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diff --git a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma
index fd3034456..1b0c88e02 100644
--- a/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/grammar/lenv_px.ma
@@ -18,59 +18,53 @@ include "basic_2/grammar/lenv_length.ma".
 
 inductive lpx (R:relation term): relation lenv ≝
 | lpx_stom: lpx R (⋆) (⋆)
-| lpx_pair: ∀K1,K2,I,V1,V2.
+| lpx_pair: ∀I,K1,K2,V1,V2.
             lpx R K1 K2 → R V1 V2 → lpx R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
 .
 
-(* Basic properties *********************************************************)
-
-lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R).
-#R #HR #L elim L -L // /2 width=1/
-qed.
-
 (* Basic inversion lemmas ***************************************************)
 
 fact lpx_inv_atom1_aux: ∀R,L1,L2. lpx R L1 L2 → L1 = ⋆ → L2 = ⋆.
 #R #L1 #L2 * -L1 -L2
 [ //
-| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
 ]
 qed-.
 
 lemma lpx_inv_atom1: ∀R,L2. lpx R (⋆) L2 → L2 = ⋆.
 /2 width=4 by lpx_inv_atom1_aux/ qed-.
 
-fact lpx_inv_pair1_aux: ∀R,L1,L2. lpx R L1 L2 → ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+fact lpx_inv_pair1_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
                         ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
 #R #L1 #L2 * -L1 -L2
-[ #K1 #I #V1 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct /2 width=5/
+[ #J #K1 #V1 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #L #W #H destruct /2 width=5/
 ]
 qed-.
 
-lemma lpx_inv_pair1: ∀R,K1,I,V1,L2. lpx R (K1. ⓑ{I} V1) L2 →
+lemma lpx_inv_pair1: ∀R,I,K1,V1,L2. lpx R (K1. ⓑ{I} V1) L2 →
                      ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
 /2 width=3 by lpx_inv_pair1_aux/ qed-.
 
 fact lpx_inv_atom2_aux: ∀R,L1,L2. lpx R L1 L2 → L2 = ⋆ → L1 = ⋆.
 #R #L1 #L2 * -L1 -L2
 [ //
-| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
+| #I #K1 #K2 #V1 #V2 #_ #_ #H destruct
 ]
 qed-.
 
 lemma lpx_inv_atom2: ∀R,L1. lpx R L1 (⋆) → L1 = ⋆.
 /2 width=4 by lpx_inv_atom2_aux/ qed-.
 
-fact lpx_inv_pair2_aux: ∀R,L1,L2. lpx R L1 L2 → ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+fact lpx_inv_pair2_aux: ∀R,L1,L2. lpx R L1 L2 → ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
                         ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
 #R #L1 #L2 * -L1 -L2
-[ #K2 #I #V2 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct /2 width=5/
+[ #J #K2 #V2 #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #J #K #W #H destruct /2 width=5/
 ]
 qed-.
 
-lemma lpx_inv_pair2: ∀R,L1,K2,I,V2. lpx R L1 (K2. ⓑ{I} V2) →
+lemma lpx_inv_pair2: ∀R,I,L1,K2,V2. lpx R L1 (K2. ⓑ{I} V2) →
                      ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
 /2 width=3 by lpx_inv_pair2_aux/ qed-.
 
@@ -79,3 +73,59 @@ lemma lpx_inv_pair2: ∀R,L1,K2,I,V2. lpx R L1 (K2. ⓑ{I} V2) →
 lemma lpx_fwd_length: ∀R,L1,L2. lpx R L1 L2 → |L1| = |L2|.
 #R #L1 #L2 #H elim H -L1 -L2 normalize //
 qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R).
+#R #HR #L elim L -L // /2 width=1/
+qed.
+
+lemma lpx_trans: ∀R. Transitive ? R → Transitive … (lpx R).
+#R #HR #L1 #L #H elim H -L //
+#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
+elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
+qed.
+
+lemma lpx_conf: ∀R. Confluent ? R → Confluent … (lpx R).
+#R #HR #L0 #L1 #H elim H -L1
+[ #X #H >(lpx_inv_atom1 … H) -X /2 width=3/
+| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H
+  elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
+  elim (IHK01 … HK02) -K0 #K #HK1 #HK2
+  elim (HR … HV01 … HV02) -HR -V0 /3 width=5/
+]
+qed.
+
+lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2. 
+#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+qed.
+
+lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L →
+                   ∀L2. lpx R L L2 → lpx (TC … R) L1 L2.
+#R #L1 #L #H elim H -L /2 width=1/
+#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
+elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
+qed.
+
+lemma TC_lpx_pair_dx: ∀R. reflexive ? R →
+                      ∀I,K,V1,V2. TC … R V1 V2 →
+                      TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2).
+#R #HR #I #K #V1 #V2 #H elim H -V2
+/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
+qed.
+
+lemma TC_lpx_pair_sn: ∀R. reflexive ? R →
+                      ∀I,V,K1,K2. TC … (lpx R) K1 K2 →
+                      TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V).
+#R #HR #I #V #K1 #K2 #H elim H -K2
+/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
+qed.
+
+lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2. 
+#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/
+qed.
+
+lemma lpx_inv_TC: ∀R. reflexive ? R →
+                  ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2 /2 width=1/ /3 width=3/
+qed.