L):
-      ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] T0.
+lemma nta_typecheck (h) (a) (G) (L):
+      ∀T,U. ❪G,L❫ ⊢ T :[h,a] U → ∃T0. ❪G,L❫ ⊢ ⓝU.T :[h,a] T0.
 /3 width=1 by cnv_cast, cnv_nta_sn/ qed-.
 
 (* Basic_1: was: ty3_correct *)
 (* Basic_2A1: was: ntaa_fwd_correct *)
-lemma nta_fwd_correct (a) (h) (G) (L):
-      ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ U :[a,h] T0.
+lemma nta_fwd_correct (h) (a) (G) (L):
+      ∀T,U. ❪G,L❫ ⊢ T :[h,a] U → ∃T0. ❪G,L❫ ⊢ U :[h,a] T0.
 /3 width=2 by nta_fwd_cnv_dx, cnv_nta_sn/ qed-.
 
 lemma nta_pure_cnv (h) (G) (L):
-      ∀T,U. ⦃G,L⦄ ⊢ T :*[h] U →
-      ∀V. ⦃G,L⦄ ⊢ ⓐV.U !*[h] → ⦃G,L⦄ ⊢ ⓐV.T :*[h] ⓐV.U.
+      ∀T,U. ❪G,L❫ ⊢ T :[h,𝛚] U →
+      ∀V. ❪G,L❫ ⊢ ⓐV.U ![h,𝛚] → ❪G,L❫ ⊢ ⓐV.T :[h,𝛚] ⓐV.U.
 #h #G #L #T #U #H1 #V #H2
 elim (cnv_inv_cast … H1) -H1 #X0 #HU #HT #HUX0 #HTX0
 elim (cnv_inv_appl … H2) #n #p #X1 #X2 #_ #HV #_ #HVX1 #HUX2
 elim (cnv_cpms_conf … HU … HUX0 … HUX2) -HU -HUX2
 <minus_O_n <minus_n_O #X #HX0 #H
 elim (cpms_inv_abst_sn … H) -H #X3 #X4 #HX13 #HX24 #H destruct
-@(cnv_cast … (ⓐV.X0)) [2:|*: /2 width=1 by cpms_appl_dx/ ]
-@(cnv_appl … X3) [4: |*: /2 width=7 by cpms_trans, cpms_cprs_trans/ ]
-#H destruct
+@(cnv_cast … (ⓐV.X0)) [2:|*: /2 width=1 by cpms_appl_dx/ ] (**) (* full auto a bit slow *)
+/3 width=10 by cnv_appl, cpms_trans, cpms_cprs_trans/
 qed.
 
 (* Basic_1: uses: ty3_sred_wcpr0_pr0 *)
-lemma nta_cpr_conf_lpr (a) (h) (G):
-      ∀L1,T1,U. ⦃G,L1⦄ ⊢ T1 :[a,h] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
-      ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
+lemma nta_cpr_conf_lpr (h) (a) (G):
+      ∀L1,T1,U. ❪G,L1❫ ⊢ T1 :[h,a] U → ∀T2. ❪G,L1❫ ⊢ T1 ➡[h] T2 →
+      ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T2 :[h,a] U.
+#h #a #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
 /3 width=6 by cnv_cpm_trans_lpr, cpm_cast/
 qed-.
 
 (* Basic_1: uses: ty3_sred_pr2 ty3_sred_pr0 *)
-lemma nta_cpr_conf (a) (h) (G) (L):
-      ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
-      ∀T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cpr_conf (h) (a) (G) (L):
+      ∀T1,U. ❪G,L❫ ⊢ T1 :[h,a] U →
+      ∀T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → ❪G,L❫ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
 /3 width=6 by cnv_cpm_trans, cpm_cast/
 qed-.
 
 (* Note: this is the preservation property *)
 (* Basic_1: uses: ty3_sred_pr3 ty3_sred_pr1 *)
-lemma nta_cprs_conf (a) (h) (G) (L):
-      ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
-      ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cprs_conf (h) (a) (G) (L):
+      ∀T1,U. ❪G,L❫ ⊢ T1 :[h,a] U →
+      ∀T2. ❪G,L❫ ⊢ T1 ➡*[h] T2 → ❪G,L❫ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
 /3 width=6 by cnv_cpms_trans, cpms_cast/
 qed-.
 
 (* Basic_1: uses: ty3_cred_pr2 *)
-lemma nta_lpr_conf (a) (h) (G):
-      ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
-      ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lpr_conf (h) (a) (G):
+      ∀L1,T,U. ❪G,L1❫ ⊢ T :[h,a] U →
+      ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
 /2 width=3 by cnv_lpr_trans/
 qed-.
 
 (* Basic_1: uses: ty3_cred_pr3 *)
-lemma nta_lprs_conf (a) (h) (G):
-      ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
-      ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lprs_conf (h) (a) (G):
+      ∀L1,T,U. ❪G,L1❫ ⊢ T :[h,a] U →
+      ∀L2. ❪G,L1❫ ⊢ ➡*[h] L2 → ❪G,L2❫ ⊢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
 /2 width=3 by cnv_lprs_trans/
 qed-.
 
 (* Inversion lemmas based on preservation ***********************************)
 
-lemma nta_inv_ldef_sn (a) (h) (G) (K) (V):
-      ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 :[a,h] X2 →
-      ∃∃W,U. ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[1] W ≘ U & ⦃G,K.ⓓV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓓV⦄ ⊢ X2 ![a,h].
-#a #h #G #Y #X #X2 #H
+lemma nta_inv_ldef_sn (h) (a) (G) (K) (V):
+      ∀X2. ❪G,K.ⓓV❫ ⊢ #0 :[h,a] X2 →
+      ∃∃W,U. ❪G,K❫ ⊢ V :[h,a] W & ⇧*[1] W ≘ U & ❪G,K.ⓓV❫ ⊢ U ⬌*[h] X2 & ❪G,K.ⓓV❫ ⊢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
 elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
 elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
 elim (cpms_inv_delta_sn … H2) -H2 *
@@ -112,10 +112,10 @@ elim (cpms_inv_delta_sn … H2) -H2 *
 ]
 qed-.
 
-lemma nta_inv_lref_sn (a) (h) (G) (L):
-      ∀X2,i. ⦃G,L⦄ ⊢ #↑i :[a,h] X2 →
-      ∃∃I,K,T2,U2. ⦃G,K⦄ ⊢ #i :[a,h] T2 & ⬆*[1] T2 ≘ U2 & ⦃G,K.ⓘ{I}⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,K.ⓘ{I}⦄ ⊢ X2 ![a,h] & L = K.ⓘ{I}.
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn (h) (a) (G) (L):
+      ∀X2,i. ❪G,L❫ ⊢ #↑i :[h,a] X2 →
+      ∃∃I,K,T2,U2. ❪G,K❫ ⊢ #i :[h,a] T2 & ⇧*[1] T2 ≘ U2 & ❪G,K.ⓘ[I]❫ ⊢ U2 ⬌*[h] X2 & ❪G,K.ⓘ[I]❫ ⊢ X2 ![h,a] & L = K.ⓘ[I].
+#h #a #G #L #X2 #i #H
 elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
 elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
 elim (cpms_inv_lref_sn … H2) -H2 *
@@ -125,11 +125,11 @@ elim (cpms_inv_lref_sn … H2) -H2 *
 ]
 qed-.
 
-lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L): 
-      ∀X2, i. ⦃G,L⦄ ⊢ #i :[a,h] X2 →
-      ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h]
-       | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![a,h] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn_drops_cnv (h) (a) (G) (L):
+      ∀X2,i. ❪G,L❫ ⊢ #i :[h,a] X2 →
+      ∨∨ ∃∃K,V,W,U. ⇩*[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V :[h,a] W & ⇧*[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a]
+       | ∃∃K,W,U. ⇩*[i] L ≘ K. ⓛW & ❪G,K❫ ⊢ W ![h,a] & ⇧*[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #i #H
 elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
 elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV
 elim (cpms_inv_lref1_drops … H2) -H2 *
@@ -139,17 +139,17 @@ elim (cpms_inv_lref1_drops … H2) -H2 *
   /4 width=8 by cnv_cpms_nta, cpcs_cprs_sn, ex5_4_intro, or_introl/
 | #n #Y #X #U #H #HVU #HUX1 #H0 destruct
   lapply (drops_mono … H … HLK) -H #H destruct
-  elim (lifts_total V (𝐔❴↑i❵)) #W #HVW
+  elim (lifts_total V (𝐔❨↑i❩)) #W #HVW
   lapply (cpms_lifts_bi … HVU (Ⓣ) … L … HVW … HUX1) -U
   [ /2 width=2 by drops_isuni_fwd_drop2/ ] #HWX1
   /4 width=9 by cprs_div, ex5_3_intro, or_intror/
 ]
 qed-.
 
-lemma nta_inv_bind_sn_cnv (a) (h) (p) (I) (G) (K) (X2):
-      ∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] X2 →
-      ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![a,h].
-#a #h #p * #G #K #X2 #V #T #H
+lemma nta_inv_bind_sn_cnv (h) (a) (p) (I) (G) (K) (X2):
+      ∀V,T. ❪G,K❫ ⊢ ⓑ[p,I]V.T :[h,a] X2 →
+      ∃∃U. ❪G,K❫ ⊢ V ![h,a] & ❪G,K.ⓑ[I]V❫ ⊢ T :[h,a] U & ❪G,K❫ ⊢ ⓑ[p,I]V.U ⬌*[h] X2 & ❪G,K❫ ⊢ X2 ![h,a].
+#h #a #p * #G #K #X2 #V #T #H
 elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
 elim (cnv_inv_bind … H1) -H1 #HV #HT
 [ elim (cpms_inv_abbr_sn_dx … H2) -H2 *
@@ -165,32 +165,26 @@ qed-.
 
 (* Basic_1: uses: ty3_gen_appl *)
 lemma nta_inv_appl_sn (h) (G) (L) (X2):
-      ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h] X2 →
-      ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h] W & ⦃G,L⦄ ⊢ T :[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h].
+      ∀V,T. ❪G,L❫ ⊢ ⓐV.T :[h,𝟐] X2 →
+      ∃∃p,W,U. ❪G,L❫ ⊢ V :[h,𝟐] W & ❪G,L❫ ⊢ T :[h,𝟐] ⓛ[p]W.U & ❪G,L❫ ⊢ ⓐV.ⓛ[p]W.U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,𝟐].
 #h #G #L #X2 #V #T #H
 elim (cnv_inv_cast … H) -H #X #HX2 #H1 #HX2 #H2
-elim (cnv_inv_appl … H1) * [ | #n ] #p #W #U #Hn #HV #HT #HVW #HTU
-[ lapply (cnv_cpms_trans … HT … HTU) #H
-  elim (cnv_inv_bind … H) -H #_ #HU
-  elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
-  lapply (cpms_step_dx … HTU 1 (ⓛ{p}W.U0) ?) -HTU [ /2 width=1 by cpm_bind/ ] #HTU
-| lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Hn #H destruct
-]
+elim (cnv_inv_appl … H1) #n #p #W #U #H <H -n #HV #HT #HVW #HTU
 /5 width=11 by cnv_cpms_nta, cnv_cpms_conf_eq, cpcs_cprs_div, cpms_appl_dx, ex4_3_intro/
 qed-.
 
 (* Basic_2A1: uses: nta_fwd_pure1 *)
 lemma nta_inv_pure_sn_cnv (h) (G) (L) (X2):
-                          ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :*[h] X2 →
-                          ∨∨ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :*[h] W & ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h]
-                           | ∃∃U. ⦃G,L⦄ ⊢ T :*[h] U & ⦃G,L⦄ ⊢ ⓐV.U !*[h] & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h].
+      ∀V,T. ❪G,L❫ ⊢ ⓐV.T :[h,𝛚] X2 →
+      ∨∨ ∃∃p,W,U. ❪G,L❫ ⊢ V :[h,𝛚] W & ❪G,L❫ ⊢ T :[h,𝛚] ⓛ[p]W.U & ❪G,L❫ ⊢ ⓐV.ⓛ[p]W.U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,𝛚]
+       | ∃∃U. ❪G,L❫ ⊢ T :[h,𝛚] U & ❪G,L❫ ⊢ ⓐV.U ![h,𝛚] & ❪G,L❫ ⊢ ⓐV.U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,𝛚].
 #h #G #L #X2 #V #T #H
 elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H
 elim (cnv_inv_appl … H1) * [| #n ] #p #W0 #T0 #Hn #HV #HT #HW0 #HT0
 [ lapply (cnv_cpms_trans … HT … HT0) #H0
   elim (cnv_inv_bind … H0) -H0 #_ #HU
   elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
-  lapply (cpms_step_dx … HT0 1 (ⓛ{p}W0.U0) ?) -HT0 [ /2 width=1 by cpm_bind/ ] #HT0
+  lapply (cpms_step_dx … HT0 1 (ⓛ[p]W0.U0) ?) -HT0 [ /2 width=1 by cpm_bind/ ] #HT0
   lapply (cpms_appl_dx … V V … HT0) [ // ] #HTU0
   lapply (cnv_cpms_conf_eq … H1 … HTU0 … H) -H1 -H -HTU0 #HU0X1
   /4 width=8 by cnv_cpms_nta, cpcs_cprs_div, ex4_3_intro, or_introl/
@@ -207,10 +201,10 @@ elim (cnv_inv_appl … H1) * [| #n ] #p #W0 #T0 #Hn #HV #HT #HW0 #HT0
 qed-.
 
 (* Basic_2A1: uses: nta_inv_cast1 *)
-lemma nta_inv_cast_sn (a) (h) (G) (L) (X2):
-      ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] X2 →
-      ∧∧ ⦃G,L⦄ ⊢ T :[a,h] U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #U #T #H
+lemma nta_inv_cast_sn (h) (a) (G) (L) (X2):
+      ∀U,T. ❪G,L❫ ⊢ ⓝU.T :[h,a] X2 →
+      ∧∧ ❪G,L❫ ⊢ T :[h,a] U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #U #T #H
 elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
 elim (cnv_inv_cast … H1) #X #HU #HT #HUX #HTX
 elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
@@ -226,10 +220,10 @@ elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
 qed-.
 
 (* Basic_1: uses: ty3_gen_cast *)
-lemma nta_inv_cast_sn_old (a) (h) (G) (L) (X2):
-      ∀T0,T1. ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] X2 →
-      ∃∃T2. ⦃G,L⦄ ⊢ T0 :[a,h] T1 & ⦃G,L⦄ ⊢ T1 :[a,h] T2 & ⦃G,L⦄ ⊢ ⓝT2.T1 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #T0 #T1 #H
+lemma nta_inv_cast_sn_old (h) (a) (G) (L) (X2):
+      ∀T0,T1. ❪G,L❫ ⊢ ⓝT1.T0 :[h,a] X2 →
+      ∃∃T2. ❪G,L❫ ⊢ T0 :[h,a] T1 & ❪G,L❫ ⊢ T1 :[h,a] T2 & ❪G,L❫ ⊢ ⓝT2.T1 ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #T0 #T1 #H
 elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
 elim (cnv_inv_cast … H1) #X #HT1 #HT0 #HT1X #HT0X
 elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
@@ -248,12 +242,12 @@ elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
 qed-.
 
 (* Basic_1: uses: ty3_gen_lift *)
-(* Note: "⦃G,L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G,L⦄ ⊢ X2 ➡*[h] U2" *)
-lemma nta_inv_lifts_sn (a) (h) (G):
-      ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[a,h] X2 →
-      ∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. ⬆*[f] T1 ≘ T2 →
-      ∃∃U1,U2. ⦃G,K⦄ ⊢ T1 :[a,h] U1 & ⬆*[f] U1 ≘ U2 & ⦃G,L⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
+(* Note: "❪G, L❫ ⊢ U2 ⬌*[h] X2" can be "❪G, L❫ ⊢ X2 ➡*[h] U2" *)
+lemma nta_inv_lifts_sn (h) (a) (G):
+      ∀L,T2,X2. ❪G,L❫ ⊢ T2 :[h,a] X2 →
+      ∀b,f,K. ⇩*[b,f] L ≘ K → ∀T1. ⇧*[f] T1 ≘ T2 →
+      ∃∃U1,U2. ❪G,K❫ ⊢ T1 :[h,a] U1 & ⇧*[f] U1 ≘ U2 & ❪G,L❫ ⊢ U2 ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
+#h #a #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
 elim (cnv_inv_cast … H) -H #U2 #HX2 #HT2 #HXU2 #HTU2
 lapply (cnv_inv_lifts … HT2 … HLK … HT12) -HT2 #HT1
 elim (cpms_inv_lifts_sn … HTU2 … HLK … HT12) -T2 -HLK #U1 #HU12 #HTU1
@@ -263,9 +257,9 @@ qed-.
 (* Forward lemmas based on preservation *************************************)
 
 (* Basic_1: was: ty3_unique *)
-theorem nta_mono (a) (h) (G) (L) (T):
-        ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 → ∀U2. ⦃G,L⦄ ⊢ T :[a,h] U2 → ⦃G,L⦄  ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #H2
+theorem nta_mono (h) (a) (G) (L) (T):
+        ∀U1. ❪G,L❫ ⊢ T :[h,a] U1 → ∀U2. ❪G,L❫ ⊢ T :[h,a] U2 → ❪G,L❫  ⊢ U1 ⬌*[h] U2.
+#h #a #G #L #T #U1 #H1 #U2 #H2
 elim (cnv_inv_cast … H1) -H1 #X1 #_ #_ #HUX1 #HTX1
 elim (cnv_inv_cast … H2) -H2 #X2 #_ #HT #HUX2 #HTX2
 lapply (cnv_cpms_conf_eq … HT … HTX1 … HTX2) -T #HX12
@@ -275,10 +269,10 @@ qed-.
 (* Advanced properties ******************************************************)
 
 (* Basic_1: uses: ty3_sconv_pc3 *)
-lemma nta_cpcs_bi (a) (h) (G) (L):
-      ∀T1,U1. ⦃G,L⦄ ⊢ T1 :[a,h] U1 → ∀T2,U2. ⦃G,L⦄ ⊢ T2 :[a,h] U2 →
-      ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+lemma nta_cpcs_bi (h) (a) (G) (L):
+      ∀T1,U1. ❪G,L❫ ⊢ T1 :[h,a] U1 → ∀T2,U2. ❪G,L❫ ⊢ T2 :[h,a] U2 →
+      ❪G,L❫ ⊢ T1 ⬌*[h] T2 → ❪G,L❫ ⊢ U1 ⬌*[h] U2.
+#h #a #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
 /3 width=6 by nta_mono, nta_cprs_conf/
 qed-.