update in delayed_updating

[helm.git] / matita / matita / contribs / lambdadelta / delayed_updating / unwind / unwind2_rmap_head.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
index 579cbd514cf20c90426d0d9dcb320a2e474e9833..6b159e8c560087e4bb20415c84fe708b56a7f0ee 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
@@ -13,26 +13,67 @@
 (**************************************************************************)
 
 include "delayed_updating/unwind/unwind2_rmap_labels.ma".
-include "delayed_updating/unwind/nap.ma".
+include "delayed_updating/unwind/xap.ma".
 include "delayed_updating/syntax/path_head_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
+include "ground/arith/nat_le_plus.ma".
 
 (* UNWIND MAP FOR PATH ******************************************************)
 
 (* Constructions with path_head *********************************************)
 
-lemma unwind2_rmap_head_push (f) (p) (n):
-      n + ♯(↳[n]p) = (▶[⫯f]↳[n]p)@↑❨n❩.
+lemma unwind2_rmap_head_xap_closed (f) (p) (n) (k):
+      (∃q. p = (↳[n]p)●q) → k ≤ n →
+      (▶[f]p)＠❨k❩ = (▶[f]↳[n]p)＠❨k❩.
+#f #p elim p -p
+[ #n #k * #q #Hq #Hk
+  elim (eq_inv_list_empty_append … Hq) -Hq * #_ //
+| #l #p #IH #n @(nat_ind_succ … n) -n
+  [ #k #_ #Hk <(nle_inv_zero_dx … Hk) -k -IH
+   <path_head_zero <unwind2_rmap_empty //
+  | #n #_ #k cases l [ #m ] * #q
+    [ <path_head_d_sn <list_append_lcons_sn #Hq #Hk
+      elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+      <unwind2_rmap_d_sn <unwind2_rmap_d_sn
+      <tr_compose_xap <tr_compose_xap
+      @IH [ /2 width=2 by ex_intro/ ] (**) (* auto too slow *)
+      @nle_trans [| @tr_uni_xap ]
+      /2 width=1 by nle_plus_bi_dx/
+    | <path_head_m_sn <list_append_lcons_sn #Hq #Hk
+      elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+      <unwind2_rmap_m_sn <unwind2_rmap_m_sn
+      /3 width=2 by ex_intro/
+    | <path_head_L_sn <list_append_lcons_sn #Hq
+      @(nat_ind_succ … k) -k // #k #_ #Hk
+      lapply (nle_inv_succ_bi … Hk) -Hk #Hk
+      elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+      <unwind2_rmap_L_sn <unwind2_rmap_L_sn
+      <tr_xap_push <tr_xap_push
+      /4 width=2 by ex_intro, eq_f/
+    | <path_head_A_sn <list_append_lcons_sn #Hq #Hk
+      elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+      <unwind2_rmap_A_sn <unwind2_rmap_A_sn
+      /3 width=2 by ex_intro/
+    | <path_head_S_sn <list_append_lcons_sn #Hq #Hk
+      elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+      <unwind2_rmap_S_sn <unwind2_rmap_S_sn
+      /3 width=2 by ex_intro/
+    ]
+  ]
+]
+qed-.
+
+lemma unwind2_rmap_head_xap (f) (p) (n):
+      n + ♯(↳[n]p) = (▶[f]↳[n]p)＠❨n❩.
 #f #p elim p -p
 [ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
-  >tr_pushs_swap <tr_nap_pushs_lt //
+  <tr_xap_pushs_le //
 | #l #p #IH #n @(nat_ind_succ … n) -n //
   #n #_ cases l [ #m ]
   [ <unwind2_rmap_d_sn <path_head_d_sn <height_d_sn
-    <nplus_assoc >IH -IH <tr_compose_nap //
+    <nplus_assoc >IH -IH <tr_compose_xap <tr_uni_xap_succ //
   | <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
   | <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
-    <tr_nap_push <npred_succ //
+    <tr_xap_push <npred_succ //
   | <unwind2_rmap_A_sn <path_head_A_sn <height_A_sn //
   | <unwind2_rmap_S_sn <path_head_S_sn <height_S_sn //
   ]