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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/rt_computation/fpbg_fqus.ma".
16 include "basic_2/rt_computation/fpbs_cpxs.ma".
17 include "basic_2/rt_computation/fpbg_fpbs.ma".
19 (* EXAMPLES *****************************************************************)
21 (* Reflexivity of proper rst-computation: the term ApplOmega ****************)
23 definition ApplDelta (s0) (s): term ≝ +ⓛ⋆s0.ⓐ⋆s.ⓐ#0.#0.
25 definition ApplOmega1 (s0) (s): term ≝ ⓐ(ApplDelta s0 s).(ApplDelta s0 s).
27 definition ApplOmega2 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.ⓐ#0.#0.
29 definition ApplOmega3 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.(ApplOmega1 s0 s).
31 definition ApplOmega4 (s0) (s): term ≝ ⓐ⋆s.(ApplOmega1 s0 s).
33 (* Basic properties *********************************************************)
35 lemma ApplDelta_lifts (f) (s0) (s):
36 ⇧*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s).
37 /5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
39 lemma cpr_ApplOmega_12 (G) (L) (s0) (s):
40 ❪G,L❫ ⊢ ApplOmega1 s0 s ⬈ ApplOmega2 s0 s.
41 /2 width=1 by cpx_beta/ qed.
43 lemma cpr_ApplOmega_23 (G) (L) (s0) (s):
44 ❪G,L❫ ⊢ ApplOmega2 s0 s ⬈ ApplOmega3 s0 s.
45 /6 width=3 by cpx_eps, cpx_flat, cpx_bind, cpx_delta, ApplDelta_lifts/ qed.
47 lemma cpr_ApplOmega_34 (G) (L) (s0) (s):
48 ❪G,L❫ ⊢ ApplOmega3 s0 s ⬈ ApplOmega4 s0 s.
49 /4 width=3 by cpx_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed.
51 lemma cpxs_ApplOmega_14 (G) (L) (s0) (s):
52 ❪G,L❫ ⊢ ApplOmega1 s0 s ⬈* ApplOmega4 s0 s.
53 /5 width=5 by cpxs_strap1, cpx_cpxs/ qed.
55 lemma fqup_ApplOmega_41 (G) (L) (s0) (s):
56 ❪G,L,ApplOmega4 s0 s❫ ⬂+ ❪G,L,ApplOmega1 s0 s❫.
59 (* Main properties **********************************************************)
61 theorem fpbg_refl (G) (L) (s0) (s):
62 ❪G,L,ApplOmega1 s0 s❫ > ❪G,L,ApplOmega1 s0 s❫.
63 /3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.