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diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma
index 03566d556..6f7bf6038 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma
@@ -19,89 +19,89 @@ include "basic_2/computation/lpxs_cpxs.ma".
 
 (* Forward lemmas involving same top term constructor ***********************)
 
-lemma cpxs_fwd_cnx: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → T ≃ U.
-#h #g #L #T #HT #U #H
->(cpxs_inv_cnx1 … H HT) -L -T //
+lemma cpxs_fwd_cnx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → T ≂ U.
+#h #g #G #L #T #HT #U #H
+>(cpxs_inv_cnx1 … H HT) -G -L -T //
 qed-.
 
-lemma cpxs_fwd_sort: ∀h,g,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
-                     ⋆k ≃ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
-#h #g #L #U #k #H
+lemma cpxs_fwd_sort: ∀h,g,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
+                     ⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
+#h #g #G #L #U #k #H
 elim (cpxs_inv_sort1 … H) -H #n #l generalize in match k; -k @(nat_ind_plus … n) -n
-[ #k #_ #H -l destruct /2 width=1/
+[ #k #_ #H -l destruct /2 width=1 by or_introl/
 | #n #IHn #k >plus_plus_comm_23 #Hnl #H destruct
   lapply (deg_next_SO … Hnl) -Hnl #Hnl
   elim (IHn … Hnl) -IHn
-  [ #H lapply (tstc_inv_atom1 … H) -H #H >H -H /2 width=1/
-  | generalize in match Hnl; -Hnl @(nat_ind_plus … n) -n /2 width=1/
-    #n #_ /4 width=3/
+  [ #H lapply (tstc_inv_atom1 … H) -H #H >H -H /2 width=1 by or_intror/
+  | generalize in match Hnl; -Hnl @(nat_ind_plus … n) -n
+    /4 width=3 by cpxs_strap2, cpx_st, or_intror/
   | >iter_SO >iter_n_Sm //
   ]
 ]
 qed-.
 
 (* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,g,a,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, g] U →
-                     ⓐV.ⓛ{a}W.T ≃ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, g] U.
-#h #g #a #L #V #W #T #U #H
+lemma cpxs_fwd_beta: ∀h,g,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, g] U →
+                     ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, g] U.
+#h #g #a #G #L #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
-[ #V0 #T0 #_ #_ #H destruct /2 width=1/
+[ #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
 | #b #W0 #T0 #HT0 #HU
   elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
-  lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1 /2 width=1/ #HT1
-  @or_intror @(cpxs_trans … HU) -U /3 width=1/ (**) (* explicit constructor *)  
+  lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1
+  /5 width=3 by cpxs_trans, cpxs_bind, cpxs_pair_sn, lsubr_abst, or_intror/
 | #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
   elim (cpxs_inv_abst1 … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
 ]
 qed-.
 
 (* Note: probably this is an inversion lemma *)
-lemma cpxs_fwd_delta: ∀h,g,I,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
+lemma cpxs_fwd_delta: ∀h,g,I,G,L,K,V1,i. ⇩[i] L ≡ K.ⓑ{I}V1 →
                       ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
                       ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, g] U →
-                      #i ≃ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, g] U.
-#h #g #I #L #K #V1 #i #HLK #V2 #HV12 #U #H
-elim (cpxs_inv_lref1 … H) -H /2 width=1/
+                      #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, g] U.
+#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
+elim (cpxs_inv_lref1 … H) -H /2 width=1 by or_introl/
 * #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
-lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK) -HLK /3 width=9/
+lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+/4 width=10 by cpxs_lift, drop_fwd_drop2, or_intror/
 qed-.
 
-lemma cpxs_fwd_theta: ∀h,g,a,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, g] U →
-                      ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≃ U ∨
+lemma cpxs_fwd_theta: ∀h,g,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, g] U →
+                      ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≂ U ∨
                       ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U.
-#h #g #a #L #V1 #V #T #U #H #V2 #HV12
+#h #g #a #G #L #V1 #V #T #U #H #V2 #HV12
 elim (cpxs_inv_appl1 … H) -H *
-[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1/
+[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or_introl/
 | #b #W #T0 #HT0 #HU
   elim (cpxs_inv_abbr1 … HT0) -HT0 *
   [ #V3 #T3 #_ #_ #H destruct
   | #X #HT2 #H #H0 destruct
     elim (lift_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
     @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
-    @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{b}W2.T2)) [ /3 width=1/ ] -T
-    @(cpxs_strap2 … (ⓐV1.ⓛ{b}W.T0)) [2: /2 width=1/ ]
-    /4 width=7 by cpx_zeta, lift_bind, lift_flat/ (**) (* auto too slow without trace *) 
+    @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{b}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+    @(cpxs_strap2 … (ⓐV1.ⓛ{b}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
+    /4 width=7 by cpx_zeta, lift_bind, lift_flat/
   ]
 | #b #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
   @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
   elim (cpxs_inv_abbr1 … HT0) -HT0 *
   [ #V5 #T5 #HV5 #HT5 #H destruct
-    lapply (cpxs_lift … HV13 (L.ⓓV) … HV12 … HV34) -V1 -V3 /2 width=1/
-    /3 width=1/
+    lapply (cpxs_lift … HV13 (L.ⓓV) … HV12 … HV34) -V1 -V3
+    /3 width=2 by cpxs_flat, cpxs_bind, drop_drop/
   | #X #HT1 #H #H0 destruct
     elim (lift_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
-    lapply (cpxs_lift … HV13 (L.ⓓV0) … HV12 … HV34) -V3 /2 width=1/ #HV24
-    @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{b}V5.T5)) [ /3 width=1/ ] -T
-    @(cpxs_strap2 … (ⓐV1.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V5 -T5 (**) (* auto too slow without trace *)
-    @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2.T0)) [ /3 width=3/ ] -V1 /3 width=1/
+    lapply (cpxs_lift … HV13 (L.ⓓV0) … HV12 … HV34) -V3 /2 width=2 by drop_drop/ #HV24
+    @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{b}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+    @(cpxs_strap2 … (ⓐV1.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V5 -T5
+    @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
   ]
 ]
 qed-.
 
-lemma cpxs_fwd_cast: ∀h,g,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, g] U →
-                     ∨∨ ⓝW. T ≃ U | ⦃G, L⦄ ⊢ T ➡*[h, g] U | ⦃G, L⦄ ⊢ W ➡*[h, g] U.
-#h #g #L #W #T #U #H
-elim (cpxs_inv_cast1 … H) -H /2 width=1/ *
-#W0 #T0 #_ #_ #H destruct /2 width=1/
+lemma cpxs_fwd_cast: ∀h,g,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, g] U →
+                     ∨∨ ⓝW. T ≂ U | ⦃G, L⦄ ⊢ T ➡*[h, g] U | ⦃G, L⦄ ⊢ W ➡*[h, g] U.
+#h #g #G #L #W #T #U #H
+elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
+#W0 #T0 #_ #_ #H destruct /2 width=1 by tstc_pair, or3_intro0/
 qed-.