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_xap_le_closed (f) (p) (q) (n) (k):
- p = (↳[n]p)●q → k ≤ n →
- (▶[f]p)＠❨k❩ = (▶[f]↳[n]p)＠❨k❩.
-#f #p elim p -p
-[ #q #n #k #Hq #Hk
- elim (eq_inv_list_empty_append … Hq) -Hq * #_ //
-| #l #p #IH #q #n @(nat_ind_succ … n) -n
- [ #k #_ #Hk <(nle_inv_zero_dx … Hk) -k -IH
- <path_head_zero <unwind2_rmap_empty //
- | #n #_ #k cases l [ #m ]
- [ <path_head_d_sn <list_append_lcons_sn #Hq #Hk
+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_sn <unwind2_rmap_d_sn
+ <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_sn <list_append_lcons_sn #Hq #Hk
+ | <path_head_m_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_m_sn <unwind2_rmap_m_sn
+ <unwind2_rmap_m_dx <unwind2_rmap_m_dx
/2 width=2 by/
- | <path_head_L_sn <list_append_lcons_sn #Hq
- @(nat_ind_succ … k) -k // #k #_ #Hk
- lapply (nle_inv_succ_bi … Hk) -Hk #Hk
+ | <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_sn <unwind2_rmap_L_sn
+ <unwind2_rmap_L_dx <unwind2_rmap_L_dx
<tr_xap_push <tr_xap_push
/3 width=2 by eq_f/
- | <path_head_A_sn <list_append_lcons_sn #Hq #Hk
+ | <path_head_A_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_A_sn <unwind2_rmap_A_sn
+ <unwind2_rmap_A_dx <unwind2_rmap_A_dx
/2 width=2 by/
- | <path_head_S_sn <list_append_lcons_sn #Hq #Hk
+ | <path_head_S_dx #Hq #Hm
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_S_sn <unwind2_rmap_S_sn
+ <unwind2_rmap_S_dx <unwind2_rmap_S_dx
/2 width=2 by/
]
]
]
qed-.
-lemma unwind2_rmap_head_append_xap_closed (f) (p) (q) (n):
- p = ↳[n](p●q) →
+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-.
[ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
<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_xap <tr_uni_xap_succ //
- | <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
- | <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
- <tr_xap_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) (n:pnat):
- p = ↳[n](p●q) →
- ♭p = ninj (▶[f](p●q)＠⧣❨n❩).
-#f #p #q #n #Hn
->tr_xap_ninj >Hn in ⊢ (??%?);
->(unwind2_rmap_head_append_xap_closed … Hn) -Hn
+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) (k):
- p = (↳[n]p)●q →
- ⇂*[k]▶[f]q ≗ ⇂*[n+k]▶[f]p.
-#f #p elim p -p
-[ #q #n #k #Hq
- elim (eq_inv_list_empty_append … Hq) -Hq #Hn #H0 destruct
- <path_head_empty in Hn; #Hn
- <(eq_inv_empty_labels … Hn) -n //
-| #l #p #IH #q #n @(nat_ind_succ … n) -n //
- #n #_ #k cases l [ #m ]
- [ <path_head_d_sn <list_append_lcons_sn #Hq
+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_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_sn <list_append_lcons_sn #Hq
+ | <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_sn <list_append_lcons_sn #Hq
+ | <path_head_L_dx #Hq
elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <unwind2_rmap_L_sn <nplus_succ_sn /2 width=1 by/
- | <path_head_A_sn <list_append_lcons_sn #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_sn /2 width=2 by/
- | <path_head_S_sn <list_append_lcons_sn #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_sn /2 width=2 by/
+ <unwind2_rmap_S_dx /2 width=2 by/
]
]
qed-.
-lemma tls_unwind2_rmap_append_closed (f) (p) (q) (n):
- p = ↳[n](p●q) →
- ▶[f]q ≗ ⇂*[n]▶[f](p●q).
+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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