- uniform relocations

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_uni.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_uni.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_uni.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2/notation/functions/uniform_1.ma".
+include "ground_2/relocation/rtmap_id.ma".
+include "ground_2/relocation/rtmap_isuni.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+rec definition uni (n:nat) on n: rtmap  ≝ match n with
+[ O   ⇒ 𝐈𝐝
+| S n ⇒ ⫯(uni n)
+].
+
+interpretation "uniform relocation (rtmap)"
+   'Uniform n = (uni n).
+
+(* Basic properties *********************************************************)
+
+lemma uni_zero: 𝐈𝐝 = 𝐔❴0❵.
+// qed.
+
+lemma uni_succ: ∀n. ⫯𝐔❴n❵ = 𝐔❴⫯n❵.
+// qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma uni_inv_push_dx: ∀f,n. 𝐔❴n❵ ≗ ↑f → 0 = n ∧ 𝐈𝐝 ≗ f.
+#f * /3 width=5 by eq_inv_pp, conj/
+#n <uni_succ #H elim (eq_inv_np … H) -H //
+qed-.
+
+lemma uni_inv_push_sn: ∀f,n. ↑f ≗ 𝐔❴n❵ → 0 = n ∧ 𝐈𝐝 ≗ f.
+/3 width=1 by uni_inv_push_dx, eq_sym/ qed-.
+
+lemma uni_inv_id_dx: ∀n. 𝐔❴n❵ ≗ 𝐈𝐝 → 0 = n.
+#n <id_rew #H elim (uni_inv_push_dx … H) -H //
+qed-.
+
+lemma uni_inv_id_sn: ∀n.  𝐈𝐝 ≗ 𝐔❴n❵ → 0 = n.
+/3 width=1 by uni_inv_id_dx, eq_sym/ qed-.
+
+lemma uni_inv_next_dx: ∀f,n. 𝐔❴n❵ ≗ ⫯f → ∃∃m. 𝐔❴m❵ ≗ f & ⫯m = n.
+#f *
+[ <uni_zero <id_rew #H elim (eq_inv_pn … H) -H //
+| #n <uni_succ /3 width=5 by eq_inv_nn, ex2_intro/
+]
+qed-.
+
+lemma uni_inv_next_sn: ∀f,n. ⫯f ≗ 𝐔❴n❵ → ∃∃m. 𝐔❴m❵ ≗ f & ⫯m = n.
+/3 width=1 by uni_inv_next_dx, eq_sym/ qed-.
+
+(* Properties with test for identity ****************************************)
+
+lemma uni_isid: ∀f. 𝐈⦃f⦄ → 𝐔❴0❵ ≗ f.
+/2 width=1 by eq_id_inv_isid/ qed-.
+
+(* Inversion lemmas with test for identity **********************************)
+
+lemma uni_inv_isid: ∀f. 𝐔❴0❵ ≗ f → 𝐈⦃f⦄.
+/2 width=1 by eq_id_isid/ qed-.
+
+(* Properties with finite colength assignment ***************************)
+
+lemma fcla_uni: ∀n. 𝐂⦃𝐔❴n❵⦄ ≡ n.
+#n elim n -n /2 width=1 by fcla_isid, fcla_next/
+qed.
+
+(* Properties with test for finite colength ***************************)
+
+lemma isfin_uni: ∀n. 𝐅⦃𝐔❴n❵⦄.
+/3 width=2 by ex_intro/ qed.
+
+(* Properties with test for uniformity **************************************)
+
+lemma isuni_uni: ∀n. 𝐔⦃𝐔❴n❵⦄.
+#n elim n -n /3 width=3 by isuni_isid, isuni_next/
+qed.
+
+lemma uni_isuni: ∀f. 𝐔⦃f⦄ → ∃n. 𝐔❴n❵ ≗ f.
+#f #H elim H -f /3 width=2 by uni_isid, ex_intro/
+#f #_ #g #H * /3 width=6 by eq_next, ex_intro/
+qed-.
+
+(* Inversion lemmas with test for uniformity ********************************)
+
+lemma uni_inv_isuni: ∀n,f. 𝐔❴n❵ ≗ f →  𝐔⦃f⦄.
+#n elim n -n /3 width=1 by uni_inv_isid, isuni_isid/
+#n #IH #x <uni_succ #H elim (eq_inv_nx … H) -H /3 width=3 by isuni_next/
+qed-.