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 c0d7dcbf5387fe01018c316c242d53e22afb3369..0273a3eb88560a961ced247647283b95fa33cb85 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
@@ -18,7 +18,7 @@ include "ground/notation/relations/ringeq_3.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. ∀f,p. k1 f p = k2 f p.
 
 interpretation
   "extensional equivalence (lift continuation)"
@@ -27,50 +27,58 @@ interpretation
 (* 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 ≗{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 //
-| * [ #n * [| #l0 ]] [|*: #q ] #IH #k1 #k2 #f #Hk /2 width=1 by/
+|*: #IH #k1 #k2 #f #Hk /2 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❩.
+      ↑❨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 *)
 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❩.
+      ↑❨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 //
 qed.
 
-(* Basic constructions with proj_path ***************************************)
+(* Advanced 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 //
-  ]
+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.