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 *)
13 (**************************************************************************)
15 include "ground/lib/subset_equivalence.ma".
16 include "delayed_updating/syntax/preterm.ma".
17 include "delayed_updating/substitution/lift_structure.ma".
18 include "delayed_updating/substitution/lift_prototerm.ma".
22 #p @(list_ind_rcons … p) -p
34 (* LIFT FOR PRETERM *********************************************************)
36 (* Constructions with subset_equivalence ************************************)
38 lemma lift_grafted_sn (f) (t) (p): p ϵ 𝐈 →
39 ↑[↑[p]f](t⋔p) ⊆ (↑[f]t)⋔(⊗p).
40 #f #t #p #Hp #q * #r #Hr #H0 destruct
42 <lift_append_inner_sn //
45 lemma lift_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
46 (↑[f]t)⋔(⊗p) ⊆ ↑[↑[p]f](t⋔p).
47 #f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
48 elim (lift_inv_append_inner_sn … (sym_eq … H0)) -H0 //
49 #p0 #q0 #Hp0 #Hq0 #H0 destruct
50 <(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
51 /2 width=1 by in_comp_lift_bi/
54 lemma lift_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
55 ↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(⊗p).
56 /3 width=1 by lift_grafted_sn, conj, lift_grafted_dx/ qed.
60 -lemma lift_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ ������ →
61 - (↑[f]t)⋔((⊗p)◖������) ⊆ ↑[↑[p]f](t⋔(p◖������)).
62 -#f #t #p #Hp #Ht #q * #r #Hr
63 -<list_append_rcons_sn #H0
64 -elim (lift_inv_append_proper_dx … (sym_eq … H0)) -H0 //
65 +lemma lift_grafted_dx (f) (t) (p): p ϵ ������ → p ϵ ▵t → t ϵ ������ →
66 + (↑[f]t)⋔(⊗p) ⊆ ↑[↑[p]f](t⋔p).
67 +#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
68 +elim (lift_inv_append_inner_sn … (sym_eq … H0)) -H0 //
69 #p0 #q0 #Hp0 #Hq0 #H0 destruct
70 <(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
71 -elim (lift_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
72 -#r1 #r2 #Hr1 #Hr2 #H0 destruct
73 -lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
74 -[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
75 /2 width=1 by in_comp_lift_bi/
78 -lemma lift_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ ������ →
79 - ↑[↑[p]f](t⋔(p◖������)) ⇔ (↑[f]t)⋔((⊗p)◖������).
80 -/3 width=1 by lift_grafted_S_sn, conj, lift_grafted_S_dx/ qed.
81 +lemma lift_grafted (f) (t) (p): p ϵ ������ → p ϵ ▵t → t ϵ ������ →
82 + ↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(⊗p).
83 +/3 width=1 by lift_grafted_sn, conj, lift_grafted_dx/ qed.