(**************************************************************************)
include "delayed_updating/substitution/lift.ma".
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_eq.ma".
+include "ground/relocation/tr_pap_eq.ma".
include "ground/relocation/tr_pn_eq.ma".
(* LIFT FOR PATH ***********************************************************)
lemma lift_eq_repl (A) (p) (k1) (k2):
k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
-#A #p @(path_ind_lift … p) -p [| #n #IH | #n #l0 #q #IH |*: #q #IH ]
-#k1 #k2 #f1 #f2 #Hk #Hf
+#A #p elim p -p [| * [ #n ] #q #IH ]
+#k1 #k2 #Hk #f1 #f2 #Hf
[ <lift_empty <lift_empty /2 width=1 by/
-| <lift_d_empty_sn <lift_d_empty_sn <(tr_pap_eq_repl … Hf)
- /3 width=1 by tr_compose_eq_repl, stream_eq_refl/
-| <lift_d_lcons_sn <lift_d_lcons_sn
- /3 width=1 by tr_compose_eq_repl, stream_eq_refl/
-| /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/
lemma lift_path_append_sn (p) (f) (q):
q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
-#p @(path_ind_lift … p) -p // [ #n #l #p |*: #p ] #IH #f #q
-[ <lift_d_lcons_sn <lift_d_lcons_sn <IH -IH //
-| <lift_m_sn <lift_m_sn //
-| <lift_L_sn <lift_L_sn >lift_lcons_alt // >lift_append_rcons_sn //
- <IH <IH -IH <list_append_rcons_sn //
-| <lift_A_sn <lift_A_sn >lift_lcons_alt >lift_append_rcons_sn //
- <IH <IH -IH <list_append_rcons_sn //
-| <lift_S_sn <lift_S_sn >lift_lcons_alt >lift_append_rcons_sn //
- <IH <IH -IH <list_append_rcons_sn //
-]
+#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):
>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.
(𝗦◗↑[f]p) = ↑[f](𝗦◗p).
// qed.
-lemma lift_path_after (p) (f1) (f2):
- ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p @(path_ind_lift … p) -p // [ #n #l #p | #p ] #IH #f1 #f2
-[ <lift_path_d_lcons_sn <lift_path_d_lcons_sn
- >(lift_path_eq_repl … (tr_compose_assoc …)) //
-| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
- >tr_compose_push_bi //
+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):
<lift_rmap_d_dx >nrplus_inj_dx
/2 width=1 by tr_tls_compose_uni_dx/
qed.
+*)