(* *)
(**************************************************************************)
-include "ground/relocation/tr_compose.ma".
-include "ground/relocation/tr_uni.ma".
-include "delayed_updating/syntax/path.ma".
include "delayed_updating/notation/functions/uparrow_4.ma".
include "delayed_updating/notation/functions/uparrow_2.ma".
+include "delayed_updating/syntax/path.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/relocation/tr_pap_tls.ma".
(* LIFT FOR PATH ***********************************************************)
definition lift_continuation (A:Type[0]) ≝
- path → tr_map → A.
+ tr_map → path → A.
-(* Note: inner numeric labels are not liftable, so they are removed *)
-rec definition lift_gen (A:Type[0]) (k:lift_continuation 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 ⇒
- match q with
- [ list_empty ⇒ lift_gen (A) (λp. k (𝗱❨f@❨n❩❩◗p)) q (f∘𝐮❨n❩)
- | list_lcons _ _ ⇒ lift_gen (A) k q (f∘𝐮❨n❩)
- ]
- | 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_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (⇂*[n]f) q
+ | label_m ⇒ lift_gen (A) (λg,p. k g (𝗺◗p)) 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_empty (A) (k) (f):
- k 𝐞 f = ↑{A}❨k, 𝐞, f❩.
+ k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
// qed.
-lemma lift_d_empty_sn (A) (k) (n) (f):
- ↑❨(λp. k (𝗱❨f@❨n❩❩◗p)), 𝐞, f∘𝐮❨ninj n❩❩ = ↑{A}❨k, 𝗱❨n❩◗𝐞, f❩.
+lemma lift_d_sn (A) (k) (p) (n) (f):
+ ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), ⇂*[n]f, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
// qed.
-lemma lift_d_lcons_sn (A) (k) (p) (l) (n) (f):
- ↑❨k, l◗p, f∘𝐮❨ninj n❩❩ = ↑{A}❨k, 𝗱❨n❩◗l◗p, f❩.
+lemma lift_m_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗺◗p)), f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
// qed.
lemma lift_L_sn (A) (k) (p) (f):
- ↑❨(λp. k (𝗟◗p)), p, ⫯f❩ = ↑{A}❨k, 𝗟◗p, f❩.
+ ↑❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ↑{A}❨k, f, 𝗟◗p❩.
// qed.
lemma lift_A_sn (A) (k) (p) (f):
- ↑❨(λp. k (𝗔◗p)), p, f❩ = ↑{A}❨k, 𝗔◗p, f❩.
+ ↑❨(λg,p. k g (𝗔◗p)), f, p❩ = ↑{A}❨k, f, 𝗔◗p❩.
// qed.
lemma lift_S_sn (A) (k) (p) (f):
- ↑❨(λp. k (𝗦◗p)), p, f❩ = ↑{A}❨k, 𝗦◗p, f❩.
+ ↑❨(λg,p. k g (𝗦◗p)), f, p❩ = ↑{A}❨k, f, 𝗦◗p❩.
// qed.
(* Basic constructions with proj_path ***************************************)
-lemma lift_path_d_empty_sn (f) (n):
- 𝗱❨f@❨n❩❩◗𝐞 = ↑[f](𝗱❨n❩◗𝐞).
-// qed.
-
-lemma lift_path_d_lcons_sn (f) (p) (l) (n):
- ↑[f∘𝐮❨ninj n❩](l◗p) = ↑[f](𝗱❨n❩◗l◗p).
+lemma lift_path_empty (f):
+ (𝐞) = ↑[f]𝐞.
// 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❩) = ↑[𝗱❨n❩◗p]f.
-#f * // qed.
+ ↑[p](⇂*[ninj n]f) = ↑[𝗱n◗p]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.
lemma lift_rmap_append (p2) (p1) (f):
↑[p2]↑[p1]f = ↑[p1●p2]f.
#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <lift_rmap_A_sn <lift_rmap_A_sn //
+[ <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 eliminations with path ******************************************)
-
-lemma path_ind_lift (Q:predicate …):
- Q 𝐞 →
- (∀n. Q 𝐞 → Q (𝗱❨n❩◗𝐞)) →
- (∀n,l,p. Q (l◗p) → Q (𝗱❨n❩◗l◗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 #p
-elim p -p [| * [ #n * ] ]
-/2 width=1 by/
-qed-.
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma lift_rmap_d_dx (f) (p) (n):
+ ⇂*[ninj n](↑[p]f) = ↑[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.
+
+lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
+ ↑[p]f@❨m+n❩ = ↑[p◖𝗱n]f@❨m❩+↑[p]f@❨n❩.
+// qed.
projects / helm.git / blobdiff