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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "delayed_updating/unwind/unwind2_rmap_labels.ma".
16 include "delayed_updating/unwind/unwind2_rmap_eq.ma".
17 include "delayed_updating/unwind/xap.ma".
18 include "delayed_updating/syntax/path_head_depth.ma".
19 include "ground/lib/stream_eq_eq.ma".
20 include "ground/arith/nat_le_plus.ma".
22 (* UNWIND MAP FOR PATH ******************************************************)
24 (* Constructions with path_head *********************************************)
26 lemma unwind2_rmap_head_xap_le_closed (f) (p) (q) (n) (k):
27 p = (↳[n]p)●q → k ≤ n →
28 (▶[f]p)@❨k❩ = (▶[f]↳[n]p)@❨k❩.
31 elim (eq_inv_list_empty_append … Hq) -Hq * #_ //
32 | #l #p #IH #q #n @(nat_ind_succ … n) -n
33 [ #k #_ #Hk <(nle_inv_zero_dx … Hk) -k -IH
34 <path_head_zero <unwind2_rmap_empty //
35 | #n #_ #k cases l [ #m ]
36 [ <path_head_d_sn <list_append_lcons_sn #Hq #Hk
37 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
38 <unwind2_rmap_d_sn <unwind2_rmap_d_sn
39 <tr_compose_xap <tr_compose_xap
40 @(IH … Hq) -IH -Hq (**) (* auto too slow *)
41 @nle_trans [| @tr_uni_xap ]
42 /2 width=1 by nle_plus_bi_dx/
43 | <path_head_m_sn <list_append_lcons_sn #Hq #Hk
44 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
45 <unwind2_rmap_m_sn <unwind2_rmap_m_sn
47 | <path_head_L_sn <list_append_lcons_sn #Hq
48 @(nat_ind_succ … k) -k // #k #_ #Hk
49 lapply (nle_inv_succ_bi … Hk) -Hk #Hk
50 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
51 <unwind2_rmap_L_sn <unwind2_rmap_L_sn
52 <tr_xap_push <tr_xap_push
54 | <path_head_A_sn <list_append_lcons_sn #Hq #Hk
55 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
56 <unwind2_rmap_A_sn <unwind2_rmap_A_sn
58 | <path_head_S_sn <list_append_lcons_sn #Hq #Hk
59 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
60 <unwind2_rmap_S_sn <unwind2_rmap_S_sn
67 lemma unwind2_rmap_head_append_xap_closed (f) (p) (q) (n):
69 ▶[f](p●q)@❨n❩ = ▶[f]↳[n](p●q)@❨n❩.
70 /2 width=2 by unwind2_rmap_head_xap_le_closed/
73 lemma unwind2_rmap_head_xap (f) (p) (n):
74 n + ♯(↳[n]p) = (▶[f]↳[n]p)@❨n❩.
76 [ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
78 | #l #p #IH #n @(nat_ind_succ … n) -n //
80 [ <unwind2_rmap_d_sn <path_head_d_sn <height_d_sn
81 <nplus_assoc >IH -IH <tr_compose_xap <tr_uni_xap_succ //
82 | <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
83 | <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
84 <tr_xap_push <npred_succ //
85 | <unwind2_rmap_A_sn <path_head_A_sn <height_A_sn //
86 | <unwind2_rmap_S_sn <path_head_S_sn <height_S_sn //
91 lemma unwind2_rmap_append_pap_closed (f) (p) (q) (n:pnat):
93 ♭p = ninj (▶[f](p●q)@⧣❨n❩).
95 >tr_xap_ninj >Hn in ⊢ (??%?);
96 >(unwind2_rmap_head_append_xap_closed … Hn) -Hn //
99 lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (k):
101 ⇂*[k]▶[f]q ≗ ⇂*[n+k]▶[f]p.
104 elim (eq_inv_list_empty_append … Hq) -Hq #Hn #H0 destruct
105 <path_head_empty in Hn; #Hn
106 <(eq_inv_empty_labels … Hn) -n //
107 | #l #p #IH #q #n @(nat_ind_succ … n) -n //
108 #n #_ #k cases l [ #m ]
109 [ <path_head_d_sn <list_append_lcons_sn #Hq
110 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
111 @(stream_eq_trans … (tls_unwind2_rmap_d_sn …))
112 >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
114 | <path_head_m_sn <list_append_lcons_sn #Hq
115 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
116 <unwind2_rmap_m_sn /2 width=1 by/
117 | <path_head_L_sn <list_append_lcons_sn #Hq
118 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
119 <unwind2_rmap_L_sn <nplus_succ_sn /2 width=1 by/
120 | <path_head_A_sn <list_append_lcons_sn #Hq
121 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
122 <unwind2_rmap_A_sn /2 width=2 by/
123 | <path_head_S_sn <list_append_lcons_sn #Hq
124 elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
125 <unwind2_rmap_S_sn /2 width=2 by/
130 lemma tls_unwind2_rmap_append_closed (f) (p) (q) (n):
132 ▶[f]q ≗ ⇂*[n]▶[f](p●q).
133 /2 width=1 by tls_unwind2_rmap_plus_closed/