--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/unwind_k/unwind2_path.ma".
+include "delayed_updating/syntax/path_inner.ma".
+include "delayed_updating/syntax/path_proper.ma".
+include "ground/xoa/ex_4_2.ma".
+
+(* TAILED UNWIND FOR PATH ***************************************************)
+
+(* Constructions with pic ***************************************************)
+
+lemma unwind2_path_pic (f) (p):
+ p ϵ 𝐈 → ⊗p = ▼[f]p.
+#f * // * // #k #q #Hq
+elim (pic_inv_d_dx … Hq)
+qed-.
+
+(* Constructions with append and pic ****************************************)
+
+lemma unwind2_path_append_pic_sn (f) (p) (q): p ϵ 𝐈 →
+ (⊗p)●(▼[▶[f]p]q) = ▼[f](p●q).
+#f #p * [ #Hp | * [ #k ] #q #_ ] //
+[ <(unwind2_path_pic … Hp) -Hp //
+| <unwind2_path_d_dx <unwind2_path_d_dx
+ /2 width=3 by trans_eq/
+| <unwind2_path_L_dx <unwind2_path_L_dx //
+| <unwind2_path_A_dx <unwind2_path_A_dx //
+| <unwind2_path_S_dx <unwind2_path_S_dx //
+]
+qed.
+
+(* Constructions with append and ppc ****************************************)
+
+lemma unwind2_path_append_ppc_dx (f) (p) (q): q ϵ 𝐏 →
+ (⊗p)●(▼[▶[f]p]q) = ▼[f](p●q).
+#f #p * [ #Hq | * [ #k ] #q #_ ] //
+[ elim (ppc_inv_empty … Hq)
+| <unwind2_path_d_dx <unwind2_path_d_dx
+ /2 width=3 by trans_eq/
+| <unwind2_path_L_dx <unwind2_path_L_dx //
+| <unwind2_path_A_dx <unwind2_path_A_dx //
+| <unwind2_path_S_dx <unwind2_path_S_dx //
+]
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma unwind2_path_d_empty (f) (k):
+ (𝗱(f＠⧣❨k❩)◗𝐞) = ▼[f](𝗱k◗𝐞).
+// qed.
+
+lemma unwind2_path_d_lcons (f) (p) (l) (k:pnat):
+ ▼[f∘𝐮❨k❩](l◗p) = ▼[f](𝗱k◗l◗p).
+#f #p #l #k <unwind2_path_append_ppc_dx in ⊢ (???%); //
+qed.
+
+lemma unwind2_path_m_sn (f) (p):
+ ▼[f]p = ▼[f](𝗺◗p).
+#f #p <unwind2_path_append_pic_sn //
+qed.
+
+lemma unwind2_path_L_sn (f) (p):
+ (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
+#f #p <unwind2_path_append_pic_sn //
+qed.
+
+lemma unwind2_path_A_sn (f) (p):
+ (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
+#f #p <unwind2_path_append_pic_sn //
+qed.
+
+lemma unwind2_path_S_sn (f) (p):
+ (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
+#f #p <unwind2_path_append_pic_sn //
+qed.
+
+(* Destructions with pic ****************************************************)
+
+lemma unwind2_path_des_pic (f) (p):
+ ▼[f]p ϵ 𝐈 → p ϵ 𝐈.
+#f * // * [ #k ] #p //
+<unwind2_path_d_dx #H0
+elim (pic_inv_d_dx … H0)
+qed-.
+
+(* Destructions with append and pic *****************************************)
+
+lemma unwind2_path_des_append_pic_sn (f) (p) (q1) (q2):
+ q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. q1 = ⊗p1 & q2 = ▼[▶[f]p1]p2 & p1●p2 = p.
+#f #p #q1 * [| * [ #k ] #q2 ] #Hq1
+[ <list_append_empty_sn #H0 destruct
+ lapply (unwind2_path_des_pic … Hq1) -Hq1 #Hp
+ <(unwind2_path_pic … Hp) -Hp
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_d_dx_unwind2_path … H0) -H0 #r #h #Hr #H1 #H2 destruct
+ elim (eq_inv_append_structure … Hr) -Hr #s1 #s2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_m_dx_unwind2_path … H0)
+| #H0 elim (eq_inv_L_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗟◗r2)) //
+ <unwind2_path_append_ppc_dx //
+ <unwind2_path_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗔◗r2)) //
+ <unwind2_path_append_ppc_dx //
+ <unwind2_path_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗦◗r2)) //
+ <unwind2_path_append_ppc_dx //
+ <unwind2_path_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with append and ppc *******************************************)
+
+lemma unwind2_path_inv_append_ppc_dx (f) (p) (q1) (q2):
+ q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
+ ∃∃p1,p2. q1 = ⊗p1 & q2 = ▼[▶[f]p1]p2 & p1●p2 = p.
+#f #p #q1 * [| * [ #k ] #q2 ] #Hq1
+[ <list_append_empty_sn #H0 destruct
+ elim (ppc_inv_empty … Hq1)
+| #H0 elim (eq_inv_d_dx_unwind2_path … H0) -H0 #r #h #Hr #H1 #H2 destruct
+ elim (eq_inv_append_structure … Hr) -Hr #s1 #s2 #H1 #H2 #H3 destruct
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_m_dx_unwind2_path … H0)
+| #H0 elim (eq_inv_L_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗟◗r2)) //
+ <unwind2_path_append_ppc_dx //
+ <unwind2_path_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗔◗r2)) //
+ <unwind2_path_append_ppc_dx //
+ <unwind2_path_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (eq_inv_append_structure … Hr1) -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗦◗r2)) //
+ <unwind2_path_append_ppc_dx //
+ <unwind2_path_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with path_lcons ***********************************************)
+
+lemma eq_inv_d_sn_unwind2_path (f) (q) (p) (k):
+ (𝗱k◗q) = ▼[f]p →
+ ∃∃r,h. 𝐞 = ⊗r & ▶[f]r＠⧣❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
+#f * [| #l #q ] #p #k
+[ <list_cons_comm #H0
+ elim (eq_inv_d_dx_unwind2_path … H0) -H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+ /2 width=5 by ex4_2_intro/
+| >list_cons_comm #H0
+ elim (unwind2_path_inv_append_ppc_dx … H0) -H0 // #r1 #r2 #Hr1 #_ #_ -r2
+ elim (eq_inv_d_dx_structure … Hr1)
+]
+qed-.
+
+lemma eq_inv_m_sn_unwind2_path (f) (q) (p):
+ (𝗺◗q) = ▼[f]p → ⊥.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_pic_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_m_dx_structure … Hr1)
+qed-.
+
+lemma eq_inv_L_sn_unwind2_path (f) (q) (p):
+ (𝗟◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▶[f]r1]r2 & r1●𝗟◗r2 = p.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_pic_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_L_dx_structure … Hr1) -Hr1 #s1 #s2 #H1 #_ #H3 destruct
+<list_append_assoc in H0; <list_append_assoc
+<unwind2_path_append_ppc_dx //
+<unwind2_path_L_sn <H1 <list_append_empty_dx #H0
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H0 #_
+/2 width=5 by ex3_2_intro/
+qed-.
+
+lemma eq_inv_A_sn_unwind2_path (f) (q) (p):
+ (𝗔◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1]r2 & r1●𝗔◗r2 = p.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_pic_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_A_dx_structure … Hr1) -Hr1 #s1 #s2 #H1 #_ #H3 destruct
+<list_append_assoc in H0; <list_append_assoc
+<unwind2_path_append_ppc_dx //
+<unwind2_path_A_sn <H1 <list_append_empty_dx #H0
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H0 #_
+/2 width=5 by ex3_2_intro/
+qed-.
+
+lemma eq_inv_S_sn_unwind2_path (f) (q) (p):
+ (𝗦◗q) = ▼[f]p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1]r2 & r1●𝗦◗r2 = p.
+#f #q #p
+>list_cons_comm #H0
+elim (unwind2_path_des_append_pic_sn … H0) <list_cons_comm in H0; //
+#H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+elim (eq_inv_S_dx_structure … Hr1) -Hr1 #s1 #s2 #H1 #_ #H3 destruct
+<list_append_assoc in H0; <list_append_assoc
+<unwind2_path_append_ppc_dx //
+<unwind2_path_S_sn <H1 <list_append_empty_dx #H0
+elim (eq_inv_list_rcons_bi ????? H0) -H0 #H0 #_
+/2 width=5 by ex3_2_intro/
+qed-.
+
+(* Advanced eliminations with path ******************************************)
+
+lemma path_ind_unwind (Q:predicate …):
+ Q (𝐞) →
+ (∀k. Q (𝐞) → Q (𝗱k◗𝐞)) →
+ (∀k,l,p. Q (l◗p) → Q (𝗱k◗l◗p)) →
+ (∀p. Q p → Q (𝗺◗p)) →
+ (∀p. Q p → Q (𝗟◗p)) →
+ (∀p. Q p → Q (𝗔◗p)) →
+ (∀p. Q p → Q (𝗦◗p)) →
+ ∀p. Q p.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
+@(list_ind_rcons … p) -p // #p * [ #k ]
+[ @(list_ind_rcons … p) -p ]
+/2 width=1 by/
+qed-.
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