(* TAILED UNWIND FOR PATH ***************************************************)
-(* Constructions with inner condition for path ******************************)
+(* Constructions with pic ***************************************************)
-lemma unwind2_path_inner (f) (p):
+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 inner condition for path *******************)
+(* Constructions with append and pic ****************************************)
-lemma unwind2_path_append_inner_sn (f) (p) (q): p ϵ 𝐈 →
+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_inner … Hp) -Hp //
+[ <(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 //
]
qed.
-(* Constructions with append and proper condition for path ******************)
+(* Constructions with append and ppc ****************************************)
-lemma unwind2_path_append_proper_dx (f) (p) (q): q ϵ 𝐏 →
+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)
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_proper_dx in ⊢ (???%); //
+#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_inner_sn //
+#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_inner_sn //
+#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_inner_sn //
+#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_inner_sn //
+#f #p <unwind2_path_append_pic_sn //
qed.
-(* Destructions with inner condition for path *******************************)
+(* Destructions with pic ****************************************************)
-lemma unwind2_path_des_inner (f) (p):
+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 inner condition for path ********************)
+(* Destructions with append and pic *****************************************)
-lemma unwind2_path_des_append_inner_sn (f) (p) (q1) (q2):
+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_inner … Hq1) -Hq1 #Hp
- <(unwind2_path_inner … Hp) -Hp
+ 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
| #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_proper_dx //
+ <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_proper_dx //
+ <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_proper_dx //
+ <unwind2_path_append_ppc_dx //
<unwind2_path_S_sn <Hr2 -Hr2 //
]
qed-.
-(* Inversions with append and proper condition for path *********************)
+(* Inversions with append and ppc *******************************************)
-lemma unwind2_path_inv_append_proper_dx (f) (p) (q1) (q2):
+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
| #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_proper_dx //
+ <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_proper_dx //
+ <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_proper_dx //
+ <unwind2_path_append_ppc_dx //
<unwind2_path_S_sn <Hr2 -Hr2 //
]
qed-.
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_proper_dx … H0) -H0 // #r1 #r2 #Hr1 #_ #_ -r2
+ elim (unwind2_path_inv_append_ppc_dx … H0) -H0 // #r1 #r2 #Hr1 #_ #_ -r2
elim (eq_inv_d_dx_structure … Hr1)
]
qed-.
(𝗺◗q) = ▼[f]p → ⊥.
#f #q #p
>list_cons_comm #H0
-elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in 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-.
∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▶[f]r1]r2 & r1●𝗟◗r2 = p.
#f #q #p
>list_cons_comm #H0
-elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in 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_proper_dx //
+<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/
∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1]r2 & r1●𝗔◗r2 = p.
#f #q #p
>list_cons_comm #H0
-elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in 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_proper_dx //
+<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/
∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▶[f]r1]r2 & r1●𝗦◗r2 = p.
#f #q #p
>list_cons_comm #H0
-elim (unwind2_path_des_append_inner_sn … H0) <list_cons_comm in 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_proper_dx //
+<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/