- advances on lfxs for lfpxs

[helm.git] / matita / matita / contribs / lambdadelta / apps_2 / examples / ex_cpr_omega.ma

diff --git a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cpr_omega.ma b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cpr_omega.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 "basic_2/rt_transition/cpr.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* A reduction cycle in two steps: the term Omega ***************************)
+
+definition Delta: term → term ≝ λW. +ⓛW.ⓐ#0.#0.
+
+definition Omega1: term → term ≝ λW. ⓐ(Delta W).(Delta W).
+
+definition Omega2: term → term ≝ λW. +ⓓⓝW.(Delta W).ⓐ#0.#0.
+
+(* Basic properties *********************************************************)
+
+lemma Delta_lifts: ∀W1,W2,f. ⬆*[f] W1 ≡ W2 →
+                   ⬆*[f] (Delta W1) ≡ (Delta W2).
+/4 width=1 by lifts_lref, lifts_bind, lifts_flat/ qed.
+
+(* Main properties **********************************************************)
+
+theorem cpr_Omega_12: ∀h,G,L,W. ⦃G, L⦄ ⊢ Omega1 W ➡[h] Omega2 W.
+/2 width=1 by cpm_beta/ qed.
+
+theorem cpr_Omega_21: ∀h,G,L,W. ⦃G, L⦄ ⊢ Omega2 W ➡[h] Omega1 W.
+#h #G #L #W1 elim (lifts_total W1 (𝐔❴1❵))
+/5 width=5 by lifts_flat, cpm_zeta, cpm_eps, cpm_appl, cpm_delta, Delta_lifts/
+qed.