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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/csx_cpxs.ma".
16 include "basic_2/rt_computation/csx_lsubr.ma".
17 include "basic_2/rt_computation/lfsx_lpxs.ma".
19 include "basic_2/rt_computation/lsubsx_lfsx.ma".
21 (* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
23 axiom lpxs_trans: ∀h,G. Transitive … (lpxs h G).
26 (* Advanced properties ******************************************************)
28 (* Basic_2A1: uses: lsx_lref_be_lpxs *)
29 lemma lfsx_pair_lpxs: ∀h,o,G,K1,V. ⦃G, K1⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
30 ∀K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
31 ∀I. G ⊢ ⬈*[h, o, #0] 𝐒⦃K2.ⓑ{I}V⦄.
33 @(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
34 @(lfsx_ind_lpxs … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
35 @lfsx_intro_lpxs #Y #HY #HnY
36 elim (lpxs_inv_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
37 elim (tdeq_dec h o V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
38 [ /5 width=5 by lfsx_lfdeq_trans, lpxs_trans, lfdeq_pair/
39 | @(IHV0 … HnV02) -IHV0 -HnV02
41 | /3 width=3 by lfsx_lpxs_trans, lfsx_cpxs_trans/
42 | /2 width=3 by lpxs_trans/
46 @(lfsx_lpxs_trans … (K0.ⓑ{I}V2))
47 [ @(IHV0 … HnV02 … HK10) -IHV0 -HnV02
49 | /2 width=3 by lfsx_cpxs_trans/
55 (* Basic_2A1: uses: lsx_lref_be *)
56 lemma lfsx_lref_pair: ∀h,o,G,K,V. ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄ →
57 ∀I,L,i. ⬇*[i] L ≡ K.ⓑ{I}V → G ⊢ ⬈*[h, o, #i] 𝐒⦃L⦄.
58 #h #o #G #K #V #HV #HK #I #L #i #HLK
59 @(lfsx_lifts … (#0) … HLK) -L /2 width=3 by lfsx_pair_lpxs/
62 (* Main properties **********************************************************)
64 theorem csx_lsx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
65 #h #o #G #L #T @(fqup_wf_ind_eq (Ⓕ) … G L T) -G -L -T
66 #Z #Y #X #IH #G #L * * //
67 [ #i #HG #HL #HT #H destruct
68 elim (csx_inv_lref … H) -H [ |*: * ]
69 [ /2 width=1 by lfsx_lref_atom/
70 | /2 width=3 by lfsx_lref_unit/
71 | /4 width=6 by lfsx_lref_pair, fqup_lref/
73 | #a #I #V #T #HG #HL #HT #H destruct
74 elim (csx_fwd_bind_unit … H Void) -H /3 width=1 by lfsx_bind_void/
75 | #I #V #T #HG #HL #HT #H destruct
76 elim (csx_fwd_flat … H) -H /3 width=1 by lfsx_flat/