+++ /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/notation/functions/uparrow_4.ma".
-include "delayed_updating/notation/functions/uparrow_2.ma".
-include "delayed_updating/syntax/path.ma".
-include "ground/relocation/tr_id_pap.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-definition lift_continuation (A:Type[0]) ≝
- tr_map → path → A.
-
-rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
-match p with
-[ list_empty ⇒ k f (𝐞)
-| list_lcons l q ⇒
- match l with
- [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (𝐢) 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 f p = (lift_gen A k f p).
-
-definition proj_path: lift_continuation … ≝
- λf,p.p.
-
-definition proj_rmap: lift_continuation … ≝
- λf,p.f.
-
-interpretation
- "lift (path)"
- 'UpArrow f p = (lift_gen ? proj_path f p).
-
-interpretation
- "lift (relocation map)"
- 'UpArrow p f = (lift_gen ? proj_rmap f p).
-
-(* Basic constructions ******************************************************)
-
-lemma lift_empty (A) (k) (f):
- k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
-// qed.
-
-lemma lift_d_sn (A) (k) (p) (n) (f):
- ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), 𝐢, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
-// qed.
-
-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):
- ↑❨(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_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]𝐢 = ↑[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.
-// 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 cinstructionswith proj_rmap and tr_id ***************************)
-
-lemma lift_rmap_id (p):
- (𝐢) = ↑[p]𝐢.
-#p elim p -p //
-* [ #n ] #p #IH //
-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_d_sn <lift_rmap_d_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 constructions with proj_rmap and path_rcons *********************)
-
-lemma lift_rmap_d_dx (f) (p) (n):
- (𝐢) = ↑[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):
- m = ↑[p◖𝗱n]f@❨m❩.
-// qed.
+++ /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/substitution/lift_eq.ma".
-include "ground/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_compose_pn.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with tr_after *********************************************)
-
-lemma lift_path_after (p) (f1) (f2):
- ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p elim p -p [| * ] // #p #IH #f1 #f2
-<lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
->tr_compose_push_bi //
-qed.
+++ /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/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/substitution/lift_uni.ma".
-include "delayed_updating/syntax/prototerm_constructors.ma".
-
-(* LIFT FOR PROTOTERM *******************************************************)
-
-lemma lift_iref_bi (t1) (t2) (n):
- t1 ⇔ t2 → 𝛗n.t1 ⇔ 𝛗n.t2.
-/2 width=1 by subset_equivalence_ext_f1_bi/
-qed.
-
-lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
- (𝛗f@❨n❩.t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q #Hq #H0 destruct
-@(ex2_intro … (𝗱n◗𝗺◗q))
-/2 width=1 by in_comp_iref/
-qed-.
-
-lemma lift_iref_dx (f) (t) (n:pnat):
- ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.t.
-#f #t #n #p * #q #Hq #H0 destruct
-elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
-/2 width=1 by in_comp_iref/
-qed-.
-
-lemma lift_iref (f) (t) (n:pnat):
- (𝛗f@❨n❩.t) ⇔ ↑[f](𝛗n.t).
-/3 width=1 by conj, lift_iref_sn, lift_iref_dx/
-qed.
-
-lemma lift_iref_uni (t) (m) (n):
- (𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
-// qed.
+++ /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/substitution/lift.ma".
-include "ground/relocation/tr_pap_eq.ma".
-include "ground/relocation/tr_pn_eq.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
- λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
-
-interpretation
- "extensional equivalence (lift continuation)"
- 'RingEq A k1 k2 = (lift_exteq A k1 k2).
-
-(* Constructions with lift_exteq ********************************************)
-
-lemma lift_eq_repl (A) (p) (k1) (k2):
- k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
-#A #p elim p -p [| * [ #n ] #q #IH ]
-#k1 #k2 #Hk #f1 #f2 #Hf
-[ <lift_empty <lift_empty /2 width=1 by/
-| <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
- /3 width=1 by stream_eq_refl/
-| /3 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /3 width=1 by/
-| /3 width=1 by/
-]
-qed-.
-
-(* Advanced constructions ***************************************************)
-
-lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
- ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
-#A #k #f #p #l #Hk
-@lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
-qed.
-
-lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
- ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
-#A #k #f #p1 #p #l #Hk
-@lift_eq_repl // #g1 #g2 #p2 #Hg
-<list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
-qed.
-
-(* Advanced constructions with proj_path ************************************)
-
-lemma proj_path_proper:
- proj_path ≗ proj_path.
-// qed.
-
-lemma lift_path_eq_repl (p):
- stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
-/2 width=1 by lift_eq_repl/ qed.
-
-lemma lift_path_append_sn (p) (f) (q):
- q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
-#p elim p -p // * [ #n ] #p #IH #f #q
-[ <lift_d_sn <lift_d_sn
-| <lift_m_sn <lift_m_sn
-| <lift_L_sn <lift_L_sn
-| <lift_A_sn <lift_A_sn
-| <lift_S_sn <lift_S_sn
-]
->lift_lcons_alt // >lift_append_rcons_sn //
-<IH <IH -IH <list_append_rcons_sn //
-qed.
-
-lemma lift_path_lcons (f) (p) (l):
- l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
-#f #p #l
->lift_lcons_alt <lift_path_append_sn //
-qed.
-
-lemma lift_path_d_sn (f) (p) (n):
- (𝗱(f@❨n❩)◗↑[𝐢]p) = ↑[f](𝗱n◗p).
-// qed.
-
-lemma lift_path_m_sn (f) (p):
- (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
-// qed.
-
-lemma lift_path_L_sn (f) (p):
- (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
-// qed.
-
-lemma lift_path_A_sn (f) (p):
- (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
-// qed.
-
-lemma lift_path_S_sn (f) (p):
- (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
-// qed.
-
-lemma lift_path_id (p):
- p = ↑[𝐢]p.
-#p elim p -p //
-* [ #n ] #p #IH //
-[ <lift_path_d_sn //
-| <lift_path_L_sn //
-]
-qed.
-
-lemma lift_path_append (p2) (p1) (f):
- (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
-#p2 #p1 elim p1 -p1 //
-* [ #n1 ] #p1 #IH #f
-[ <lift_path_d_sn <lift_path_d_sn <IH //
-| <lift_path_m_sn <lift_path_m_sn <IH //
-| <lift_path_L_sn <lift_path_L_sn <IH //
-| <lift_path_A_sn <lift_path_A_sn <IH //
-| <lift_path_S_sn <lift_path_S_sn <IH //
-]
-qed.
-
-lemma lift_path_d_dx (n) (p) (f):
- (↑[f]p)◖𝗱((↑[p]f)@❨n❩) = ↑[f](p◖𝗱n).
-#n #p #f <lift_path_append //
-qed.
-
-lemma lift_path_m_dx (p) (f):
- (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
-#p #f <lift_path_append //
-qed.
-
-lemma lift_path_L_dx (p) (f):
- (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
-#p #f <lift_path_append //
-qed.
-
-lemma lift_path_A_dx (p) (f):
- (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
-#p #f <lift_path_append //
-qed.
-
-lemma lift_path_S_dx (p) (f):
- (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
-#p #f <lift_path_append //
-qed.
-
-(* COMMENT
-
-(* Advanced constructions with proj_rmap and stream_tls *********************)
-
-lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
- ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
-#f #p #m #n
-<lift_rmap_d_dx >nrplus_inj_dx
-/2 width=1 by tr_tls_compose_uni_dx/
-qed.
-*)
+++ /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/substitution/lift_eq.ma".
-include "ground/lib/list_length.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with list_length ******************************************)
-
-lemma lift_path_length (p) (f):
- ❘p❘ = ❘↑[f]p❘.
-#p elim p -p // * [ #n ] #p #IH #f //
-[ <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-]
->list_length_lcons >list_length_lcons //
-qed.
+++ /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/substitution/lift.ma".
-include "delayed_updating/syntax/prototerm.ma".
-include "ground/lib/subset_ext.ma".
-
-(* LIFT FOR PROTOTERM *******************************************************)
-
-interpretation
- "lift (prototerm)"
- 'UpArrow f t = (subset_ext_f1 ? ? (lift_gen ? proj_path f) t).
-
-(* Basic constructions ******************************************************)
-
-lemma in_comp_lift_bi (f) (p) (t):
- p ϵ t → ↑[f]p ϵ ↑[f]t.
-/2 width=1 by subset_in_ext_f1_dx/
-qed.
+++ /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 "ground/lib/subset_ext_equivalence.ma".
-include "delayed_updating/substitution/lift_after.ma".
-include "delayed_updating/substitution/lift_prototerm.ma".
-
-(* LIFT FOR PROTOTERM *******************************************************)
-
-(* Constructions with subset_equivalence ************************************)
-
-lemma lift_term_eq_repl_sn (f1) (f2) (t):
- f1 ≗ f2 → ↑[f1]t ⇔ ↑[f2]t.
-/3 width=1 by subset_equivalence_ext_f1_exteq, lift_path_eq_repl/
-qed.
-
-lemma lift_term_eq_repl_dx (f) (t1) (t2):
- t1 ⇔ t2 → ↑[f]t1 ⇔ ↑[f]t2.
-/2 width=1 by subset_equivalence_ext_f1_bi/
-qed.
-
-lemma lift_term_after (f1) (f2) (t):
- ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
-#f1 #f2 #t @subset_eq_trans
-[
-| @subset_inclusion_ext_f1_compose
-| @subset_equivalence_ext_f1_exteq /2 width=5/
-]
-qed.
-
-lemma lift_term_id_sn (t):
- t ⊆ ↑[𝐢]t.
-#t #p #Hp
->(lift_path_id p)
-/2 width=1 by in_comp_lift_bi/
-qed-.
-
-lemma lift_term_id_dx (t):
- ↑[𝐢]t ⊆ t.
-#t #p * #q #Hq #H destruct //
-qed-.
-
-lemma lift_term_id (t):
- t ⇔ ↑[𝐢]t.
-/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
-qed.
+++ /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/substitution/lift_eq.ma".
-include "delayed_updating/syntax/path_structure.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with structure ********************************************)
-
-lemma structure_lift (p) (f):
- ⊗p = ⊗↑[f]p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-[ <lift_path_m_sn <structure_m_sn <structure_m_sn //
-| <lift_path_L_sn <structure_L_sn <structure_L_sn //
-]
-qed.
-
-lemma lift_structure (p) (f):
- ⊗p = ↑[f]⊗p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-qed.
+++ /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/substitution/lift_eq.ma".
-include "ground/relocation/tr_uni_pap.ma".
-
-(* LIFT FOR PATH ***********************************************************)
-
-(* Constructions with tr_uni ***********************************************)
-
-lemma lift_path_d_sn_uni (p) (m) (n):
- (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
-// qed.
--- /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/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]) ≝
+ tr_map → path → A.
+
+rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
+match p with
+[ list_empty ⇒ k f (𝐞)
+| list_lcons l q ⇒
+ match l with
+ [ 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 f p = (lift_gen A k f p).
+
+definition proj_path: lift_continuation … ≝
+ λf,p.p.
+
+definition proj_rmap: lift_continuation … ≝
+ λf,p.f.
+
+interpretation
+ "lift (path)"
+ 'UpArrow f p = (lift_gen ? proj_path f p).
+
+interpretation
+ "lift (relocation map)"
+ 'UpArrow p f = (lift_gen ? proj_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma lift_empty (A) (k) (f):
+ k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
+// qed.
+
+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_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):
+ ↑❨(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_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](⇂*[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.
+// 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):
+ ⇂*[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.
--- /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/substitution/lift_prototerm_id.ma".
+include "delayed_updating/substitution/lift_path_uni.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+lemma lift_iref_bi (t1) (t2) (n):
+ t1 ⇔ t2 → 𝛗n.t1 ⇔ 𝛗n.t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
+ (𝛗f＠⧣❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro … (𝗱n◗𝗺◗r))
+/2 width=1 by in_comp_iref/
+qed-.
+
+lemma lift_iref_dx (f) (t) (n:pnat):
+ ↑[f](𝛗n.t) ⊆ 𝛗f＠⧣❨n❩.↑[⇂*[n]f]t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
+/3 width=1 by in_comp_iref, in_comp_lift_bi/
+qed-.
+
+lemma lift_iref (f) (t) (n:pnat):
+ (𝛗f＠⧣❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
+/3 width=1 by conj, lift_iref_sn, lift_iref_dx/
+qed.
+
+lemma lift_iref_uni (t) (m) (n):
+ (𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
+#t #m #n
+@(subset_eq_trans … (lift_iref …))
+<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
+/3 width=1 by lift_iref_bi, lift_term_id/
+qed.
--- /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/substitution/lift.ma".
+include "ground/relocation/tr_pap_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+include "ground/lib/stream_tls_eq.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
+ λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
+
+interpretation
+ "extensional equivalence (lift continuation)"
+ 'RingEq A k1 k2 = (lift_exteq A k1 k2).
+
+(* Constructions with lift_exteq ********************************************)
+
+lemma lift_eq_repl (A) (p) (k1) (k2):
+ k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
+#A #p elim p -p [| * [ #n ] #q #IH ]
+#k1 #k2 #Hk #f1 #f2 #Hf
+[ <lift_empty <lift_empty /2 width=1 by/
+| <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
+ /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
+| /3 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /3 width=1 by/
+| /3 width=1 by/
+]
+qed-.
+
+(* Advanced constructions ***************************************************)
+
+lemma lift_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
+ ↑❨λg,p2. k g (l◗p2), f, p❩ = ↑{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
+#A #k #f #p #l #Hk
+@lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
+qed.
+
+lemma lift_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
+ ↑❨λg,p2. k g (p1●l◗p2), f, p❩ = ↑{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
+#A #k #f #p1 #p #l #Hk
+@lift_eq_repl // #g1 #g2 #p2 #Hg
+<list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
+qed.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma proj_path_proper:
+ proj_path ≗ proj_path.
+// qed.
+
+lemma lift_path_eq_repl (p):
+ stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
+/2 width=1 by lift_eq_repl/ qed.
+
+lemma lift_path_append_sn (p) (f) (q):
+ q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
+#p elim p -p // * [ #n ] #p #IH #f #q
+[ <lift_d_sn <lift_d_sn
+| <lift_m_sn <lift_m_sn
+| <lift_L_sn <lift_L_sn
+| <lift_A_sn <lift_A_sn
+| <lift_S_sn <lift_S_sn
+]
+>lift_lcons_alt // >lift_append_rcons_sn //
+<IH <IH -IH <list_append_rcons_sn //
+qed.
+
+lemma lift_path_lcons (f) (p) (l):
+ l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
+#f #p #l
+>lift_lcons_alt <lift_path_append_sn //
+qed.
+
+lemma lift_path_d_sn (f) (p) (n):
+ (𝗱(f＠⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
+// qed.
+
+lemma lift_path_m_sn (f) (p):
+ (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
+// qed.
+
+lemma lift_path_L_sn (f) (p):
+ (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
+// qed.
+
+lemma lift_path_A_sn (f) (p):
+ (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
+// qed.
+
+lemma lift_path_S_sn (f) (p):
+ (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
+// qed.
+
+lemma lift_path_append (p2) (p1) (f):
+ (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
+#p2 #p1 elim p1 -p1 //
+* [ #n1 ] #p1 #IH #f
+[ <lift_path_d_sn <lift_path_d_sn <IH //
+| <lift_path_m_sn <lift_path_m_sn <IH //
+| <lift_path_L_sn <lift_path_L_sn <IH //
+| <lift_path_A_sn <lift_path_A_sn <IH //
+| <lift_path_S_sn <lift_path_S_sn <IH //
+]
+qed.
+
+lemma lift_path_d_dx (f) (p) (n):
+ (↑[f]p)◖𝗱((↑[p]f)＠⧣❨n❩) = ↑[f](p◖𝗱n).
+#f #p #n <lift_path_append //
+qed.
+
+lemma lift_path_m_dx (f) (p):
+ (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
+#f #p <lift_path_append //
+qed.
+
+lemma lift_path_L_dx (f) (p):
+ (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
+#f #p <lift_path_append //
+qed.
+
+lemma lift_path_A_dx (f) (p):
+ (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
+#f #p <lift_path_append //
+qed.
+
+lemma lift_path_S_dx (f) (p):
+ (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
+#f #p <lift_path_append //
+qed.
+
+(* COMMENT
+
+(* Advanced constructions with proj_rmap and stream_tls *********************)
+
+lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
+ ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
+#f #p #m #n
+<lift_rmap_d_dx >nrplus_inj_dx
+/2 width=1 by tr_tls_compose_uni_dx/
+qed.
+
+*)
--- /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/substitution/lift_eq.ma".
+include "ground/relocation/tr_compose_pap.ma".
+include "ground/relocation/tr_compose_pn.ma".
+include "ground/relocation/tr_compose_tls.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with tr_after **********************************************)
+
+lemma lift_path_after (p) (f1) (f2):
+ ↑[f2]↑[f1]p = ↑[f2∘f1]p.
+#p elim p -p [| * ] // [ #n ] #p #IH #f1 #f2
+[ <lift_path_d_sn <lift_path_d_sn <lift_path_d_sn //
+| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
+ >tr_compose_push_bi //
+]
+qed.
--- /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/substitution/lift_eq.ma".
+include "ground/relocation/tr_id_pap.ma".
+include "ground/relocation/tr_id_tls.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with tr_id *************************************************)
+
+lemma lift_path_id (p):
+ p = ↑[𝐢]p.
+#p elim p -p //
+* [ #n ] #p #IH //
+[ <lift_path_d_sn //
+| <lift_path_L_sn //
+]
+qed.
--- /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/substitution/lift_eq.ma".
+include "delayed_updating/syntax/path_proper.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with proper condition for path *****************************)
+
+lemma lift_path_proper (f) (p):
+ p ϵ 𝐏 → ↑[f]p ϵ 𝐏.
+#f *
+[ #H0 elim (ppc_inv_empty … H0)
+| * [ #n ] #p #_
+ [ <lift_path_d_sn /2 width=3 by ppc_lcons/
+ | <lift_path_m_sn /2 width=3 by ppc_lcons/
+ | <lift_path_L_sn /2 width=3 by ppc_lcons/
+ | <lift_path_A_sn /2 width=3 by ppc_lcons/
+ | <lift_path_S_sn /2 width=3 by ppc_lcons/
+ ]
+]
+qed.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma lift_path_inv_proper (f) (p):
+ ↑[f]p ϵ 𝐏 → p ϵ 𝐏.
+#f * //
+#H0 elim (ppc_inv_empty … H0)
+qed-.
--- /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/substitution/lift_path_id.ma".
+include "ground/relocation/tr_uni_pap.ma".
+include "ground/relocation/tr_uni_tls.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with tr_uni ************************************************)
+
+lemma lift_path_d_sn_uni (p) (m) (n):
+ (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
+#p #m #n
+<lift_path_d_sn <tr_uni_pap >nsucc_pnpred
+<tr_tls_succ_uni //
+qed.
--- /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/substitution/lift.ma".
+include "delayed_updating/syntax/prototerm.ma".
+include "ground/lib/subset_ext.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+interpretation
+ "lift (prototerm)"
+ 'UpArrow f t = (subset_ext_f1 ? ? (lift_gen ? proj_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_lift_bi (f) (p) (t):
+ p ϵ t → ↑[f]p ϵ ↑[f]t.
+/2 width=1 by subset_in_ext_f1_dx/
+qed.
--- /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 "ground/lib/subset_ext_equivalence.ma".
+include "delayed_updating/substitution/lift_path_after.ma".
+include "delayed_updating/substitution/lift_prototerm.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma lift_term_eq_repl_sn (f1) (f2) (t):
+ f1 ≗ f2 → ↑[f1]t ⇔ ↑[f2]t.
+/3 width=1 by subset_equivalence_ext_f1_exteq, lift_path_eq_repl/
+qed.
+
+lemma lift_term_eq_repl_dx (f) (t1) (t2):
+ t1 ⇔ t2 → ↑[f]t1 ⇔ ↑[f]t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma lift_term_after (f1) (f2) (t):
+ ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
+#f1 #f2 #t @subset_eq_trans
+[
+| @subset_inclusion_ext_f1_compose
+| @subset_equivalence_ext_f1_exteq /2 width=5/
+]
+qed.
--- /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/substitution/lift_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_path_id.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with tr_id *************************************************)
+
+lemma lift_term_id_sn (t):
+ t ⊆ ↑[𝐢]t.
+#t #p #Hp
+>(lift_path_id p)
+/2 width=1 by in_comp_lift_bi/
+qed-.
+
+lemma lift_term_id_dx (t):
+ ↑[𝐢]t ⊆ t.
+#t #p * #q #Hq #H destruct //
+qed-.
+
+lemma lift_term_id (t):
+ t ⇔ ↑[𝐢]t.
+/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
+qed.
--- /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/substitution/lift_prototerm.ma".
+include "delayed_updating/substitution/lift_path_proper.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with proper condition for path *****************************)
+
+lemma lift_term_proper (f) (t):
+ t ϵ 𝐏 → ↑[f]t ϵ 𝐏.
+#f #t #Ht #p * #q #Hq #H0 destruct
+@lift_path_proper @Ht -Ht // (**) (* auto fails *)
+qed.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma lift_term_inv_proper (f) (t):
+ ↑[f]t ϵ 𝐏 → t ϵ 𝐏.
+#f #t #Ht #p #Hp
+@(lift_path_inv_proper f)
+@Ht -Ht @in_comp_lift_bi // (**) (* auto fails *)
+qed-.
--- /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/syntax/prototerm.ma".
+include "delayed_updating/syntax/path_inner.ma".
+include "ground/lib/subset_ext_equivalence.ma".
+
+(* INNER CONDITION FOR PROTOTERM ********************************************)
+
+interpretation
+ "inner condition (prototerm)"
+ 'ClassI = (subset_ext_p1 path pic).
+
+(* Basic destructions *******************************************************)
+
+lemma in_comp_tic_trans (t) (p):
+ p ϵ t → t ϵ 𝐈 → p ϵ 𝐈.
+#t #p #Hp #Ht
+@(Ht … Hp)
+qed-.
+
+(* Basic inversions *********************************************************)
+
+lemma tic_inv_d_dx (t) (p) (n):
+ p◖𝗱n ϵ t → t ϵ 𝐈 → ⊥.
+/2 width=7 by in_comp_tic_trans/ qed-.
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