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diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma
index b5ebbe8af..31a8d0b49 100644
--- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma
@@ -65,19 +65,46 @@ lemma cpcs_inv_abst_shift: ∀L,W1,W2,T1,T2. L ⊢ ⓛW1.T1 ⬌* ⓛW2.T2 → 
 lapply (cpcs_inv_abst … H Abst W) -H //
 qed.
 
+lemma cpcs_inv_abst1: ∀L,W1,T1,T. L ⊢ ⓛW1.T1 ⬌* T →
+                      ∃∃W2,T2. L ⊢ T ➡* ⓛW2.T2 & L ⊢ ⓛW1.T1 ➡* ⓛW2.T2.
+#L #W1 #T1 #T #H
+elim (cpcs_inv_cprs … H) -H #X #H1 #H2
+elim (cprs_inv_abst1 Abst W1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
+@(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *)
+qed-.
+
+lemma cpcs_inv_abst2: ∀L,W1,T1,T. L ⊢ T ⬌* ⓛW1.T1 →
+                      ∃∃W2,T2. L ⊢ T ➡* ⓛW2.T2 & L ⊢ ⓛW1.T1 ➡* ⓛW2.T2.
+/3 width=1 by cpcs_inv_abst1, cpcs_sym/ 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 →
                      L ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
 elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
-elim (cprs_inv_lift … HLK … HTU1 … HU1) -U1 #T #HTU #HT1
-elim (cprs_inv_lift … HLK … HTU2 … HU2) -L -U2 #X #HXU
+elim (cprs_inv_lift1 … HLK … HTU1 … HU1) -U1 #T #HTU #HT1
+elim (cprs_inv_lift1 … HLK … HTU2 … HU2) -L -U2 #X #HXU
 >(lift_inj … HXU … HTU) -X -U -d -e /2 width=3/
 qed-.
 
 (* Advanced properties ******************************************************)
 
+lemma cpr_cprs_conf: ∀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/
+qed-.
+
+lemma cprs_cpr_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1.
+#L #T #T1 #T2 #HT1 #HT2
+elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/
+qed-.
+
+lemma cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
+#L #T #T1 #T2 #HT1 #HT2
+elim (cprs_conf … HT1 … HT2) /2 width=3/
+qed-.
+
 (* Basic_1: was only: pc3_thin_dx *)
 lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
                  ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
@@ -86,6 +113,10 @@ elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2
 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *)
 qed.
 
+lemma cpcs_flat_dx_tpr_rev: ∀L,V1,V2. V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
+                            ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
+/3 width=1/ qed.
+
 lemma cpcs_abst: ∀L,V1,V2. L ⊢ V1 ⬌* V2 →
                  ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛV1. T1 ⬌* ⓛV2. T2.
 #L #V1 #V2 #HV12 #V #T1 #T2 #HT12
@@ -108,6 +139,32 @@ lemma cpcs_abbr_sn: ∀L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓓV1. T ⬌* ⓓV2
 elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *)
 qed.
 
+lemma cpcs_bind_sn: ∀I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{I}V1. T ⬌* ⓑ{I}V2. T.
+* /2 width=1/ /2 width=2/ qed.
+
+lemma cpcs_beta_dx: ∀L,V1,V2,W,T1,T2.
+                    L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ⬌* T2 → L ⊢ ⓐV1.ⓛW.T1 ⬌* ⓓV2.T2.
+#L #V1 #V2 #W #T1 #T2 #HV12 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
+lapply (cprs_beta_dx … HV12 HT1) -HV12 -HT1 #HT1
+lapply (cprs_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+@(cprs_div … HT1) /2 width=1/
+qed.
+
+lemma cpcs_beta_dx_tpr_rev: ∀L,V1,V2,W,T1,T2.
+                            V1 ➡ V2 → L.ⓛW ⊢ T2 ⬌* T1 →
+                            L ⊢ ⓓV2.T2 ⬌* ⓐV1.ⓛW.T1.
+/4 width=1/ qed.
+
+(* Note: it does not hold replacing |L1| with |L2| *)
+lemma cpcs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
+                        ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ⬌* T2.
+#L1 #T1 #T2 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, cprs_lsubs_trans/ (**) (* /3 width=5/ is a bit slow *)
+qed.
+
+
 (* Basic_1: was: pc3_lift *)
 lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                  ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →