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diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
index 784c75470..7d60ed1b9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/computation/cprs_cprs.ma".
+include "basic_2/computation/lprs_cprs.ma".
 include "basic_2/conversion/cpc_cpc.ma".
 include "basic_2/equivalence/cpcs_cprs.ma".
 
@@ -40,35 +40,11 @@ elim (cpcs_inv_cprs … H) -H #T #H1
 lapply (cprs_inv_sort1 … H2) -L #H destruct //
 qed-.
 
-(* Basic_1: was: pc3_gen_sort_abst *)
-lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. L ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
-#a #L #W #T #k #H
-elim (cpcs_inv_cprs … H) -H #X #H1
->(cprs_inv_sort1 … H1) -X #H2
-elim (cprs_fwd_abst1 … H2 Abst W) -H2 #W0 #T0 #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: pc3_gen_abst *)
-lemma cpcs_inv_abst: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → ∀I,V.
-                     ∧∧ L ⊢ W1 ⬌* W2 & L. ②{I}V ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #L #W1 #W2 #T1 #T2 #H #I #V
-elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (cprs_fwd_abst1 … H1 I V) -H1 #W0 #T0 #HW10 #HT10 #H destruct
-elim (cprs_fwd_abst1… H2 I V) -H2 #W #T #HW2 #HT2 #H destruct /3 width=3/
-qed-.
-
-(* Basic_1: was: pc3_gen_abst_shift *)
-lemma cpcs_inv_abst_shift: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 → ∀W.
-                           ∧∧ L ⊢ W1 ⬌* W2 & L. ⓛW ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #L #W1 #W2 #T1 #T2 #H #W
-lapply (cpcs_inv_abst … H Abst W) -H //
-qed.
-
 lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. L ⊢ ⓛ{a}W1.T1 ⬌* T →
                       ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 #a #L #W1 #T1 #T #H
 elim (cpcs_inv_cprs … H) -H #X #H1 #H2
-elim (cprs_fwd_abst1 … H1 Abst W1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
+elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
 @(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *)
 qed-.
 
@@ -76,6 +52,14 @@ lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. L ⊢ T ⬌* ⓛ{a}W1.T1 →
                       ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 /3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
 
+(* Basic_1: was: pc3_gen_sort_abst *)
+lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. L ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
+#a #L #W #T #k #H
+elim (cpcs_inv_cprs … H) -H #X #H1
+>(cprs_inv_sort1 … H1) -X #H2
+elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
+qed-.
+
 (* Basic_1: was: pc3_gen_lift *)
 lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                      ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
@@ -96,6 +80,13 @@ lapply (lpr_cprs_trans … HT1 … HL12) -HT1
 lapply (lpr_cprs_trans … HT2 … HL12) -L2 /2 width=3/
 qed-.
 
+lemma lprs_cpcs_trans: ∀L1,L2. L1 ⊢ ➡* L2 → ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
+#L1 #L2 #HL12 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
+lapply (lprs_cprs_trans … HT1 … HL12) -HT1
+lapply (lprs_cprs_trans … HT2 … HL12) -L2 /2 width=3/
+qed-.
+
 lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/
@@ -111,12 +102,20 @@ lemma cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L 
 elim (cprs_conf … HT1 … HT2) /2 width=3/
 qed-.
 
+lemma lprs_cprs_conf: ∀L1,L2. L1 ⊢ ➡* L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
+#L1 #L2 #HL12 #T1 #T2 #HT12
+elim (lprs_cprs_conf_dx … HT12 … HL12) -L1 /2 width=3/
+qed-.
+
+(* Basic_1: was: pc3_wcpr0_t *)
+(* Basic_1: note: pc3_wcpr0_t should be renamed *)
+lemma lpr_cprs_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
+/3 width=5 by lprs_cprs_conf, lpr_lprs/ qed-.
+
 (* Basic_1: was only: pc3_pr0_pr2_t *)
 (* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
 lemma lpr_cpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #HT12
-elim (lpr_cpr_conf_dx … HT12 … HL12) -L1 /3 width=3/
-qed-.
+/3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
 
 (* Basic_1: was only: pc3_thin_dx *)
 lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
@@ -130,14 +129,6 @@ lemma cpcs_flat_dx_cpr_rev: ∀L,V1,V2. L ⊢ V2 ➡ V1 → ∀T1,T2. L ⊢ T1 
                             ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
 /3 width=1/ qed.
 
-lemma cpcs_ext_bind: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 →
-                     ∀a,I. L ⊢ ⓑ{a,I}V1.T1 ⬌* ⓑ{a,I}V2.T2.
-#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I
-elim (cpcs_inv_cprs … HV12) -HV12
-elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=6 by cprs_div, cprs_ext_bind/ (**) (* /3 width=6/ is a bit slow *)
-qed.
-
 lemma cpcs_bind_dx: ∀a,I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 →
                     L ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2.
 #a #I #L #V #T1 #T2 #HT12
@@ -149,25 +140,11 @@ lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{a,I}V1. T 
 elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
 qed.
 
-lemma cpcs_beta_dx_cpr: ∀a,L,V1,V2,W,T1,T2.
-                        L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ⬌* T2 → L ⊢ ⓐV1.ⓛ{a}W.T1 ⬌* ⓓ{a}V2.T2.
-#a #L #V1 #V2 #W #T1 #T2 #HV12 #HT12
-elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
-lapply (cprs_beta_dx a … HV12 HT1) -HV12 -HT1 #HT1
-lapply (cprs_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
-@(cprs_div … HT1) /2 width=1/
-qed.
-
-lemma cpcs_beta_dx_rev_cpr: ∀a,L,V1,V2,W,T1,T2.
-                            L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T2 ⬌* T1 →
-                            L ⊢ ⓓ{a}V2.T2 ⬌* ⓐV1.ⓛ{a}W.T1.
-/4 width=1/ qed.
-
-lemma cpcs_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
-                        ∀L2. L2 ⊑ L1 → L2 ⊢ T1 ⬌* T2.
+lemma lsubx_cpcs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
+                        ∀L2. L2 ⓝ⊑ L1 → L2 ⊢ T1 ⬌* T2.
 #L1 #T1 #T2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_div, cprs_lsubr_trans/ (**) (* /3 width=5/ is a bit slow *)
+/3 width=5 by cprs_div, lsubx_cprs_trans/ (**) (* /3 width=5/ is a bit slow *)
 qed-.
 
 (* Basic_1: was: pc3_lift *)
@@ -185,6 +162,26 @@ lemma cpcs_strip: ∀L,T1,T. L ⊢ T ⬌* T1 → ∀T2. L ⊢ T ⬌ T2 →
                   ∃∃T0. L ⊢ T1 ⬌ T0 & L ⊢ T2 ⬌* T0.
 #L #T1 #T @TC_strip1 /2 width=3/ qed-.
 
+(* More inversion lemmas ****************************************************)
+
+lemma cpcs_inv_abst_sn: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ L ⊢ W1 ⬌* W2 & L.ⓛW1 ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #L #W1 #W2 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
+elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
+lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1/ -HT2 #HT2
+lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1/ -HT2 #HT2
+/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
+qed-.
+
+lemma cpcs_inv_abst_dx: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ L ⊢ W1 ⬌* W2 & L. ⓛW2 ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #L #W1 #W2 #T1 #T2 #HT12
+lapply (cpcs_sym … HT12) -HT12 #HT12
+elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1/
+qed-.
+
 (* Main properties **********************************************************)
 
 (* Basic_1: was pc3_t *)
@@ -209,15 +206,23 @@ lemma cpcs_bind2: ∀a,I,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V2 ⊢
 @(cpcs_trans … (ⓑ{a,I}V2.T1)) /2 width=1/
 qed.
 
-(* Basic_1: was: pc3_wcpr0_t *)
-(* Basic_1: note: pc3_wcpr0_t should be renamed *)
-lemma cprs_lpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
-/3 width=5 by cpcs_trans, lpr_cpr_conf/
-qed-.
+lemma cpcs_beta_dx: ∀a,L,V1,V2,W1,W2,T1,T2.
+                    L ⊢ V1 ⬌* V2 → L ⊢ W1 ⬌* W2 → L.ⓛW2 ⊢ T1 ⬌* T2 →
+                    L ⊢ ⓐV1.ⓛ{a}W1.T1 ⬌* ⓓ{a}ⓝW2.V2.T2.
+#a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12
+@(cpcs_cpr_strap1 … (ⓐV2.ⓛ{a}W2.T2)) /2 width=1/ /3 width=1/
+qed.
+
+lemma cpcs_beta_sn: ∀a,L,V1,V2,W1,W2,T1,T2.
+                    L ⊢ V1 ⬌* V2 → L ⊢ W1 ⬌* W2 → L.ⓛW1 ⊢ T1 ⬌* T2 →
+                    L ⊢ ⓐV1.ⓛ{a}W1.T1 ⬌* ⓓ{a}ⓝW2.V2.T2.
+#a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12
+lapply (lsubx_cpcs_trans … HT12 (L.ⓓⓝW1.V1) ?) /2 width=1/ #H2T12
+@(cpcs_cpr_strap2 … (ⓓ{a}ⓝW1.V1.T1)) /2 width=1/ -HT12 /3 width=1/
+qed.
 
 (* Basic_1: was: pc3_wcpr0 *)
-lemma cpcs_lpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
+lemma lpr_cpcs_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
 #L1 #L2 #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, cprs_lpr_conf/
+elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
 qed-.