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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/cpms_aaa.ma".
16 include "basic_2/dynamic/cnv.ma".
18 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
20 (* Forward lemmas on atomic arity assignment for terms **********************)
22 (* Basic_2A1: uses: snv_fwd_aaa *)
23 lemma cnv_fwd_aaa (h) (a):
24 ∀G,L,T. ❨G,L❩ ⊢ T ![h,a] → ∃A. ❨G,L❩ ⊢ T ⁝ A.
25 #h #a #G #L #T #H elim H -G -L -T
26 [ /2 width=2 by aaa_sort, ex_intro/
27 | #I #G #L #V #_ * /3 width=2 by aaa_zero, ex_intro/
28 | #I #G #L #K #_ * /3 width=2 by aaa_lref, ex_intro/
29 | #p * #G #L #V #T #_ #_ * #B #HV * #A #HA
30 /3 width=2 by aaa_abbr, aaa_abst, ex_intro/
31 | #n #p #G #L #V #W #T0 #U0 #_ #_ #_ #HVW #HTU0 * #B #HV * #X #HT
32 lapply (cpms_aaa_conf … HV … HVW) -HVW #H1W
33 lapply (cpms_aaa_conf … HT … HTU0) -HTU0 #H
34 elim (aaa_inv_abst … H) -H #B0 #A #H2W #HU #H destruct
35 lapply (aaa_mono … H2W … H1W) -W #H destruct
36 /3 width=4 by aaa_appl, ex_intro/
37 | #G #L #U #T #U0 #_ #_ #HU0 #HTU0 * #B #HU * #A #HT
38 lapply (cpms_aaa_conf … HU … HU0) -HU0 #HU0
39 lapply (cpms_aaa_conf … HT … HTU0) -HTU0 #H
40 lapply (aaa_mono … H … HU0) -U0 #H destruct
41 /3 width=3 by aaa_cast, ex_intro/
45 (* Forward lemmas with t_bound rt_transition for terms **********************)
47 lemma cnv_fwd_cpm_SO (h) (a) (G) (L):
48 ∀T. ❨G,L❩ ⊢ T ![h,a] → ∃U. ❨G,L❩ ⊢ T ➡[h,1] U.
50 elim (cnv_fwd_aaa … H) -H #A #HA
51 /2 width=2 by aaa_cpm_SO/
54 (* Forward lemmas with t_bound rt_computation for terms *********************)
56 lemma cnv_fwd_cpms_total (h) (a) (n) (G) (L):
57 ∀T. ❨G,L❩ ⊢ T ![h,a] → ∃U. ❨G,L❩ ⊢ T ➡*[h,n] U.
59 elim (cnv_fwd_aaa … H) -H #A #HA
60 /2 width=2 by cpms_total_aaa/
63 lemma cnv_fwd_cpms_abst_dx_le (h) (a) (G) (L) (W) (p):
64 ∀T. ❨G,L❩ ⊢ T ![h,a] →
65 ∀n1,U1. ❨G,L❩ ⊢ T ➡*[h,n1] ⓛ[p]W.U1 → ∀n2. n1 ≤ n2 →
66 ∃∃U2. ❨G,L❩ ⊢ T ➡*[h,n2] ⓛ[p]W.U2 & ❨G,L.ⓛW❩ ⊢ U1 ➡*[h,n2-n1] U2.
67 #h #a #G #L #W #p #T #H
68 elim (cnv_fwd_aaa … H) -H #A #HA
69 /2 width=2 by cpms_abst_dx_le_aaa/
72 (* Advanced properties ******************************************************)
74 lemma cnv_appl_ge (h) (a) (n1) (p) (G) (L):
75 ∀n2. n1 ≤ n2 → ad a n2 →
76 ∀V. ❨G,L❩ ⊢ V ![h,a] → ∀T. ❨G,L❩ ⊢ T ![h,a] →
77 ∀X. ❨G,L❩ ⊢ V ➡*[h,1] X → ∀W. ❨G,L❩ ⊢ W ➡*[h,0] X →
78 ∀U. ❨G,L❩ ⊢ T ➡*[h,n1] ⓛ[p]W.U → ❨G,L❩ ⊢ ⓐV.T ![h,a].
79 #h #a #n1 #p #G #L #n2 #Hn12 #Ha #V #HV #T #HT #X #HVX #W #HW #X #HTX
80 elim (cnv_fwd_cpms_abst_dx_le … HT … HTX … Hn12) #U #HTU #_ -n1
81 /4 width=11 by cnv_appl, cpms_bind, cpms_cprs_trans/