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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfpxs_cpxs.ma
index e565bdc67..472e94af1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfpxs_cpxs.ma
@@ -19,18 +19,16 @@ include "basic_2/rt_computation/lfpxs_fqup.ma".
 
 (* Properties with uncounted context-sensitive rt-computation for terms *****)
 
-lemma lfpxs_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
-                  ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
-#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
-/3 width=3 by lfpxs_strap1, lfpx_pair/
-qed.
+(* Basic_2A1: uses: lpxs_pair lpxs_pair_refl *)
+lemma lfpxs_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+                       ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
+/2 width=1 by tc_lfxs_pair_refl/ qed.
 
-(* Basic_2A1: was just: lpxs_cpx_trans *)
 lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
-/3 width=5 by s_r_trans_LTC2, lfpx_cpxs_trans/ qed-.
+#h #G @s_r_trans_LTC2 @lfpx_cpxs_trans (**) (* auto fails *)
+qed-.
 
 (* Note: lfpxs_cpx_conf does not hold, thus we cannot invoke s_r_trans_LTC1 *)
-(* Basic_2A1: was just: lpxs_cpxs_trans *)
 lemma lfpxs_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpxs h G).
 #h #G @s_r_to_s_rs_trans @s_r_trans_LTC2
 @s_rs_trans_TC1 /2 width=3 by lfpx_cpxs_trans/ (**) (* full auto too slow *)
@@ -41,7 +39,7 @@ qed-.
 lemma cpxs_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
                   ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
                   ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
-/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair, cpxs_bind/ qed.
+/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair_refl, cpxs_bind/ qed.
 
 (* Advanced inversion lemmas on uncounted rt-computation for terms **********)
 
@@ -52,7 +50,7 @@ lemma cpxs_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2
 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
 elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
 lapply (lfpxs_cpx_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
+/3 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro/
 qed-.
 
 lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → (
@@ -66,13 +64,13 @@ lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2
   elim (cpx_inv_abbr1 … HU02) -HU02 *
   [ #V2 #T2 #HV02 #HT02 #H destruct
     lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?)
-    /4 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
+    /4 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
   | #T2 #HT02 #HUT2
     lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
-    /4 width=3 by lfpxs_pair, cpxs_trans, ex3_intro, or_intror/
+    /4 width=3 by lfpxs_pair_refl, cpxs_trans, ex3_intro, or_intror/
   ]
 | #U1 #HTU1 #HU01
-  elim (cpx_lifts … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
+  elim (cpx_lifts_sn … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
   /4 width=3 by cpxs_strap1, drops_refl, drops_drop, ex3_intro, or_intror/
 ]
 qed-.