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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma
index 299b0f8c8..381233f04 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpr.ma
@@ -34,8 +34,8 @@ lemma lfpr_sort: ∀h,I,G,L1,L2,V1,V2,s.
                  ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2.
 /2 width=1 by lfxs_sort/ qed.
 
-lemma lfpr_zero: ∀h,I,G,L1,L2,V.
-                 ⦃G, L1⦄ ⊢ ➡[h, V] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V.
+lemma lfpr_zero: ∀h,I,G,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 →
+                 ⦃G, L1⦄ ⊢ V1 ➡[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #0] L2.ⓑ{I}V2.
 /2 width=1 by lfxs_zero/ qed.
 
 lemma lfpr_lref: ∀h,I,G,L1,L2,V1,V2,i.
@@ -60,6 +60,12 @@ lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 =
 lemma lfpr_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ➡[h, ⓪{I}] ⋆ → Y1 = ⋆.
 /2 width=3 by lfxs_inv_atom_dx/ qed-.
 
+lemma lfpr_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] Y2 →
+                     (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+                     ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 &
+                                      Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_sort/ qed-.
+
 lemma lfpr_inv_zero: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ➡[h, #0] Y2 →
                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
                      ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 &
@@ -73,6 +79,12 @@ lemma lfpr_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ➡[h, #⫯i] Y2 →
                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 /2 width=1 by lfxs_inv_lref/ qed-.
 
+lemma lfpr_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ➡[h, §l] Y2 →
+                     (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+                     ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 &
+                                      Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by lfxs_inv_gref/ qed-.
+
 lemma lfpr_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 →
                      ⦃G, L1⦄ ⊢ ➡[h, V] L2 ∧ ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V.
 /2 width=2 by lfxs_inv_bind/ qed-.
@@ -83,6 +95,14 @@ lemma lfpr_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 
 
 (* Advanced inversion lemmas ************************************************)
 
+lemma lfpr_inv_sort_pair_sn: ∀h,I,G,Y2,L1,V1,s. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] Y2 →
+                             ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_sort_pair_sn/ qed-.
+
+lemma lfpr_inv_sort_pair_dx: ∀h,I,G,Y1,L2,V2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2 →
+                             ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_sort_pair_dx/ qed-.
+
 lemma lfpr_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, #0] Y2 →
                              ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 &
                                       Y2 = L2.ⓑ{I}V2.
@@ -101,6 +121,14 @@ lemma lfpr_inv_lref_pair_dx: ∀h,I,G,Y1,L2,V2,i. ⦃G, Y1⦄ ⊢ ➡[h, #⫯i]
                              ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 & Y1 = L1.ⓑ{I}V1.
 /2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
 
+lemma lfpr_inv_gref_pair_sn: ∀h,I,G,Y2,L1,V1,l. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, §l] Y2 →
+                             ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_gref_pair_sn/ qed-.
+
+lemma lfpr_inv_gref_pair_dx: ∀h,I,G,Y1,L2,V2,l. ⦃G, Y1⦄ ⊢ ➡[h, §l] L2.ⓑ{I}V2 →
+                             ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, §l] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_gref_pair_dx/ qed-.
+
 (* Basic forward lemmas *****************************************************)
 
 lemma lfpr_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.