1 (**************************************************************************)
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/substitution/lift_path.ma".
16 include "delayed_updating/syntax/path_head.ma".
17 include "ground/relocation/xap.ma".
19 (* LIFT FOR PATH ************************************************************)
21 (* Constructions with head for path *****************************************)
23 lemma lift_path_head_closed (f) (p) (q) (n):
25 ↳[↑[p●q]f@❨n❩]↑[↑[p]f]q = ↑[↑[p]f]q.
28 <(eq_inv_path_empty_head … H0) -H0
30 | #l #q #IH #n @(nat_ind_succ …n) -n [| #n #_ ]
31 [ <path_head_zero #H0 destruct
34 elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
35 <lift_rmap_d_dx <lift_path_d_dx
36 <tr_xap_succ_nap <path_head_d_dx
37 <(IH … H0) in ⊢ (???%); -IH -H0 <tr_xap_plus //
39 elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
40 <lift_rmap_m_dx <lift_path_m_dx
41 <tr_xap_succ_nap <path_head_m_dx
42 <(IH … H0) in ⊢ (???%); -IH -H0 //
44 elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
45 <lift_rmap_L_dx <lift_path_L_dx
46 <tr_xap_succ_nap <path_head_L_dx
47 <(IH … H0) in ⊢ (???%); -IH -H0 //
49 elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
50 <lift_rmap_A_dx <lift_path_A_dx
51 <tr_xap_succ_nap <path_head_A_dx
52 <(IH … H0) in ⊢ (???%); -IH -H0 //
54 elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
55 <lift_rmap_S_dx <lift_path_S_dx
56 <tr_xap_succ_nap <path_head_S_dx
57 <(IH … H0) in ⊢ (???%); -IH -H0 //