precommit for rtmap ...

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

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isid.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isid.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/relations/isidentity_1.ma".
+include "ground_2/relocation/rtmap_eq.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+coinductive isid: predicate rtmap ≝
+| isid_push: ∀f,g. isid f → ↑f = g → isid g
+.
+
+interpretation "test for identity (rtmap)"
+   'IsIdentity f = (isid f).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma isid_inv_gen: ∀g. 𝐈⦃g⦄ → ∃∃f. 𝐈⦃f⦄ & ↑f = g.
+#g * -g
+#f #g #Hf * /2 width=3 by ex2_intro/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma isid_inv_push: ∀g. 𝐈⦃g⦄ → ∀f. ↑f = g → 𝐈⦃f⦄.
+#g #H elim (isid_inv_gen … H) -H
+#f #Hf * -g #g #H >(injective_push … H) -H //
+qed-.
+
+lemma isid_inv_next: ∀g. 𝐈⦃g⦄ → ∀f. ⫯f = g → ⊥.
+#g #H elim (isid_inv_gen … H) -H
+#f #Hf * -g #g #H elim (discr_next_push … H)
+qed-.
+
+let corec isid_inv_eq_repl: ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → f1 ≗ f2 ≝ ?.
+#f1 #f2 #H1 #H2
+cases (isid_inv_gen … H1) -H1
+cases (isid_inv_gen … H2) -H2
+/3 width=5 by eq_push/
+qed-.
+
+(* Basic properties *********************************************************)
+
+let corec isid_eq_repl_back: eq_repl_back … isid ≝ ?.
+#f1 #H cases (isid_inv_gen … H) -H
+#g1 #Hg1 #H1 #f2 #Hf cases (eq_inv_px … Hf … H1) -f1
+/3 width=3 by isid_push/
+qed-.
+
+lemma isid_eq_repl_fwd: eq_repl_fwd … isid.
+/3 width=3 by isid_eq_repl_back, eq_repl_sym/ qed-.
+
+(* Alternative definition ***************************************************)
+
+let corec eq_push_isid: ∀f. ↑f ≗ f → 𝐈⦃f⦄ ≝ ?.
+#f #H cases (eq_inv_px … H) -H /4 width=3 by isid_push, eq_trans/
+qed.
+
+let corec eq_push_inv_isid: ∀f. 𝐈⦃f⦄ → ↑f ≗ f ≝ ?.
+#f * -f
+#f #g #Hf #Hg @(eq_push … Hg) [2: @eq_push_inv_isid // | skip ]
+@eq_f //
+qed-.