(**************************************************************************)
include "delayed_updating/syntax/path.ma".
-include "ground/arith/nat_succ.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 ***********************************************************)
[ list_empty โ ๐
| list_lcons l q โ
match l with
- [ label_d _ โ depth q
+ [ label_d k โ depth q
| label_m โ depth q
| label_L โ โ(depth q)
| label_A โ depth q
interpretation
"depth (path)"
- 'VerticalBars p = (depth p).
+ 'Flat p = (depth p).
(* Basic constructions ******************************************************)
-lemma depth_empty: รฐ\9d\9f\8e = รข\9d\98รฐ\9d\90\9eรข\9d\98.
+lemma depth_empty: รฐ\9d\9f\8e = รข\99ยญรฐ\9d\90\9e.
// qed.
-lemma depth_d (q) (n): โqโ = โ๐ฑnโqโ.
+lemma depth_d_dx (p) (k):
+ โญp = โญ(pโ๐ฑk).
// qed.
-lemma depth_m (q): โqโ = โ๐บโqโ.
+lemma depth_m_dx (p):
+ โญp = โญ(pโ๐บ).
// qed.
-lemma depth_L (q): โโqโ = โ๐โqโ.
+lemma depth_L_dx (p):
+ โโญp = โญ(pโ๐).
// qed.
-lemma depth_A (q): โqโ = โ๐โqโ.
+lemma depth_A_dx (p):
+ โญp = โญ(pโ๐).
// qed.
-lemma depth_S (q): โqโ = โ๐ฆโqโ.
+lemma depth_S_dx (p):
+ โญp = โญ(pโ๐ฆ).
+// qed.
+
+(* Main constructions *******************************************************)
+
+theorem depth_append (p) (q):
+ (โญp)+(โญq) = โญ(pโq).
+#p #q elim q -q //
+* [ #k ] #q #IH <list_append_lcons_sn
+[ <depth_d_dx <depth_d_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):
+ โญp = โญ(๐ฑkโp).
+// qed.
+
+lemma depth_m_sn (p):
+ โญp = โญ(๐บโp).
+// qed.
+
+lemma depth_L_sn (p):
+ โโญp = โญ(๐โp).
+// qed.
+
+lemma depth_A_sn (p):
+ โญp = โญ(๐โp).
+// qed.
+
+lemma depth_S_sn (p):
+ โญp = โญ(๐ฆโp).
// qed.