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/arith/nat_pred_succ.ma".
16 include "ground/relocation/pr_tl_eq.ma".
17 include "ground/relocation/pr_uni.ma".
19 (* UNIFORM ELEMENTS FOR PARTIAL RELOCATION MAPS *****************************)
21 (* Inversions with pr_eq ****************************************************)
23 (*** uni_inv_push_dx *)
24 lemma pr_eq_inv_uni_push (n) (g): š®āØnā© ā ā«Æg ā ā§ā§ š = n & š¢ ā g.
25 #n @(nat_ind_succ ā¦ n) -n
26 [ /3 width=5 by pr_eq_inv_push_bi, conj/
27 | #n #_ #f <pr_uni_succ #H elim (pr_eq_inv_next_push ā¦ H) -H //
31 (*** uni_inv_push_sn *)
32 lemma pr_eq_inv_push_uni (n) (g): ā«Æg ā š®āØnā© ā ā§ā§ š = n & š¢ ā g.
33 /3 width=1 by pr_eq_inv_uni_push, pr_eq_sym/ qed-.
35 (*** uni_inv_next_dx *)
36 lemma pr_eq_inv_uni_next (n) (g): š®āØnā© ā āg ā ā§ā§ š®āØānā© ā g & āān = n.
37 #n @(nat_ind_succ ā¦ n) -n
38 [ #g <pr_uni_zero <pr_id_unfold #H elim (pr_eq_inv_push_next ā¦ H) -H //
39 | #n #_ #g <pr_uni_succ /3 width=5 by pr_eq_inv_next_bi, conj/
43 (*** uni_inv_next_sn *)
44 lemma pr_eq_inv_next_uni (n) (g): āg ā š®āØnā© ā ā§ā§ š®āØānā© ā g & āān = n.
45 /3 width=1 by pr_eq_inv_uni_next, pr_eq_sym/ qed-.
47 (* Inversions with pr_id and pr_eq ******************************************)
50 lemma pr_eq_inv_uni_id (n): š®āØnā© ā š¢ ā š = n.
51 #n <pr_id_unfold #H elim (pr_eq_inv_uni_push ā¦ H) -H //
55 lemma pr_eq_inv_id_uni (n): š¢ ā š®āØnā© ā š = n.
56 /3 width=1 by pr_eq_inv_uni_id, pr_eq_sym/ qed-.