(* *)
(**************************************************************************)
-include "delayed_updating/syntax/path_structure.ma".
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 ***********************************************************)
-(* Constructions with structure ********************************************)
+(* Basic constructions with structure **************************************)
+
+lemma structure_lift (p) (f):
+ ⊗p = ⊗↑[f]p.
+#p @(path_ind_lift … p) -p // #p #IH #f
+<lift_path_L_sn //
+qed.
+
+lemma lift_structure (p) (f):
+ ⊗p = ↑[f]⊗p.
+#p @(path_ind_lift … p) -p //
+qed.
-lemma lift_d_empty_dx (n) (p) (f):
- (⊗p)◖𝗱❨(↑[p◖𝗱❨n❩]f)@❨n❩❩ = ↑[f](p◖𝗱❨n❩).
-#n #p @(path_ind_lift … p) -p // [ #m #l #p |*: #p ] #IH #f
-[ <lift_rmap_d_lcons_sn <lift_path_d_lcons_sn //
-| <lift_rmap_L_sn <lift_path_L_sn <IH //
-| <lift_rmap_A_sn <lift_path_A_sn <IH //
-| <lift_rmap_S_sn <lift_path_S_sn <IH //
+(* Destructions with structure **********************************************)
+
+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-.
+
+(* Constructions with inner condition for path ******************************)
+
+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).
+#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).
-#p @(path_ind_lift … p) -p // #m #l #p #IH #f
-<lift_path_d_lcons_sn //
+#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).
-#p @(path_ind_lift … p) -p // #m #l #p #IH #f
-<lift_path_d_lcons_sn //
+#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).
-#p @(path_ind_lift … p) -p // #m #l #p #IH #f
-<lift_path_d_lcons_sn //
+#p #f <lift_append_proper_dx //
qed.
-lemma structure_lift (p) (f):
- ⊗p = ⊗↑[f]p.
-#p @(path_ind_lift … p) -p // #p #IH #f
-<lift_path_L_sn //
-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-.
-lemma lift_structure (p) (f):
- ⊗p = ↑[f]⊗p.
-#p @(path_ind_lift … p) -p //
-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-.
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