include "ground/relocation/xap.ma".
(* LIFT FOR PATH ************************************************************)
(* Constructions with head for path *****************************************)
include "ground/relocation/xap.ma".
(* LIFT FOR PATH ************************************************************)
(* Constructions with head for path *****************************************)
-lemma lift_path_head (f) (p) (q) (n):
- pᴿ = ↳[n](pᴿ) →
- ↳[↑[q●p]f@❨n❩](↑[↑[q]f]p)ᴿ = (↑[↑[q]f]p)ᴿ.
-#f #p @(list_ind_rcons … p) -p
-[ #q #n #H0
+lemma lift_path_head_closed (f) (p) (q) (n):
+ q = ↳[n]q →
+ ↳[↑[p●q]f@❨n❩]↑[↑[p]f]q = ↑[↑[p]f]q.
+#f #p #q elim q -q
+[ #n #H0
-| #p #l #IH #q #n @(nat_ind_succ …n) -n [| #m #_ ]
- [ <reverse_rcons <path_head_zero #H0 destruct
- | cases l [ #n ]
- [ <reverse_rcons <path_head_d_sn #H0
+| #l #q #IH #n @(nat_ind_succ …n) -n [| #n #_ ]
+ [ <path_head_zero #H0 destruct
+ | cases l [ #k ]
+ [ <path_head_d_dx #H0