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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 *)
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15 include "basic_2/rt_computation/fpbs_cpxs.ma".
16 include "basic_2/rt_computation/fpbg_fqus.ma".
18 (* EXAMPLES *****************************************************************)
20 (* Reflexivity of proper rst-computation: the term ApplOmega ****************)
22 definition ApplDelta (s0) (s): term ≝ +ⓛ⋆s0.ⓐ⋆s.ⓐ#0.#0.
24 definition ApplOmega1 (s0) (s): term ≝ ⓐ(ApplDelta s0 s).(ApplDelta s0 s).
26 definition ApplOmega2 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.ⓐ#0.#0.
28 definition ApplOmega3 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.(ApplOmega1 s0 s).
30 definition ApplOmega4 (s0) (s): term ≝ ⓐ⋆s.(ApplOmega1 s0 s).
32 (* Basic properties *********************************************************)
34 lemma ApplDelta_lifts (f) (s0) (s):
35 ⇧*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s).
36 /5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
38 lemma cpr_ApplOmega_12 (G) (L) (s0) (s):
39 ❨G,L❩ ⊢ ApplOmega1 s0 s ⬈ ApplOmega2 s0 s.
40 /2 width=1 by cpx_beta/ qed.
42 lemma cpr_ApplOmega_23 (G) (L) (s0) (s):
43 ❨G,L❩ ⊢ ApplOmega2 s0 s ⬈ ApplOmega3 s0 s.
44 /6 width=3 by cpx_eps, cpx_flat, cpx_bind, cpx_delta, ApplDelta_lifts/ qed.
46 lemma cpr_ApplOmega_34 (G) (L) (s0) (s):
47 ❨G,L❩ ⊢ ApplOmega3 s0 s ⬈ ApplOmega4 s0 s.
48 /4 width=3 by cpx_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed.
50 lemma cpxs_ApplOmega_14 (G) (L) (s0) (s):
51 ❨G,L❩ ⊢ ApplOmega1 s0 s ⬈* ApplOmega4 s0 s.
52 /5 width=5 by cpxs_strap1, cpx_cpxs/ qed.
54 lemma fqup_ApplOmega_41 (G) (L) (s0) (s):
55 ❨G,L,ApplOmega4 s0 s❩ ⬂+ ❨G,L,ApplOmega1 s0 s❩.
58 (* Main properties **********************************************************)
60 theorem fpbg_refl (G) (L) (s0) (s):
61 ❨G,L,ApplOmega1 s0 s❩ > ❨G,L,ApplOmega1 s0 s❩.
62 /3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.