ground_2 released and permanently renamed as ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / rtmap_isuni.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/notation/relations/isuniform_1.ma".
+include "ground/relocation/rtmap_isfin.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+inductive isuni: predicate rtmap ≝
+| isuni_isid: ∀f. 𝐈❪f❫ → isuni f
+| isuni_next: ∀f. isuni f → ∀g. ↑f = g → isuni g
+.
+
+interpretation "test for uniformity (rtmap)"
+   'IsUniform f = (isuni f).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma isuni_inv_push: ∀g. 𝐔❪g❫ → ∀f. ⫯f = g → 𝐈❪f❫.
+#g * -g /2 width=3 by isid_inv_push/
+#f #_ #g #H #x #Hx destruct elim (discr_push_next … Hx)
+qed-.
+
+lemma isuni_inv_next: ∀g. 𝐔❪g❫ → ∀f. ↑f = g → 𝐔❪f❫.
+#g * -g #f #Hf
+[ #x #Hx elim (isid_inv_next … Hf … Hx)
+| #g #H #x #Hx destruct /2 width=1 by injective_push/
+]
+qed-.
+
+lemma isuni_split: ∀g. 𝐔❪g❫ → (∃∃f. 𝐈❪f❫ & ⫯f = g) ∨ (∃∃f.𝐔❪f❫ & ↑f = g).
+#g #H elim (pn_split g) * #f #Hf
+/4 width=3 by isuni_inv_next, isuni_inv_push, or_introl, or_intror, ex2_intro/
+qed-.
+
+(* basic forward lemmas *****************************************************)
+
+lemma isuni_fwd_push: ∀g. 𝐔❪g❫ → ∀f. ⫯f = g → 𝐔❪f❫.
+/3 width=3 by isuni_inv_push, isuni_isid/ qed-.
+
+(* Forward lemmas with test for finite colength *****************************)
+
+lemma isuni_fwd_isfin: ∀f. 𝐔❪f❫ → 𝐅❪f❫.
+#f #H elim H -f /3 width=1 by isfin_next, isfin_isid/
+qed-.