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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma
index b16c21601..8bfdf9432 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma
@@ -19,22 +19,22 @@ include "basic_2/rt_computation/lprs_lpr.ma".
 (* Properties with t-bound context-sensitive rt-computarion for terms *******)
 
 lemma lprs_cpms_trans (n) (h) (G):
-                      ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 →
-                      ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+                      ∀L2,T1,T2. ❪G,L2❫ ⊢ T1 ➡*[n,h] T2 →
+                      ∀L1. ❪G,L1❫ ⊢ ➡*[h] L2 → ❪G,L1❫ ⊢ T1 ➡*[n,h] T2.
 #n #h #G #L2 #T1 #T2 #HT12 #L1 #H
 @(lprs_ind_sn … H) -L1 /2 width=3 by lpr_cpms_trans/
 qed-.
 
 lemma lprs_cpm_trans (n) (h) (G):
-                     ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 →
-                     ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+                     ∀L2,T1,T2. ❪G,L2❫ ⊢ T1 ➡[n,h] T2 →
+                     ∀L1. ❪G,L1❫ ⊢ ➡*[h] L2 → ❪G,L1❫ ⊢ T1 ➡*[n,h] T2.
 /3 width=3 by lprs_cpms_trans, cpm_cpms/ qed-.
 
 (* Basic_2A1: includes cprs_bind2 *)
 lemma cpms_bind_dx (n) (h) (G) (L):
-                   ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
-                   ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 →
-                   ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+                   ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
+                   ∀I,T1,T2. ❪G,L.ⓑ[I]V2❫ ⊢ T1 ➡*[n,h] T2 →
+                   ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[n,h] ⓑ[p,I]V2.T2.
 /4 width=5 by lprs_cpms_trans, lprs_pair, cpms_bind/ qed.
 
 (* Inversion lemmas with t-bound context-sensitive rt-computarion for terms *)
@@ -43,9 +43,9 @@ lemma cpms_bind_dx (n) (h) (G) (L):
 (* Basic_2A1: includes: cprs_inv_abst1 *)
 (* Basic_2A1: uses: scpds_inv_abst1 *)
 lemma cpms_inv_abst_sn (n) (h) (G) (L):
-                       ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n, h] X2 →
-                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[n, h] T2 &
-                                X2 = ⓛ{p}V2.T2.
+                       ∀p,V1,T1,X2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ➡*[n,h] X2 →
+                       ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡*[h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡*[n,h] T2 &
+                                X2 = ⓛ[p]V2.T2.
 #n #h #G #L #p #V1 #T1 #X2 #H
 @(cpms_ind_dx … H) -X2 /2 width=5 by ex3_2_intro/
 #n1 #n2 #X #X2 #_ * #V #T #HV1 #HT1 #H1 #H2 destruct
@@ -53,21 +53,29 @@ elim (cpm_inv_abst1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H2 destruct
 /5 width=7 by lprs_cpm_trans, lprs_pair, cprs_step_dx, cpms_trans, ex3_2_intro/
 qed-.
 
+lemma cpms_inv_abst_sn_cprs (h) (n) (p) (G) (L) (W):
+                            ∀T,X. ❪G,L❫ ⊢ ⓛ[p]W.T ➡*[n,h] X →
+                            ∃∃U. ❪G,L.ⓛW❫⊢ T ➡*[n,h] U & ❪G,L❫ ⊢ ⓛ[p]W.U ➡*[h] X.
+#h #n #p #G #L #W #T #X #H
+elim (cpms_inv_abst_sn … H) -H #W0 #U #HW0 #HTU #H destruct
+@(ex2_intro … HTU) /2 width=1 by cpms_bind/
+qed-.
+
 (* Basic_2A1: includes: cprs_inv_abst *)
-lemma cpms_inv_abst_bi (n) (h) (G) (L):
-                       ∀p,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[n, h] ⓛ{p}W2.T2 →
-                       ∧∧ ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2.
-#n #h #G #L #p #W1 #W2 #T1 #T2 #H
+lemma cpms_inv_abst_bi (n) (h) (p1) (p2) (G) (L):
+                       ∀W1,W2,T1,T2. ❪G,L❫ ⊢ ⓛ[p1]W1.T1 ➡*[n,h] ⓛ[p2]W2.T2 →
+                       ∧∧ p1 = p2 & ❪G,L❫ ⊢ W1 ➡*[h] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2.
+#n #h #p1 #p2 #G #L #W1 #W2 #T1 #T2 #H
 elim (cpms_inv_abst_sn … H) -H #W #T #HW1 #HT1 #H destruct
-/2 width=1 by conj/
+/2 width=1 by and3_intro/
 qed-.
 
 (* Basic_1: was pr3_gen_abbr *)
 (* Basic_2A1: includes: cprs_inv_abbr1 *)
-lemma cpms_inv_abbr_sn (n) (h) (G) (L):
-                       ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n, h] X2 →
-                       ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T2 & X2 = ⓓ{p}V2.T2
-                        | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ.
+lemma cpms_inv_abbr_sn_dx (n) (h) (G) (L):
+                          ∀p,V1,T1,X2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ➡*[n,h] X2 →
+                          ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡*[h] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ➡*[n,h] T2 & X2 = ⓓ[p]V2.T2
+                           | ∃∃T2. ❪G,L.ⓓV1❫ ⊢ T1 ➡*[n ,h] T2 & ⇧*[1] X2 ≘ T2 & p = Ⓣ.
 #n #h #G #L #p #V1 #T1 #X2 #H
 @(cpms_ind_dx … H) -X2 -n /3 width=5 by ex3_2_intro, or_introl/
 #n1 #n2 #X #X2 #_ * *
@@ -75,8 +83,9 @@ lemma cpms_inv_abbr_sn (n) (h) (G) (L):
   elim (cpm_inv_abbr1 … HX2) -HX2 *
   [ #V2 #T2 #HV2 #HT2 #H destruct
     /6 width=7 by lprs_cpm_trans, lprs_pair, cprs_step_dx, cpms_trans, ex3_2_intro, or_introl/
-  | #T2 #HT2 #HXT2 #Hp
-    /6 width=7 by lprs_cpm_trans, lprs_pair, cpms_trans, ex3_intro, or_intror/
+  | #T2 #HT2 #HTX2 #Hp -V
+    elim (cpm_lifts_sn … HTX2 (Ⓣ) … (L.ⓓV1) … HT2) -T2 [| /3 width=3 by drops_refl, drops_drop/ ] #X #HX2 #HTX
+    /4 width=3 by cpms_step_dx, ex3_intro, or_intror/
   ]
 | #T #HT1 #HXT #Hp #HX2
   elim (cpm_lifts_sn … HX2 (Ⓣ) … (L.ⓓV1) … HXT) -X
@@ -86,10 +95,10 @@ qed-.
 
 (* Basic_2A1: uses: scpds_inv_abbr_abst *)
 lemma cpms_inv_abbr_abst (n) (h) (G) (L):
-                         ∀p1,p2,V1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n, h] ⓛ{p2}W2.T2 →
-                         ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ.
+                         ∀p1,p2,V1,W2,T1,T2. ❪G,L❫ ⊢ ⓓ[p1]V1.T1 ➡*[n,h] ⓛ[p2]W2.T2 →
+                         ∃∃T. ❪G,L.ⓓV1❫ ⊢ T1 ➡*[n,h] T & ⇧*[1] ⓛ[p2]W2.T2 ≘ T & p1 = Ⓣ.
 #n #h #G #L #p1 #p2 #V1 #W2 #T1 #T2 #H
-elim (cpms_inv_abbr_sn … H) -H *
+elim (cpms_inv_abbr_sn_dx … H) -H *
 [ #V #T #_ #_ #H destruct
 | /2 width=3 by ex3_intro/
 ]