]> matita.cs.unibo.it Git - helm.git/blob - matita/matita/contribs/lambdadelta/basic_2A/examples/ex_fpbg_refl.ma
update in delayed_updating
[helm.git] / matita / matita / contribs / lambdadelta / basic_2A / examples / ex_fpbg_refl.ma
1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 include "basic_2A/computation/fpbg_fpbs.ma".
16
17 (* EXAMPLES *****************************************************************)
18
19 (* Reflexivity of proper qrst-computation: the term ApplOmega ***************)
20
21 definition ApplDelta: term → nat → term ≝ λW,k. +ⓛW.ⓐ⋆k.ⓐ#0.#0.
22
23 definition ApplOmega1: term → nat → term ≝ λW,k. ⓐ(ApplDelta W k).(ApplDelta W k).
24
25 definition ApplOmega2: term → nat → term ≝ λW,k. +ⓓⓝW.(ApplDelta W k).ⓐ⋆k.ⓐ#0.#0.
26
27 definition ApplOmega3: term → nat → term ≝ λW,k. ⓐ⋆k.(ApplOmega1 W k).
28
29 (* Basic properties *********************************************************)
30
31 lemma ApplDelta_lift: ∀W1,W2,k,l,m. ⬆[l, m] W1 ≡ W2 →
32                       ⬆[l, m] (ApplDelta W1 k) ≡ (ApplDelta W2 k).
33 /5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
34
35 lemma cpr_ApplOmega_12: ∀G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega1 W k ➡ ApplOmega2 W k.
36 /2 width=1 by cpr_beta/ qed.
37
38 lemma cpr_ApplOmega_23: ∀G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega2 W k ➡ ApplOmega3 W k.
39 #G #L #W1 #k elim (lift_total W1 0 1) #W2 #HW12
40 @(cpr_zeta … (ApplOmega3 W2 k)) /4 width=1 by ApplDelta_lift, lift_flat/
41 @cpr_flat // @cpr_flat @(cpr_delta … (ApplDelta W1 k) ? 0)
42 [3,5,8,10: /2 width=2 by ApplDelta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
43 qed.
44
45 lemma cpxs_ApplOmega_13: ∀h,g,G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega1 W k ➡*[h,g] ApplOmega3 W k.
46 /4 width=3 by cpxs_strap1, cpr_cpx/ qed.
47
48 lemma fqup_ApplOmega_13: ∀G,L,W,k. ⦃G, L, ApplOmega3 W k⦄ ⊐+ ⦃G, L, ApplOmega1 W k⦄.
49 /2 width=1 by/ qed.
50
51 (* Main properties **********************************************************)
52
53 theorem fpbg_refl: ∀h,g,G,L,W,k. ⦃G, L, ApplOmega1 W k⦄ >≡[h,g] ⦃G, L, ApplOmega1 W k⦄.
54 /3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.