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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_transition/lpr.ma".
16 include "basic_2/rt_computation/cpms_fpbg.ma".
17 include "basic_2/dynamic/cnv.ma".
19 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
21 (* Inductive premises for the preservation results **************************)
23 definition IH_cnv_cpm_trans_lpr (h) (a): relation3 genv lenv term ≝
24 λG,L1,T1. ❨G,L1❩ ⊢ T1 ![h,a] →
25 ∀n,T2. ❨G,L1❩ ⊢ T1 ➡[h,n] T2 →
26 ∀L2. ❨G,L1❩ ⊢ ➡[h,0] L2 → ❨G,L2❩ ⊢ T2 ![h,a].
28 definition IH_cnv_cpms_trans_lpr (h) (a): relation3 genv lenv term ≝
29 λG,L1,T1. ❨G,L1❩ ⊢ T1 ![h,a] →
30 ∀n,T2. ❨G,L1❩ ⊢ T1 ➡*[h,n] T2 →
31 ∀L2. ❨G,L1❩ ⊢ ➡[h,0] L2 → ❨G,L2❩ ⊢ T2 ![h,a].
33 definition IH_cnv_cpm_conf_lpr (h) (a): relation3 genv lenv term ≝
34 λG,L0,T0. ❨G,L0❩ ⊢ T0 ![h,a] →
35 ∀n1,T1. ❨G,L0❩ ⊢ T0 ➡[h,n1] T1 → ∀n2,T2. ❨G,L0❩ ⊢ T0 ➡[h,n2] T2 →
36 ∀L1. ❨G,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G,L0❩ ⊢ ➡[h,0] L2 →
37 ∃∃T. ❨G,L1❩ ⊢ T1 ➡*[h,n2-n1] T & ❨G,L2❩ ⊢ T2 ➡*[h,n1-n2] T.
39 definition IH_cnv_cpms_strip_lpr (h) (a): relation3 genv lenv term ≝
40 λG,L0,T0. ❨G,L0❩ ⊢ T0 ![h,a] →
41 ∀n1,T1. ❨G,L0❩ ⊢ T0 ➡*[h,n1] T1 → ∀n2,T2. ❨G,L0❩ ⊢ T0 ➡[h,n2] T2 →
42 ∀L1. ❨G,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G,L0❩ ⊢ ➡[h,0] L2 →
43 ∃∃T. ❨G,L1❩ ⊢ T1 ➡*[h,n2-n1] T & ❨G,L2❩ ⊢ T2 ➡*[h,n1-n2] T.
45 definition IH_cnv_cpms_conf_lpr (h) (a): relation3 genv lenv term ≝
46 λG,L0,T0. ❨G,L0❩ ⊢ T0 ![h,a] →
47 ∀n1,T1. ❨G,L0❩ ⊢ T0 ➡*[h,n1] T1 → ∀n2,T2. ❨G,L0❩ ⊢ T0 ➡*[h,n2] T2 →
48 ∀L1. ❨G,L0❩ ⊢ ➡[h,0] L1 → ∀L2. ❨G,L0❩ ⊢ ➡[h,0] L2 →
49 ∃∃T. ❨G,L1❩ ⊢ T1 ➡*[h,n2-n1] T & ❨G,L2❩ ⊢ T2 ➡*[h,n1-n2] T.
51 (* Auxiliary properties for preservation ************************************)
53 fact cnv_cpms_trans_lpr_sub (h) (a):
55 (∀G1,L1,T1. ❨G0,L0,T0❩ > ❨G1,L1,T1❩ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
56 ∀G1,L1,T1. ❨G0,L0,T0❩ > ❨G1,L1,T1❩ → IH_cnv_cpms_trans_lpr h a G1 L1 T1.
57 #h #a #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
58 @(cpms_ind_dx … H) -n -T2
59 /3 width=7 by fpbg_cpms_trans/
62 fact cnv_cpm_conf_lpr_sub (h) (a):
64 (∀G1,L1,T1. ❨G0,L0,T0❩ > ❨G1,L1,T1❩ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
65 ∀G1,L1,T1. ❨G0,L0,T0❩ > ❨G1,L1,T1❩ → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
66 /3 width=8 by cpm_cpms/ qed-.
68 fact cnv_cpms_strip_lpr_sub (h) (a):
70 (∀G1,L1,T1. ❨G0,L0,T0❩ > ❨G1,L1,T1❩ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
71 ∀G1,L1,T1. ❨G0,L0,T0❩ > ❨G1,L1,T1❩ → IH_cnv_cpms_strip_lpr h a G1 L1 T1.
72 /3 width=8 by cpm_cpms/ qed-.