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diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma
index e7c4a6c61..6055f11e3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma
@@ -19,7 +19,23 @@ include "basic_2/dynamic/cnv_aaa.ma".
 
 (* Forward lemmas with strongly rst-normalizing closures ********************)
 
+(* Note: this is the "big tree" theorem *)
 (* Basic_2A1: uses: snv_fwd_fsb *)
-lemma cnv_fwd_fsb (a) (h) (o): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ≥[h, o] 𝐒⦃G, L, T⦄.
-#a #h #o #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
+lemma cnv_fwd_fsb (h) (a):
+      ∀G,L,T. ❪G,L❫ ⊢ T ![h,a] → ≥𝐒 ❪G,L,T❫.
+#h #a #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
 qed-.
+
+(* Forward lemmas with strongly rt-normalizing terms ************************)
+
+lemma cnv_fwd_csx (h) (a):
+      ∀G,L,T. ❪G,L❫ ⊢ T ![h,a] → ❪G,L❫ ⊢ ⬈*𝐒 T.
+#h #a #G #L #T #H
+/3 width=3 by cnv_fwd_fsb, fsb_inv_csx/
+qed-.
+
+(* Inversion lemmas with proper parallel rst-computation for closures *******)
+
+lemma cnv_fpbg_refl_false (h) (a):
+      ∀G,L,T. ❪G,L❫ ⊢ T ![h,a] → ❪G,L,T❫ > ❪G,L,T❫ → ⊥.
+/3 width=7 by cnv_fwd_fsb, fsb_fpbg_refl_false/ qed-.