update in delayed_updating

[helm.git] / matita / matita / contribs / lambdadelta / delayed_updating / substitution / lift_eq.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
index 0273a3eb88560a961ced247647283b95fa33cb85..9e23b2cb9a5620d08bab1658735ea0cc32344dd6 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
@@ -13,12 +13,13 @@
 (**************************************************************************)
 
 include "delayed_updating/substitution/lift.ma".
-include "ground/notation/relations/ringeq_3.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. ∀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)"
@@ -26,41 +27,57 @@ interpretation
 
 (* 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 @(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):
+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
+| <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
+| <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
+| <lift_S_sn <lift_S_sn >lift_lcons_alt >lift_append_rcons_sn //
   <IH <IH -IH <list_append_rcons_sn //
 ]
 qed.