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diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma
index d8981613d..cb2f97f55 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma
@@ -56,5 +56,37 @@ lemma cnv_fwd_cpms_total (a) (h) (n) (G) (L):
       ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
 #a #h #n #G #L #T #H
 elim (cnv_fwd_aaa … H) -H #A #HA
-/2 width=2 by aaa_cpms_total/
+/2 width=2 by cpms_total_aaa/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cnv_inv_appl_SO (a) (h) (G) (L):
+      ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
+      ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
+                   ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+* #h #G #L #V #T #H
+elim (cnv_inv_appl … H) -H [ * [| #n ] | #n ] #p #W #U #Ha #HV #HT #HVW #HTU
+[ elim (cnv_fwd_aaa … HT) #A #HA
+  elim (aaa_cpm_SO h … (ⓛ{p}W.U))
+  [|*: /2 width=8 by cpms_aaa_conf/ ] #X #HU0
+  elim (cpm_inv_abst1 … HU0) #W0 #U0 #HW0 #_ #H0 destruct
+  lapply (cpms_step_dx … HVW … HW0) -HVW -HW0 #HVW0
+  lapply (cpms_step_dx … HTU … HU0) -HTU -HU0 #HTU0
+  /2 width=7 by ex5_4_intro/
+| lapply (Ha ?) -Ha [ // ] #Ha
+  lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Ha #H destruct
+  /2 width=7 by ex5_4_intro/
+| @(ex5_4_intro … HV HT HVW HTU) #H destruct
+]
+qed-.
+
+lemma cnv_inv_appl_true (h) (G) (L):
+      ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h] →
+      ∃∃p,W0,U0. ⦃G,L⦄ ⊢ V ![h] & ⦃G,L⦄ ⊢ T ![h] &
+                   ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[1,h] ⓛ{p}W0.U0.
+#h #G #L #V #T #H
+elim (cnv_inv_appl_SO … H) -H #n #p #W #U #Hn
+>Hn -n [| // ] #HV #HT #HVW #HTU
+/2 width=5 by ex4_3_intro/
 qed-.