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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_cpms.ma".
16 include "basic_2/rt_equivalence/cpes.ma".
17 include "basic_2/dynamic/cnv.ma".
19 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
21 (* Properties with t-bound rt-equivalence for terms *************************)
23 lemma cnv_appl_cpes (a) (h) (G) (L):
25 ∀V. ⦃G, L⦄ ⊢ V ![a, h] → ∀T. ⦃G, L⦄ ⊢ T ![a, h] →
26 ∀W. ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W →
27 ∀p,U. ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U → ⦃G, L⦄ ⊢ ⓐV.T ![a, h].
28 #a #h #G #L #n #Hn #V #HV #T #HT #W *
29 /4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
32 lemma cnv_cast_cpes (a) (h) (G) (L):
33 ∀U. ⦃G, L⦄ ⊢ U ![a, h] →
34 ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ⦃G, L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G, L⦄ ⊢ ⓝU.T ![a, h].
35 #a #h #G #L #U #HU #T #HT * /2 width=3 by cnv_cast/
38 (* Inversion lemmas with t-bound rt-equivalence for terms *******************)
40 lemma cnv_inv_appl_cpes (a) (h) (G) (L):
41 ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
42 ∃∃n,p,W,U. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
43 ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U.
45 elim (cnv_inv_appl … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
46 /3 width=7 by cpms_div, ex5_4_intro/
49 lemma cnv_inv_cast_cpes (a) (h) (G) (L):
50 ∀U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
51 ∧∧ ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & ⦃G, L⦄ ⊢ U ⬌*[h,0,1] T.
53 elim (cnv_inv_cast … H) -H
54 /3 width=3 by cpms_div, and3_intro/
57 (* Eliminators with t-bound rt-equivalence for terms ************************)
59 lemma cnv_ind_cpes (a) (h) (Q:relation3 genv lenv term):
60 (∀G,L,s. Q G L (⋆s)) →
61 (∀I,G,K,V. ⦃G,K⦄ ⊢ V![a,h] → Q G K V → Q G (K.ⓑ{I}V) (#O)) →
62 (∀I,G,K,i. ⦃G,K⦄ ⊢ #i![a,h] → Q G K (#i) → Q G (K.ⓘ{I}) (#(↑i))) →
63 (∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ V![a,h] → ⦃G,L.ⓑ{I}V⦄⊢T![a,h] →
64 Q G L V →Q G (L.ⓑ{I}V) T →Q G L (ⓑ{p,I}V.T)
66 (∀n,p,G,L,V,W,T,U. (a = Ⓣ → n ≤ 1) → ⦃G,L⦄ ⊢ V![a,h] → ⦃G,L⦄ ⊢ T![a,h] →
67 ⦃G,L⦄ ⊢ V ⬌*[h,1,0]W → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U →
68 Q G L V → Q G L T → Q G L (ⓐV.T)
70 (∀G,L,U,T. ⦃G,L⦄⊢ U![a,h] → ⦃G,L⦄ ⊢ T![a,h] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T →
71 Q G L U → Q G L T → Q G L (ⓝU.T)
73 ∀G,L,T. ⦃G,L⦄⊢ T![a,h] → Q G L T.
74 #a #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #G #L #T #H
75 elim H -G -L -T [5,6: /3 width=7 by cpms_div/ |*: /2 width=1 by/ ]