X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fdelayed_updating%2Fsyntax%2Fpath_depth.ma;h=2e638b8ac43bfba05ccc9783d2045d130d302a9d;hb=797a607af83f82102033270087722a7e59ddcd17;hp=88952c2bc0b4b4edcd60a53c291f1c85a7a4866c;hpb=13584a37bbcde10e03c8a488f5b93e1e042da0a6;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth.ma
index 88952c2bc..2e638b8ac 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth.ma
@@ -13,43 +13,95 @@
 (**************************************************************************)
 
 include "delayed_updating/syntax/path.ma".
-include "ground/notation/functions/verticalbars_1.ma".
+include "delayed_updating/notation/functions/flat_1.ma".
+include "ground/arith/nat_plus.ma".
 
 (* DEPTH FOR PATH ***********************************************************)
 
-rec definition depth (p) on p: pnat ≝
+rec definition depth (p) on p: nat ≝
 match p with
-[ list_empty     ⇒ 𝟏
+[ list_empty     ⇒ 𝟎
 | list_lcons l q ⇒
   match l with
-  [ label_d _ ⇒ depth q
-  | label_m   ⇒ depth q
-  | label_L   ⇒ ↑(depth q)
-  | label_A   ⇒ depth q
-  | label_S   ⇒ depth q
+  [ label_d k    ⇒ depth q
+  | label_d2 k d ⇒ depth q
+  | label_m      ⇒ depth q
+  | label_L      ⇒ ↑(depth q)
+  | label_A      ⇒ depth q
+  | label_S      ⇒ depth q
   ]
 ].
 
 interpretation
   "depth (path)"
-  'VerticalBars p = (depth p).
+  'Flat p = (depth p).
 
 (* Basic constructions ******************************************************)
 
-lemma depth_empty: 𝟏 = ❘𝐞❘.
+lemma depth_empty: 𝟎 = ♭𝐞.
 // qed.
 
-lemma depth_d (q) (n): ❘q❘ = ❘𝗱n◗q❘.
+lemma depth_d_dx (p) (k):
+      ♭p = ♭(p◖𝗱k).
 // qed.
 
-lemma depth_m (q): ❘q❘ = ❘𝗺◗q❘.
+lemma depth_d2_dx (p) (k) (d):
+      ♭p = ♭(p◖𝗱❨k,d❩).
 // qed.
 
-lemma depth_L (q): ↑❘q❘ = ❘𝗟◗q❘.
+lemma depth_m_dx (p):
+      ♭p = ♭(p◖𝗺).
 // qed.
 
-lemma depth_A (q): ❘q❘ = ❘𝗔◗q❘.
+lemma depth_L_dx (p):
+      ↑♭p = ♭(p◖𝗟).
 // qed.
 
-lemma depth_S (q): ❘q❘ = ❘𝗦◗q❘.
+lemma depth_A_dx (p):
+      ♭p = ♭(p◖𝗔).
+// qed.
+
+lemma depth_S_dx (p):
+      ♭p = ♭(p◖𝗦).
+// qed.
+
+(* Main constructions *******************************************************)
+
+theorem depth_append (p) (q):
+        (♭p)+(♭q) = ♭(p●q).
+#p #q elim q -q //
+* [ #k | #k #d ] #q #IH <list_append_lcons_sn
+[ <depth_d_dx <depth_d_dx //
+| <depth_d2_dx <depth_d2_dx //
+| <depth_m_dx <depth_m_dx //
+| <depth_L_dx <depth_L_dx //
+| <depth_A_dx <depth_A_dx //
+| <depth_S_dx <depth_S_dx //
+]
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma depth_d_sn (p) (k):
+      ♭p = ♭(𝗱k◗p).
+// qed.
+
+lemma depth_d2_sn (p) (k) (d):
+      ♭p = ♭(𝗱❨k,d❩◗p).
+// qed.
+
+lemma depth_m_sn (p):
+      ♭p = ♭(𝗺◗p).
+// qed.
+
+lemma depth_L_sn (p):
+      ↑♭p = ♭(𝗟◗p).
+// qed.
+
+lemma depth_A_sn (p):
+      ♭p = ♭(𝗔◗p).
+// qed.
+
+lemma depth_S_sn (p):
+      ♭p = ♭(𝗦◗p).
 // qed.