X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fdelayed_updating%2Fsubstitution%2Flift.ma;h=bfb66f1bc911e23f4b6a2b445f62fe236f6552c4;hb=e0c91d8a4422da0b39aca790e5826dc8a617b303;hp=0b3dc7d5c3b7d185b153f5e420d73bb94a5a09c4;hpb=503500ff9a6d9cca363a42b5fe7f3f5de69239f9;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
index 0b3dc7d5c..bfb66f1bc 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
@@ -20,63 +20,164 @@ include "delayed_updating/notation/functions/uparrow_2.ma".
 
 (* LIFT FOR PATH ***********************************************************)
 
+definition lift_continuation (A:Type[0]) ≝
+           tr_map → path → A.
+
 (* Note: inner numeric labels are not liftable, so they are removed *)
-rec definition lift_gen (A:Type[0]) (k:?→?→A) (p) (f) on p ≝
+rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
 match p with
-[ list_empty     ⇒ k 𝐞 f
+[ list_empty     ⇒ k f (𝐞)
 | list_lcons l q ⇒
   match l with
-  [ label_node_d n ⇒
+  [ label_d n ⇒
     match q with
-    [ list_empty     ⇒ lift_gen (A) (λp. k (𝗱❨f@❨n❩❩◗p)) q f
-    | list_lcons _ _ ⇒ lift_gen (A) k q (f∘𝐮❨n❩)
+    [ list_empty     ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (f∘𝐮❨n❩) q
+    | list_lcons _ _ ⇒ lift_gen (A) k (f∘𝐮❨n❩) q
     ]
-  | label_edge_L   ⇒ lift_gen (A) (λp. k (𝗟◗p)) q (⫯f)
-  | label_edge_A   ⇒ lift_gen (A) (λp. k (𝗔◗p)) q f
-  | label_edge_S   ⇒ lift_gen (A) (λp. k (𝗦◗p)) q f
+  | label_m   ⇒ lift_gen (A) k f q
+  | label_L   ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
+  | label_A   ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
+  | label_S   ⇒ lift_gen (A) (λg,p. k g (𝗦◗p)) f q
   ]
 ].
 
 interpretation
   "lift (gneric)"
-  'UpArrow A k p f = (lift_gen A k p f).
+  'UpArrow A k f p = (lift_gen A k f p).
 
-definition proj_path (p:path) (f:tr_map) ≝ p.
+definition proj_path: lift_continuation … ≝
+           λf,p.p.
 
-definition proj_rmap (p:path) (f:tr_map) ≝ f.
+definition proj_rmap: lift_continuation … ≝
+           λf,p.f.
 
 interpretation
   "lift (path)"
-  'UpArrow f p = (lift_gen ? proj_path p f).
+  'UpArrow f p = (lift_gen ? proj_path f p).
 
 interpretation
   "lift (relocation map)"
-  'UpArrow p f = (lift_gen ? proj_rmap p f).
+  'UpArrow p f = (lift_gen ? proj_rmap f p).
 
 (* Basic constructions ******************************************************)
 
-lemma lift_L (A) (k) (p) (f):
-      ↑❨(λp. k (𝗟◗p)), p, ⫯f❩ = ↑{A}❨k, 𝗟◗p, f❩.
+lemma lift_empty (A) (k) (f):
+      k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
+// qed.
+
+lemma lift_d_empty_sn (A) (k) (n) (f):
+      ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), f∘𝐮❨ninj n❩, 𝐞❩ = ↑{A}❨k, f, 𝗱n◗𝐞❩.
+// qed.
+
+lemma lift_d_lcons_sn (A) (k) (p) (l) (n) (f):
+      ↑❨k, f∘𝐮❨ninj n❩, l◗p❩ = ↑{A}❨k, f, 𝗱n◗l◗p❩.
+// qed.
+
+lemma lift_m_sn (A) (k) (p) (f):
+      ↑❨k, f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
+// qed.
+
+lemma lift_L_sn (A) (k) (p) (f):
+      ↑❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ↑{A}❨k, f, 𝗟◗p❩.
+// qed.
+
+lemma lift_A_sn (A) (k) (p) (f):
+      ↑❨(λg,p. k g (𝗔◗p)), f, p❩ = ↑{A}❨k, f, 𝗔◗p❩.
+// qed.
+
+lemma lift_S_sn (A) (k) (p) (f):
+      ↑❨(λg,p. k g (𝗦◗p)), f, p❩ = ↑{A}❨k, f, 𝗦◗p❩.
 // qed.
 
 (* Basic constructions with proj_path ***************************************)
 
-lemma lift_append (p) (f) (q):
-      q●↑[f]p = ↑❨(λp. proj_path (q●p)), p, f❩.
-#p elim p -p
-[ //
-| #l #p #IH #f #q cases l
-  [
-  | <lift_L in ⊢ (???%);
-    >(list_append_rcons_sn ? q) in ⊢ (???(??(λ_.%)??));
-    
-     <IH 
-  normalize >IH
-  | //   
-
-(* Constructions with append ************************************************)
-
-theorem lift_append_A (p2) (p1) (f):
-        (↑[f]p1)●𝗔◗↑[↑[p1]f]p2 = ↑[f](p1●𝗔◗p2).
-#p2 #p1 elim p1 -p1
-[ #f normalize 
+lemma lift_path_empty (f):
+      (𝐞) = ↑[f]𝐞.
+// qed.
+
+lemma lift_path_d_empty_sn (f) (n):
+      𝗱(f@❨n❩)◗𝐞 = ↑[f](𝗱n◗𝐞).
+// qed.
+
+lemma lift_path_d_lcons_sn (f) (p) (l) (n):
+      ↑[f∘𝐮❨ninj n❩](l◗p) = ↑[f](𝗱n◗l◗p).
+// qed.
+
+lemma lift_path_m_sn (f) (p):
+      ↑[f]p = ↑[f](𝗺◗p).
+// qed.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma lift_rmap_empty (f):
+      f = ↑[𝐞]f.
+// qed.
+
+lemma lift_rmap_d_sn (f) (p) (n):
+      ↑[p](f∘𝐮❨ninj n❩) = ↑[𝗱n◗p]f.
+#f * // qed.
+
+lemma lift_rmap_m_sn (f) (p):
+      ↑[p]f = ↑[𝗺◗p]f.
+// qed.
+
+lemma lift_rmap_L_sn (f) (p):
+      ↑[p](⫯f) = ↑[𝗟◗p]f.
+// qed.
+
+lemma lift_rmap_A_sn (f) (p):
+      ↑[p]f = ↑[𝗔◗p]f.
+// qed.
+
+lemma lift_rmap_S_sn (f) (p):
+      ↑[p]f = ↑[𝗦◗p]f.
+// qed.
+
+(* Advanced constructions with proj_rmap and path_append ********************)
+
+lemma lift_rmap_append (p2) (p1) (f):
+      ↑[p2]↑[p1]f = ↑[p1●p2]f.
+#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
+[ <lift_rmap_m_sn <lift_rmap_m_sn //
+| <lift_rmap_A_sn <lift_rmap_A_sn //
+| <lift_rmap_S_sn <lift_rmap_S_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma lift_rmap_d_dx (f) (p) (n):
+      (↑[p]f)∘𝐮❨ninj n❩ = ↑[p◖𝗱n]f.
+// qed.
+
+lemma lift_rmap_m_dx (f) (p):
+      ↑[p]f = ↑[p◖𝗺]f.
+// qed.
+
+lemma lift_rmap_L_dx (f) (p):
+      (⫯↑[p]f) = ↑[p◖𝗟]f.
+// qed.
+
+lemma lift_rmap_A_dx (f) (p):
+      ↑[p]f = ↑[p◖𝗔]f.
+// qed.
+
+lemma lift_rmap_S_dx (f) (p):
+      ↑[p]f = ↑[p◖𝗦]f.
+// qed.
+
+(* Advanced eliminations with path ******************************************)
+
+lemma path_ind_lift (Q:predicate …):
+      Q (𝐞) →
+      (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
+      (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
+      (∀p. Q p → Q (𝗺◗p)) →
+      (∀p. Q p → Q (𝗟◗p)) →
+      (∀p. Q p → Q (𝗔◗p)) →
+      (∀p. Q p → Q (𝗦◗p)) →
+      ∀p. Q p.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
+elim p -p [| * [ #n * ] ]
+/2 width=1 by/
+qed-.