+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "delayed_updating/relocation/sbr_pap_push.ma".
-include "delayed_updating/syntax/path.ma".
-include "delayed_updating/notation/functions/black_downtriangle_4.ma".
-include "delayed_updating/notation/functions/black_downtriangle_2.ma".
-include "ground/relocation/tr_pn.ma".
-
-(* UNWIND FOR PATH **********************************************************)
-
-definition unwind_continuation (A:Type[0]) ≝
- tr_map → path → A.
-
-(* Note: f is a stack of update functions *)
-rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f:tr_map) (p) on p ≝
-match p with
-[ list_empty ⇒ k f (𝐞)
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒
- match q with
- [ list_empty ⇒ unwind_gen (A) (λg,p. k g (𝗱(f❘@❨n❩)◗p)) (f❘@❨n❩⫯❘f) q
- | list_lcons _ _ ⇒ unwind_gen (A) k (f❘@❨n❩⫯❘f) q
- ]
- | label_m ⇒ unwind_gen (A) k f q
- | label_L ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
- | label_A ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
- | label_S ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
- ]
-].
-
-interpretation
- "unwind (gneric)"
- 'BlackDownTriangle A k f p = (unwind_gen A k f p).
-
-definition proj_path: unwind_continuation … ≝
- λf,p.p.
-
-definition proj_rmap: unwind_continuation … ≝
- λf,p.f.
-
-interpretation
- "unwind (path)"
- 'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
-
-interpretation
- "unwind (relocation map)"
- 'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
-
-(* Basic constructions ******************************************************)
-
-lemma unwind_empty (A) (k) (f):
- k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
-// qed.
-
-lemma unwind_d_empty_sn (A) (k) (n) (f):
- ▼❨(λg,p. k g (𝗱(f❘@❨n❩)◗p)), f❘@❨n❩⫯❘f, 𝐞❩ = ▼{A}❨k, f, 𝗱n◗𝐞❩.
-// qed.
-
-lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
- ▼❨k, f❘@❨n❩⫯❘f, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
-// qed.
-
-lemma unwind_m_sn (A) (k) (p) (f):
- ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
-// qed.
-
-lemma unwind_L_sn (A) (k) (p) (f):
- ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
-// qed.
-
-lemma unwind_A_sn (A) (k) (p) (f):
- ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
-// qed.
-
-lemma unwind_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 unwind_path_empty (f):
- (𝐞) = ▼[f]𝐞.
-// qed.
-
-lemma unwind_path_d_empty_sn (f) (n):
- 𝗱(f❘@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
-// qed.
-
-lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
- ▼[f❘@❨n❩⫯❘f](l◗p) = ▼[f](𝗱n◗l◗p).
-// qed.
-
-lemma unwind_path_m_sn (f) (p):
- ▼[f]p = ▼[f](𝗺◗p).
-// qed.
-
-(* Basic constructions with proj_rmap ***************************************)
-
-lemma unwind_rmap_empty (f):
- f = ▼[𝐞]f.
-// qed.
-
-lemma unwind_rmap_d_sn (f) (p) (n):
- ▼[p](f❘@❨n❩⫯❘f) = ▼[𝗱n◗p]f.
-#f * // qed.
-
-lemma unwind_rmap_m_sn (f) (p):
- ▼[p]f = ▼[𝗺◗p]f.
-// qed.
-
-lemma unwind_rmap_L_sn (f) (p):
- ▼[p](⫯f) = ▼[𝗟◗p]f.
-// qed.
-
-lemma unwind_rmap_A_sn (f) (p):
- ▼[p]f = ▼[𝗔◗p]f.
-// qed.
-
-lemma unwind_rmap_S_sn (f) (p):
- ▼[p]f = ▼[𝗦◗p]f.
-// qed.
-
-(* Advanced constructions with proj_rmap and path_append ********************)
-
-lemma unwind_rmap_append (p2) (p1) (f):
- ▼[p2]▼[p1]f = ▼[p1●p2]f.
-#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <unwind_rmap_m_sn <unwind_rmap_m_sn //
-| <unwind_rmap_A_sn <unwind_rmap_A_sn //
-| <unwind_rmap_S_sn <unwind_rmap_S_sn //
-]
-qed.
-
-(* Advanced constructions with proj_rmap and path_rcons *********************)
-
-lemma unwind_rmap_d_dx (f) (p) (n):
- (▼[p]f)❘@❨n❩⫯❘▼[p]f = ▼[p◖𝗱n]f.
-// qed.
-
-lemma unwind_rmap_m_dx (f) (p):
- ▼[p]f = ▼[p◖𝗺]f.
-// qed.
-
-lemma unwind_rmap_L_dx (f) (p):
- (⫯▼[p]f) = ▼[p◖𝗟]f.
-// qed.
-
-lemma unwind_rmap_A_dx (f) (p):
- ▼[p]f = ▼[p◖𝗔]f.
-// qed.
-
-lemma unwind_rmap_S_dx (f) (p):
-▼[p]f = ▼[p◖𝗦]f.
-// qed.
-
-lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
- m+▼[p]f❘@❨n❩ = ▼[p◖𝗱n]f❘@❨m❩.
-#f #p #n #m
-<unwind_rmap_d_dx <sbr_push_pap //
-qed.
-
-(* Advanced eliminations with path ******************************************)
-
-lemma path_ind_unwind (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-.
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