1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/reduction/cpx_cpys.ma".
16 include "basic_2/computation/lpxs_llneq.ma".
17 include "basic_2/computation/csx_alt.ma".
18 include "basic_2/computation/lsx_lpxs.ma".
20 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
22 (* Advanced properties ******************************************************)
24 lemma lpxs_cpys_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
25 ∀T. ⦃G, L1⦄ ⊢ T ▶*[0, ∞] U →
27 #h #g #G #L1 #U #H @(csx_ind_alt … H) -U
28 #U #_ #IHU #T #HTU @lsx_intro
29 #L2 #HL02 #HnT elim (lpxs_nlleq_fwd_cpxs … HL02 HnT) -HnT
30 #U0 #U2 #H0 #H2 #HU02 #HnU02 elim (cpys_conf_eq … HTU … H0)
31 #X #HUX #H0X elim (eq_term_dec U X) #HnUX destruct
33 | -HnU02 @(lsx_lpxs_trans … HL02) @(IHU … HnUX)
34 [ /3 width=3 by cpys_cpx, cpx_cpxs/
35 | /2 width=3 by cpys_trans_eq/
39 lemma lpxs_cpys_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
40 ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀T. ⦃G, L2⦄ ⊢ T ▶*[0, ∞] U →
42 #h #g #G #L1 #U #H @(csx_ind_alt … H) -U
43 #U #_ #IHU #L0 #HL10 #T #HTU @lsx_intro
44 #L2 #HL02 #HnT elim (lpxs_nlleq_fwd_cpxs … HL02 HnT) -HnT
45 #U0 #U2 #H0 #H2 #HU02 #HnU02 elim (cpys_conf_eq … HTU … H0)
46 #X #HUX #H0X elim (eq_term_dec U X) #HnUX destruct
48 | -HnU02 @(IHU … HnUX)
51 -HnT /4 width=9 by lpxs_trans, lpxs_cpxs_trans, cpx_cpye_fwd_lpxs/
60 include "basic_2/reduction/cpx_cpys.ma".
61 include "basic_2/computation/lpxs_cpye.ma".
62 include "basic_2/computation/csx_alt.ma".
63 include "basic_2/computation/lsx_lpxs.ma".
65 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
67 (* Advanced properties ******************************************************)
69 axiom lpxs_cpye_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
70 ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀T. ⦃G, L2⦄ ⊢ T ▶*[0, ∞] 𝐍⦃U⦄ →
73 #h #g #G #L1 #U #H @(csx_ind_alt … H) -U
74 #U0 #_ #IHU0 #L0 #HL10 #T #H0 @lsx_intro
75 #L2 #HL02 #HnT elim (cpye_total G L2 T 0 (∞))
76 #U2 #H2 elim (eq_term_dec U0 U2) #H destruct
78 | -HnT /4 width=9 by lpxs_trans, lpxs_cpxs_trans, cpx_cpye_fwd_lpxs/
81 (* Main properties **********************************************************)
83 lemma csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → G ⊢ ⋕⬊*[h, g, T] L.
84 #h #g #G #L #T #HT elim (cpye_total G L T 0 (∞))
86 /4 width=5 by lpxs_cpye_csx_lsx, csx_cpx_trans, cpys_cpx/