(**************************************************************************)
include "delayed_updating/unwind/unwind2_rmap_labels.ma".
-include "delayed_updating/unwind/nap.ma".
+include "delayed_updating/unwind/unwind2_rmap_eq.ma".
include "delayed_updating/syntax/path_head_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
+include "ground/relocation/xap.ma".
+include "ground/lib/stream_eq_eq.ma".
+include "ground/arith/nat_le_plus.ma".
-(* UNWIND MAP FOR PATH ******************************************************)
+(* TAILED UNWIND FOR RELOCATION MAP *****************************************)
(* Constructions with path_head *********************************************)
-lemma unwind2_rmap_head_push (f) (p) (n):
- n + ♯(↳[n]p) = (▶[⫯f]↳[n]p)@↑❨n❩.
+lemma unwind2_rmap_head_xap_le_closed (f) (p) (q) (n) (m):
+ q = ↳[n]q → m ≤ n →
+ ▶[f](p●q)＠❨m❩ = ▶[f]↳[n](p●q)＠❨m❩.
+#f #p #q elim q -q
+[ #n #m #Hq
+ <(eq_inv_path_empty_head … Hq) -n #Hm
+ <(nle_inv_zero_dx … Hm) -m //
+| #l #q #IH #n @(nat_ind_succ … n) -n
+ [ #m #_ #Hm <(nle_inv_zero_dx … Hm) -m -IH //
+ | #n #_ #m cases l [ #k ]
+ [ <path_head_d_dx #Hq #Hm
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_d_dx <unwind2_rmap_d_dx
+ <tr_compose_xap <tr_compose_xap
+ @(IH … Hq) -IH -Hq (**) (* auto too slow *)
+ @nle_trans [| @tr_uni_xap ]
+ /2 width=1 by nle_plus_bi_dx/
+ | <path_head_m_dx #Hq #Hm
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_m_dx <unwind2_rmap_m_dx
+ /2 width=2 by/
+ | <path_head_L_dx #Hq
+ @(nat_ind_succ … m) -m // #m #_ #Hm
+ lapply (nle_inv_succ_bi … Hm) -Hm #Hm
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_L_dx <unwind2_rmap_L_dx
+ <tr_xap_push <tr_xap_push
+ /3 width=2 by eq_f/
+ | <path_head_A_dx #Hq #Hm
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_A_dx <unwind2_rmap_A_dx
+ /2 width=2 by/
+ | <path_head_S_dx #Hq #Hm
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_S_dx <unwind2_rmap_S_dx
+ /2 width=2 by/
+ ]
+ ]
+]
+qed-.
+
+lemma unwind2_rmap_head_xap_closed (f) (p) (q) (n):
+ q = ↳[n]q →
+ ▶[f](p●q)＠❨n❩ = ▶[f]↳[n](p●q)＠❨n❩.
+/2 width=2 by unwind2_rmap_head_xap_le_closed/
+qed-.
+
+lemma unwind2_rmap_head_xap (f) (p) (n):
+ n + ♯(↳[n]p) = (▶[f]↳[n]p)＠❨n❩.
#f #p elim p -p
[ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
- >tr_pushs_swap <tr_nap_pushs_lt //
+ <tr_xap_pushs_le //
| #l #p #IH #n @(nat_ind_succ … n) -n //
- #n #_ cases l [ #m ]
- [ <unwind2_rmap_d_sn <path_head_d_sn <height_d_sn
- <nplus_assoc >IH -IH <tr_compose_nap //
- | <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
- | <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
- <tr_nap_push <npred_succ //
- | <unwind2_rmap_A_sn <path_head_A_sn <height_A_sn //
- | <unwind2_rmap_S_sn <path_head_S_sn <height_S_sn //
+ #n #_ cases l [ #k ]
+ [ <unwind2_rmap_d_dx <path_head_d_dx <height_d_dx
+ <nplus_comm in ⊢ (??(??%)?); <nplus_assoc
+ >IH -IH <tr_compose_xap <tr_uni_xap_succ //
+ | <unwind2_rmap_m_dx <path_head_m_dx <height_m_dx //
+ | <unwind2_rmap_L_dx <path_head_L_dx <height_L_dx
+ <tr_xap_push <npred_succ <nplus_succ_sn //
+ | <unwind2_rmap_A_dx <path_head_A_dx <height_A_dx //
+ | <unwind2_rmap_S_dx <path_head_S_dx <height_S_dx //
]
]
qed.
+
+lemma unwind2_rmap_append_pap_closed (f) (p) (q) (h:pnat):
+ q = ↳[h]q →
+ ♭q = ninj (▶[f](p●q)＠⧣❨h❩).
+#f #p #q #h #Hh
+>tr_xap_ninj >(path_head_refl_append_sn p … Hh) in ⊢ (??%?);
+>(unwind2_rmap_head_xap_closed … Hh) -Hh
+<path_head_depth //
+qed.
+
+lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (m):
+ q = ↳[n]q →
+ ⇂*[m]▶[f]p ≗ ⇂*[n+m]▶[f](p●q).
+#f #p #q elim q -q
+[ #n #m #Hq
+ <(eq_inv_path_empty_head … Hq) -n //
+| #l #q #IH #n @(nat_ind_succ … n) -n //
+ #n #_ #m cases l [ #k ]
+ [ <path_head_d_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
+ @(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
+ >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
+ /2 width=1 by/
+ | <path_head_m_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_m_sn /2 width=1 by/
+ | <path_head_L_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_L_dx <nplus_succ_sn /2 width=1 by/
+ | <path_head_A_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_A_dx /2 width=2 by/
+ | <path_head_S_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_S_dx /2 width=2 by/
+ ]
+]
+qed-.
+
+lemma tls_unwind2_rmap_closed (f) (p) (q) (n):
+ q = ↳[n]q →
+ ▶[f]p ≗ ⇂*[n]▶[f](p●q).
+/2 width=1 by tls_unwind2_rmap_plus_closed/
+qed.
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