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diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2_rmap_closed_1.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2_rmap_closed_1.etc
new file mode 100644
index 000000000..df946bf8f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2_rmap_closed_1.etc
@@ -0,0 +1,110 @@
+include "ground/arith/nat_le_plus.ma".
+include "ground/arith/nat_le_pred.ma".
+
+lemma tls_succ_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
+      ▶[f]p ≗ ⇂*[↑n]▶[f](p●𝗟◗q).
+/2 width=2 by tls_succ_plus_unwind2_rmap_push_closed/
+qed-.
+
+lemma xap_le_unwind2_rmap_append_closed_dx (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ → ∀m. m ≤ n →
+      ▶[f]q＠❨m❩ = ▶[f](p●q)＠❨m❩.
+#o #f #p #q #n #Hq elim Hq -q -n
+[|*: #q #n [ #k #_ ] #_ #IH ] #m #Hm
+[ <(nle_inv_zero_dx … Hm) -m //
+| <unwind2_rmap_d_dx <unwind2_rmap_d_dx
+  <tr_compose_xap <tr_compose_xap
+  @IH -IH (**) (* auto too slow *)
+  @nle_trans [| @tr_uni_xap ]
+  /2 width=1 by nle_plus_bi_dx/
+| <unwind2_rmap_m_dx <unwind2_rmap_m_dx
+  /2 width=2 by/
+| <unwind2_rmap_L_dx <unwind2_rmap_L_dx
+  elim (nle_inv_succ_dx … Hm) -Hm // * #Hm #H0
+  >H0 -H0 <tr_xap_push <tr_xap_push
+  /3 width=2 by eq_f/
+| <unwind2_rmap_A_dx <unwind2_rmap_A_dx
+  /2 width=2 by/
+| <unwind2_rmap_S_dx <unwind2_rmap_S_dx
+  /2 width=2 by/
+]
+qed-.
+
+lemma nap_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
+      ▶[f](𝗟◗q)＠§❨n❩ = ▶[f](p●𝗟◗q)＠§❨n❩.
+#o #f #p #q #n #Hq
+lapply (pcc_L_sn … Hq) -Hq #Hq
+lapply (xap_le_unwind2_rmap_append_closed_dx o f p … Hq (↑n) ?) -Hq //
+<tr_xap_succ_nap <tr_xap_succ_nap #Hq
+/2 width=1 by eq_inv_nsucc_bi/
+qed-.
+
+lemma nap_unwind2_rmap_push_closed_depth (o) (f) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
+      ♭q = ▶[⫯f]q＠§❨n❩.
+#o #f #q #n #Hq elim Hq -q -n
+[|*: #q #n [ #k #_ ] #_ #IH ] //
+<unwind2_rmap_d_dx <tr_compose_nap //
+qed-.
+
+lemma nap_unwind2_rmap_append_closed_Lq_dx_depth (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
+      ♭q = ▶[f](p●𝗟◗q)＠§❨n❩.
+#o #f #p #q #n #Hq
+<nap_unwind2_rmap_append_closed_Lq_dx //
+/2 width=2 by nap_unwind2_rmap_push_closed_depth/
+qed-.
+
+lemma xap_unwind2_rmap_append_closed_true_dx_depth (f) (p) (q) (n):
+      q ϵ 𝐂❨Ⓣ,n❩ → ♭q = ▶[f](p●q)＠❨n❩.
+#f #p #q #n #Hq elim Hq -q -n //
+#q #n #k #Ho #_ #IH
+<unwind2_rmap_d_dx <tr_compose_xap
+>Ho // <tr_uni_xap_succ <Ho //
+qed-.
+
+lemma tls_plus_unwind2_rmap_closed_true (f) (q) (n):
+      q ϵ 𝐂❨Ⓣ,n❩ →
+      ∀m. ⇂*[m]f ≗ ⇂*[m+n]▶[f]q.
+#f #q #n #Hq elim Hq -q -n //
+#q #n #k #Ho #_ #IH #m
+>Ho // <nplus_succ_dx
+@(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
+>nrplus_inj_dx >nrplus_inj_sn >nsucc_unfold
+>nplus_succ_dx <Ho //
+qed-.
+
+lemma tls_unwind2_rmap_append_closed_true_dx (f) (p) (q) (n):
+      q ϵ 𝐂❨Ⓣ,n❩ →
+      ▶[f]p ≗ ⇂*[n]▶[f](p●q).
+/2 width=1 by tls_plus_unwind2_rmap_closed_true/
+qed-.
+
+lemma nap_plus_unwind2_rmap_append_closed_Lq_dx_depth (o) (f) (p) (q) (m) (n):
+      q ϵ 𝐂❨o,n❩ →
+      ▶[f]p＠❨m❩+♭q = ▶[f](p●𝗟◗q)＠§❨m+n❩.
+#o #f #p #q #m #n #Hq
+<tr_nap_plus @eq_f2
+[ <(tr_xap_eq_repl … (tls_succ_unwind2_rmap_append_closed_Lq_dx …)) //
+| /2 width=2 by nap_unwind2_rmap_append_closed_Lq_dx_depth/
+]
+qed-.
+
+lemma nap_plus_unwind2_rmap_append_closed_bLq_dx_depth (o) (f) (p) (b) (q) (m) (n):
+      b ϵ 𝐂❨Ⓣ,m❩ → q ϵ 𝐂❨o,n❩ →
+      ♭b+♭q = ▶[f](p●b●𝗟◗q)＠§❨m+n❩.
+#o #f #p #b #q #m #n #Hb #Hq
+>(xap_unwind2_rmap_append_closed_true_dx_depth f p … Hb) -Hb
+>(nap_plus_unwind2_rmap_append_closed_Lq_dx_depth … Hq) -Hq //
+qed-.
+
+lemma tls_succ_plus_unwind2_rmap_append_closed_bLq_dx (o) (f) (p) (b) (q) (m) (n):
+      b ϵ 𝐂❨Ⓣ,m❩ → q ϵ 𝐂❨o,n❩ →
+      ▶[f]p ≗ ⇂*[↑(m+n)]▶[f](p●b●𝗟◗q).
+#o #f #p #b #q #m #n #Hb #Hq
+>nplus_succ_dx <stream_tls_plus >list_append_assoc
+@(stream_eq_trans … (tls_unwind2_rmap_append_closed_true_dx … Hb)) -Hb
+/3 width=2 by stream_tls_eq_repl, tls_succ_unwind2_rmap_append_closed_Lq_dx/
+qed-.