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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 "ground/notation/functions/uniform_1.ma".
16 include "ground/relocation/rtmap_id.ma".
17 include "ground/relocation/rtmap_isuni.ma".
19 (* RELOCATION MAP ***********************************************************)
21 rec definition uni (n:nat) on n: rtmap ≝ match n with
26 interpretation "uniform relocation (rtmap)"
29 (* Basic properties *********************************************************)
31 lemma uni_zero: 𝐈𝐝 = 𝐔❨0❩.
34 lemma uni_succ: ∀n. ↑𝐔❨n❩ = 𝐔❨↑n❩.
37 (* Basic inversion lemmas ***************************************************)
39 lemma uni_inv_push_dx: ∀f,n. 𝐔❨n❩ ≡ ⫯f → 0 = n ∧ 𝐈𝐝 ≡ f.
40 #f * /3 width=5 by eq_inv_pp, conj/
41 #n <uni_succ #H elim (eq_inv_np … H) -H //
44 lemma uni_inv_push_sn: ∀f,n. ⫯f ≡ 𝐔❨n❩ → 0 = n ∧ 𝐈𝐝 ≡ f.
45 /3 width=1 by uni_inv_push_dx, eq_sym/ qed-.
47 lemma uni_inv_id_dx: ∀n. 𝐔❨n❩ ≡ 𝐈𝐝 → 0 = n.
48 #n <id_rew #H elim (uni_inv_push_dx … H) -H //
51 lemma uni_inv_id_sn: ∀n. 𝐈𝐝 ≡ 𝐔❨n❩ → 0 = n.
52 /3 width=1 by uni_inv_id_dx, eq_sym/ qed-.
54 lemma uni_inv_next_dx: ∀f,n. 𝐔❨n❩ ≡ ↑f → ∃∃m. 𝐔❨m❩ ≡ f & ↑m = n.
56 [ <uni_zero <id_rew #H elim (eq_inv_pn … H) -H //
57 | #n <uni_succ /3 width=5 by eq_inv_nn, ex2_intro/
61 lemma uni_inv_next_sn: ∀f,n. ↑f ≡ 𝐔❨n❩ → ∃∃m. 𝐔❨m❩ ≡ f & ↑m = n.
62 /3 width=1 by uni_inv_next_dx, eq_sym/ qed-.
64 (* Properties with test for identity ****************************************)
66 lemma uni_isid: ∀f. 𝐈❪f❫ → 𝐔❨0❩ ≡ f.
67 /2 width=1 by eq_id_inv_isid/ qed-.
69 (* Inversion lemmas with test for identity **********************************)
71 lemma uni_inv_isid: ∀f. 𝐔❨0❩ ≡ f → 𝐈❪f❫.
72 /2 width=1 by eq_id_isid/ qed-.
74 (* Properties with finite colength assignment ***************************)
76 lemma fcla_uni: ∀n. 𝐂❪𝐔❨n❩❫ ≘ n.
77 #n elim n -n /2 width=1 by fcla_isid, fcla_next/
80 (* Properties with test for finite colength ***************************)
82 lemma isfin_uni: ∀n. 𝐅❪𝐔❨n❩❫.
83 /3 width=2 by ex_intro/ qed.
85 (* Properties with test for uniformity **************************************)
87 lemma isuni_uni: ∀n. 𝐔❪𝐔❨n❩❫.
88 #n elim n -n /3 width=3 by isuni_isid, isuni_next/
91 lemma uni_isuni: ∀f. 𝐔❪f❫ → ∃n. 𝐔❨n❩ ≡ f.
92 #f #H elim H -f /3 width=2 by uni_isid, ex_intro/
93 #f #_ #g #H * /3 width=6 by eq_next, ex_intro/
96 (* Inversion lemmas with test for uniformity ********************************)
98 lemma uni_inv_isuni: ∀n,f. 𝐔❨n❩ ≡ f → 𝐔❪f❫.
99 #n elim n -n /3 width=1 by uni_inv_isid, isuni_isid/
100 #n #IH #x <uni_succ #H elim (eq_inv_nx … H) -H /3 width=3 by isuni_next/