X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fdelayed_updating%2Fsubstitution%2Flift_structure.ma;h=2a219e0ffd08d7b7545f97aaecc7767a4b44570b;hb=8a47ade5ffd1942f9d16474c547e5050caab3cc8;hp=961a5c435ef8b0f3d2cbd106a5bc80279ccf85db;hpb=ac8d0dc0c6fb995736e0c10486d996bd023f3c32;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma
index 961a5c435..2a219e0ff 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma
@@ -14,7 +14,9 @@
 
 include "delayed_updating/substitution/lift_eq.ma".
 include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_inner.ma".
 include "delayed_updating/syntax/path_proper.ma".
+include "ground/xoa/ex_4_2.ma".
 
 (* LIFT FOR PATH ***********************************************************)
 
@@ -31,38 +33,237 @@ lemma lift_structure (p) (f):
 #p @(path_ind_lift … p) -p //
 qed.
 
-(* Properties with proper condition for path ********************************)
+(* Destructions with structure **********************************************)
 
-lemma lift_append_proper_dx (p2) (p1) (f): Ꝕp2 →
+lemma lift_des_structure (q) (p) (f):
+      ⊗q = ↑[f]p → ⊗q = ⊗p.
+// qed-.
+
+(* Constructions with proper condition for path *****************************)
+
+lemma lift_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
       (⊗p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
 #p2 #p1 @(path_ind_lift … p1) -p1 //
 [ #n | #n #l #p1 |*: #p1 ] #IH #f #Hp2
 [ elim (ppc_inv_lcons … Hp2) -Hp2 #l #q #H destruct //
 | <lift_path_d_lcons_sn <IH //
+| <lift_path_m_sn <IH //
 | <lift_path_L_sn <IH //
 | <lift_path_A_sn <IH //
 | <lift_path_S_sn <IH //
 ]
 qed-.
 
-(* Advanced constructions with structure ************************************)
+(* Constructions with inner condition for path ******************************)
 
-lemma lift_d_empty_dx (n) (p) (f):
+lemma lift_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
+      (⊗p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
+#p1 @(list_ind_rcons … p1) -p1 // #p1 *
+[ #n ] #_ #p2 #f #Hp1
+[ elim (pic_inv_d_dx … Hp1)
+| <list_append_rcons_sn <lift_append_proper_dx //
+| <list_append_rcons_sn <lift_append_proper_dx //
+  <structure_L_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <lift_append_proper_dx //
+  <structure_A_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <lift_append_proper_dx //
+  <structure_S_dx <list_append_rcons_sn //
+]
+qed-.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma lift_path_d_empty_dx (n) (p) (f):
       (⊗p)◖𝗱((↑[p]f)@❨n❩) = ↑[f](p◖𝗱n).
-/3 width=3 by ppc_lcons, lift_append_proper_dx/
+#n #p #f <lift_append_proper_dx // 
+qed.
+
+lemma lift_path_m_dx (p) (f):
+      ⊗p = ↑[f](p◖𝗺).
+#p #f <lift_append_proper_dx //
 qed.
 
-lemma lift_L_dx (p) (f):
+lemma lift_path_L_dx (p) (f):
       (⊗p)◖𝗟 = ↑[f](p◖𝗟).
-/3 width=3 by ppc_lcons, lift_append_proper_dx/
+#p #f <lift_append_proper_dx //
 qed.
 
-lemma lift_A_dx (p) (f):
+lemma lift_path_A_dx (p) (f):
       (⊗p)◖𝗔 = ↑[f](p◖𝗔).
-/3 width=3 by ppc_lcons, lift_append_proper_dx/
+#p #f <lift_append_proper_dx //
 qed.
 
-lemma lift_S_dx (p) (f):
+lemma lift_path_S_dx (p) (f):
       (⊗p)◖𝗦 = ↑[f](p◖𝗦).
-/3 width=3 by ppc_lcons, lift_append_proper_dx/
+#p #f <lift_append_proper_dx //
 qed.
+
+lemma lift_path_root (f) (p):
+      ∃∃r. 𝐞 = ⊗r & ⊗p●r = ↑[f]p.
+#f #p @(list_ind_rcons … p) -p
+[ /2 width=3 by ex2_intro/
+| #p * [ #n ] /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Advanced inversions with proj_path ***************************************)
+
+lemma lift_path_inv_d_sn (k) (q) (p) (f):
+      (𝗱k◗q) = ↑[f]p →
+      ∃∃r,h. 𝐞 = ⊗r & (↑[r]f)@❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
+#k #q #p @(path_ind_lift … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <lift_path_empty #H destruct
+| <lift_path_d_empty_sn #H destruct -IH
+  /2 width=5 by ex4_2_intro/
+| <lift_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+  /2 width=5 by ex4_2_intro/
+| <lift_path_m_sn #H
+  elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+  /2 width=5 by ex4_2_intro/
+| <lift_path_L_sn #H destruct
+| <lift_path_A_sn #H destruct
+| <lift_path_S_sn #H destruct
+]
+qed-.
+
+lemma lift_path_inv_m_sn (q) (p) (f):
+      (𝗺◗q) = ↑[f]p → ⊥.
+#q #p @(path_ind_lift … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <lift_path_empty #H destruct
+| <lift_path_d_empty_sn #H destruct
+| <lift_path_d_lcons_sn #H /2 width=2 by/
+| <lift_path_m_sn #H /2 width=2 by/
+| <lift_path_L_sn #H destruct
+| <lift_path_A_sn #H destruct
+| <lift_path_S_sn #H destruct
+]
+qed-.
+
+lemma lift_path_inv_L_sn (q) (p) (f):
+      (𝗟◗q) = ↑[f]p →
+      ∃∃r1,r2. 𝐞 = ⊗r1 & q = ↑[⫯↑[r1]f]r2 & r1●𝗟◗r2 = p.
+#q #p @(path_ind_lift … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <lift_path_empty #H destruct
+| <lift_path_d_empty_sn #H destruct
+| <lift_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <lift_path_m_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <lift_path_L_sn #H destruct -IH
+  /2 width=5 by ex3_2_intro/
+| <lift_path_A_sn #H destruct
+| <lift_path_S_sn #H destruct
+]
+qed-.
+
+lemma lift_path_inv_A_sn (q) (p) (f):
+      (𝗔◗q) = ↑[f]p →
+      ∃∃r1,r2. 𝐞 = ⊗r1 & q = ↑[↑[r1]f]r2 & r1●𝗔◗r2 = p.
+#q #p @(path_ind_lift … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <lift_path_empty #H destruct
+| <lift_path_d_empty_sn #H destruct
+| <lift_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <lift_path_m_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <lift_path_L_sn #H destruct
+| <lift_path_A_sn #H destruct -IH
+  /2 width=5 by ex3_2_intro/
+| <lift_path_S_sn #H destruct
+]
+qed-.
+
+lemma lift_path_inv_S_sn (q) (p) (f):
+      (𝗦◗q) = ↑[f]p →
+      ∃∃r1,r2. 𝐞 = ⊗r1 & q = ↑[↑[r1]f]r2 & r1●𝗦◗r2 = p.
+#q #p @(path_ind_lift … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <lift_path_empty #H destruct
+| <lift_path_d_empty_sn #H destruct
+| <lift_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <lift_path_m_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/| <lift_path_L_sn #H destruct
+| <lift_path_A_sn #H destruct
+| <lift_path_S_sn #H destruct -IH
+  /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma lift_inv_append_proper_dx (q2) (q1) (p) (f):
+      q2 ϵ 𝐏 → q1●q2 = ↑[f]p →
+      ∃∃p1,p2. ⊗p1 = q1 & ↑[↑[p1]f]p2 = q2 & p1●p2 = p.
+#q2 #q1 elim q1 -q1
+[ #p #f #Hq2 <list_append_empty_sn #H destruct
+  /2 width=5 by ex3_2_intro/
+| * [ #n1 ] #q1 #IH #p #f #Hq2 <list_append_lcons_sn #H
+  [ elim (lift_path_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
+    elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
+    elim Hq2 -Hq2 //
+  | elim (lift_path_inv_m_sn … H)
+  | elim (lift_path_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+    elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+    @(ex3_2_intro … (r1●𝗟◗p1)) //
+    <structure_append <Hr1 -Hr1 //
+  | elim (lift_path_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+    elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+    @(ex3_2_intro … (r1●𝗔◗p1)) //
+    <structure_append <Hr1 -Hr1 //
+  | elim (lift_path_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+    elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+    @(ex3_2_intro … (r1●𝗦◗p1)) //
+    <structure_append <Hr1 -Hr1 //
+  ]
+]
+qed-.
+
+(* Inversions with inner condition for path *********************************)
+
+lemma lift_inv_append_inner_sn (q1) (q2) (p) (f):
+      q1 ϵ 𝐈 → q1●q2 = ↑[f]p →
+      ∃∃p1,p2. ⊗p1 = q1 & ↑[↑[p1]f]p2 = q2 & p1●p2 = p.
+#q1 @(list_ind_rcons … q1) -q1
+[ #q2 #p #f #Hq1 <list_append_empty_sn #H destruct
+  /2 width=5 by ex3_2_intro/
+| #q1 * [ #n1 ] #_ #q2 #p #f #Hq2
+  [ elim (pic_inv_d_dx … Hq2)
+  | <list_append_rcons_sn #H0
+    elim (lift_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (lift_path_inv_m_sn … (sym_eq … H2))
+  | <list_append_rcons_sn #H0
+    elim (lift_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (lift_path_inv_L_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+    @(ex3_2_intro … (p1●r2◖𝗟)) [1,3: // ]
+    [ <structure_append <structure_L_dx <Hr2 -Hr2 //
+    | <list_append_assoc <list_append_rcons_sn //
+    ]
+  | <list_append_rcons_sn #H0
+    elim (lift_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (lift_path_inv_A_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+    @(ex3_2_intro … (p1●r2◖𝗔)) [1,3: // ]
+    [ <structure_append <structure_A_dx <Hr2 -Hr2 //
+    | <list_append_assoc <list_append_rcons_sn //
+    ]
+  | <list_append_rcons_sn #H0
+    elim (lift_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (lift_path_inv_S_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+    @(ex3_2_intro … (p1●r2◖𝗦)) [1,3: // ]
+    [ <structure_append <structure_S_dx <Hr2 -Hr2 //
+    | <list_append_assoc <list_append_rcons_sn //
+    ]
+  ]
+]
+qed-.