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diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
index 381e6e939..b2907c52b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
@@ -19,41 +19,42 @@ include "basic_2/computation/lpxs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lpxs_pair: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ∀V1,V2. ⦃h, L1⦄ ⊢ V1 ➡*[h, g] V2 →
-                 ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+lemma lpxs_pair: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h, g] V2 →
+                 ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
 /2 width=1 by TC_lpx_sn_pair/ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lpxs_inv_pair1: ∀h,g,I,K1,L2,V1. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
-                      ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair1: ∀h,g,I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
+                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
 /3 width=3 by TC_lpx_sn_inv_pair1, lpx_cpxs_trans/ qed-.
 
-lemma lpxs_inv_pair2: ∀h,g,I,L1,K2,V2. ⦃h, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
-                      ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
 /3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
 
 (* Properties on context-sensitive extended parallel computation for terms **)
 
-lemma lpxs_cpx_trans: ∀h,g. s_r_trans … (cpx h g) (lpxs h g).
+lemma lpxs_cpx_trans: ∀h,g,G. s_r_trans … (cpx h g G) (lpxs h g G).
 /3 width=5 by s_r_trans_TC2, lpx_cpxs_trans/ qed-.
 
-lemma lpxs_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpxs h g).
+lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_trans … (cpx h g G) (lpxs h g G).
 /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ qed-.
 
-lemma cpxs_bind2: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                  ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
                   ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-#h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
+#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lpxs_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
 
 (* Inversion lemmas on context-sensitive ext parallel computation for terms *)
 
-lemma cpxs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
                                U2 = ⓛ{a}V2.T2.
-#h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5/
+#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5/
 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
 elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
 lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?) /2 width=1/ -HT02 #HT02
@@ -61,12 +62,12 @@ lapply (cpxs_strap1 … HV10 … HV02) -V0
 lapply (cpxs_trans … HT10 … HT02) -T0 /2 width=5/
 qed-.
 
-lemma cpxs_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
                                U2 = ⓓ{a}V2.T2
                       ) ∨
-                      ∃∃T2. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
+                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
 #U0 #U2 #_ #HU02 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
   elim (cpx_inv_abbr1 … HU02) -HU02 *
@@ -86,6 +87,6 @@ qed-.
 
 (* More advanced properties *************************************************)
 
-lemma lpxs_pair2: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 →
-                  ∀V1,V2. ⦃h, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+lemma lpxs_pair2: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                  ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
 /3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.