+++ /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/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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