(**************************************************************************)
include "delayed_updating/unwind/unwind2_prototerm_eq.ma".
-include "delayed_updating/unwind/unwind2_path_structure.ma".
+include "delayed_updating/unwind/unwind2_path_append.ma".
include "delayed_updating/substitution/fsubst.ma".
include "delayed_updating/syntax/preterm.ma".
include "delayed_updating/syntax/prototerm_proper.ma".
-(* UNWIND FOR PRETERM *******************************************************)
+(* TAILED UNWIND FOR PRETERM ************************************************)
-(* Constructions with fsubst ************************************************)
+(* Constructions with fsubst and pic ****************************************)
-lemma unwind2_term_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
- (▼[f]t)[⋔(⊗p)←▼[▶[f]pᴿ]u] ⊆ ▼[f](t[⋔p←u]).
+lemma unwind2_term_fsubst_pic_sn (f) (t) (u) (p): p ϵ 𝐈 →
+ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u] ⊆ ▼[f](t[⋔p←u]).
+#f #t #u #p #Hp #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind2_path_append_pic_sn
+ /4 width=3 by in_comp_unwind2_path_term, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro … H1) @or_intror @conj // *
+ [ <list_append_empty_sn #H2 destruct
+ elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ /3 width=2 by unwind2_path_append_pic_sn/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_pic_dx (f) (t) (u) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u].
+#f #t #u #p #Hp #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ /5 width=3 by unwind2_path_append_pic_sn, ex2_intro, or_introl/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
+ [ <list_append_empty_sn #Hr @(H0 … (𝐞)) -H0
+ <list_append_empty_sn @H2p -H2p
+ /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind2_path_inv_append_ppc_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_pic (f) (t) (u) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u] ⇔ ▼[f](t[⋔p←u]).
+/4 width=3 by unwind2_term_fsubst_pic_sn, conj, unwind2_term_fsubst_pic_dx/ qed.
+
+(* Constructions with fsubst and ppc ****************************************)
+
+lemma unwind2_term_fsubst_ppc_sn (f) (t) (u) (p): u ϵ 𝐏 →
+ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u] ⊆ ▼[f](t[⋔p←u]).
#f #t #u #p #Hu #ql * *
[ #rl * #r #Hr #H1 #H2 destruct
- >unwind2_path_append_proper_dx
+ >unwind2_path_append_ppc_dx
/4 width=5 by in_comp_unwind2_path_term, in_comp_tpc_trans, or_introl, ex2_intro/
| * #q #Hq #H1 #H0
@(ex2_intro … H1) @or_intror @conj // *
- [ <list_append_empty_dx #H2 destruct
+ [ <list_append_empty_sn #H2 destruct
elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
| #l #r #H2 destruct
- @H0 -H0 [| <unwind2_path_append_proper_dx /2 width=3 by ppc_lcons/ ]
+ @H0 -H0 [| <unwind2_path_append_ppc_dx /2 width=3 by ppc_rcons/ ]
]
]
qed-.
-lemma unwind2_term_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
- ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▶[f]pᴿ]u].
+lemma unwind2_term_fsubst_ppc_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+ ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u].
#f #t #u #p #Hu #H1p #H2p #ql * #q * *
[ #r #Hu #H1 #H2 destruct
@or_introl @ex2_intro
- [|| <unwind2_path_append_proper_dx /2 width=5 by in_comp_tpc_trans/ ]
+ [|| <unwind2_path_append_ppc_dx /2 width=5 by in_comp_tpc_trans/ ]
/2 width=3 by ex2_intro/
| #Hq #H0 #H1 destruct
@or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
- [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
- <list_append_empty_dx @H2p -H2p
- /2 width=2 by unwind_gen_des_structure, prototerm_in_comp_root/
+ [ <list_append_empty_sn #Hr @(H0 … (𝐞)) -H0
+ <list_append_empty_sn @H2p -H2p
+ /2 width=2 by unwind2_path_des_structure, prototerm_in_comp_root/
| #l #r #Hr
- elim (unwind2_path_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ elim (unwind2_path_inv_append_ppc_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
]
]
qed-.
-lemma unwind2_term_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
- (▼[f]t)[⋔(⊗p)←▼[▶[f]pᴿ]u] ⇔ ▼[f](t[⋔p←u]).
-/4 width=3 by unwind2_term_fsubst_sn, conj, unwind2_term_fsubst_dx/ qed.
+lemma unwind2_term_fsubst_ppc (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+ (▼[f]t)[⋔(⊗p)←▼[▶[f]p]u] ⇔ ▼[f](t[⋔p←u]).
+/4 width=3 by unwind2_term_fsubst_ppc_sn, conj, unwind2_term_fsubst_ppc_dx/ qed.