(**************************************************************************)
include "delayed_updating/substitution/lift.ma".
-include "ground/notation/relations/ringeq_3.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. ∀f,p. k1 f p = k2 f p.
+ 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, f, p❩ = ↑❨k2, f, p❩.
-#A #p @(path_ind_lift … p) -p [| #n | #n #l0 #q ]
-[ #k1 #k2 #f #Hk <lift_empty <lift_empty //
-|*: #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 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):
+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
-@lift_eq_repl_sn #p2 #g // (**) (* auto fails with typechecker failure *)
+#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):
+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
-@lift_eq_repl_sn #p2 #g
-<list_append_rcons_sn //
+#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 @(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.
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.
+*)
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