(**************************************************************************)
include "delayed_updating/substitution/lift.ma".
-include "ground/notation/relations/ringeq_3.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_pn_eq.ma".
(* LIFT FOR PATH ***********************************************************)
definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
- λk1,k2. ∀p,f. k1 p f = k2 p f.
+ λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
interpretation
"extensional equivalence (lift continuation)"
(* Constructions with lift_exteq ********************************************)
-lemma lift_eq_repl_sn (A) (p) (k1) (k2) (f):
- k1 ≗{A} k2 → ↑❨k1, p, f❩ = ↑❨k2, p, f❩.
-#A #p elim p -p
-[ #k1 #k2 #f #Hk <lift_empty <lift_empty //
-| * [ #n * [| #l0 ]] [|*: #q ] #IH #k1 #k2 #f #Hk /2 width=1 by/
+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
+[ <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/
+| /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):
- ↑❨λp2.k(l◗p2),p,f❩ = ↑{A}❨λp2.k((l◗𝐞)●p2),p,f❩.
-#A #k #f #p #l
-@lift_eq_repl_sn #p2 #g // (**) (* auto fails with typechecker failure *)
+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):
- ↑❨λp2.k(p1●l◗p2),p,f❩ = ↑{A}❨λp2.k(p1◖l●p2),p,f❩.
-#A #k #f #p1 #p #l
-@lift_eq_repl_sn #p2 #g
-<list_append_rcons_sn //
+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.
-(* Basic constructions with proj_path ***************************************)
-
-lemma lift_append_sn (p) (f) (q):
- q●↑[f]p = ↑❨(λp. proj_path (q●p)), p, f❩.
-#p elim p -p
-[ //
-| * [ #n * [| #l ]] [|*: #p ] #IH #f #q
- [ <lift_d_empty_sn <lift_d_empty_sn >lift_lcons_alt >lift_append_rcons_sn
- <IH <IH -IH <list_append_rcons_sn //
- | <lift_d_lcons_sn <lift_d_lcons_sn <IH -IH //
- | <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 //
- ]
+(* 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 @(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 //
]
qed.
-lemma lift_lcons (f) (p) (l):
- l◗↑[f]p = ↑❨(λp. proj_path (l◗p)), p, f❩.
+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_append_sn //
+>lift_lcons_alt <lift_path_append_sn //
+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_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 //
+]
+qed.
+
+(* 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.