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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/notation/relations/exclaim_5.ma".
16 include "basic_2/notation/relations/exclaim_4.ma".
17 include "basic_2/notation/relations/exclaimstar_4.ma".
18 include "basic_2/rt_computation/cpms.ma".
20 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
23 (* Basic_2A1: uses: snv *)
24 inductive cnv (a) (h): relation3 genv lenv term ≝
25 | cnv_sort: ∀G,L,s. cnv a h G L (⋆s)
26 | cnv_zero: ∀I,G,K,V. cnv a h G K V → cnv a h G (K.ⓑ{I}V) (#0)
27 | cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
28 | cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T)
29 | cnv_appl: ∀n,p,G,L,V,W0,T,U0. (a = Ⓣ → n ≤ 1) → cnv a h G L V → cnv a h G L T →
30 ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
31 | cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
32 ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → cnv a h G L (ⓝU.T)
35 interpretation "context-sensitive native validity (term)"
36 'Exclaim a h G L T = (cnv a h G L T).
38 interpretation "context-sensitive restricted native validity (term)"
39 'Exclaim h G L T = (cnv true h G L T).
41 interpretation "context-sensitive extended native validity (term)"
42 'ExclaimStar h G L T = (cnv false h G L T).
44 (* Basic inversion lemmas ***************************************************)
46 fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
47 ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
48 #a #h #G #L #X * -G -L -X
49 [ #G #L #s #H destruct
50 | #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
51 | #I #G #K #i #_ #H destruct
52 | #p #I #G #L #V #T #_ #_ #H destruct
53 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #H destruct
54 | #G #L #U #T #U0 #_ #_ #_ #_ #H destruct
58 lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] →
59 ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
60 /2 width=3 by cnv_inv_zero_aux/ qed-.
62 fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) →
63 ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
64 #a #h #G #L #X * -G -L -X
65 [ #G #L #s #j #H destruct
66 | #I #G #K #V #_ #j #H destruct
67 | #I #G #L #i #Hi #j #H destruct /2 width=4 by ex2_2_intro/
68 | #p #I #G #L #V #T #_ #_ #j #H destruct
69 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #j #H destruct
70 | #G #L #U #T #U0 #_ #_ #_ #_ #j #H destruct
74 lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] →
75 ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
76 /2 width=3 by cnv_inv_lref_aux/ qed-.
78 fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥.
79 #a #h #G #L #X * -G -L -X
80 [ #G #L #s #l #H destruct
81 | #I #G #K #V #_ #l #H destruct
82 | #I #G #K #i #_ #l #H destruct
83 | #p #I #G #L #V #T #_ #_ #l #H destruct
84 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #l #H destruct
85 | #G #L #U #T #U0 #_ #_ #_ #_ #l #H destruct
89 (* Basic_2A1: uses: snv_inv_gref *)
90 lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥.
91 /2 width=8 by cnv_inv_gref_aux/ qed-.
93 fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
94 ∀p,I,V,T. X = ⓑ{p,I}V.T →
96 & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
97 #a #h #G #L #X * -G -L -X
98 [ #G #L #s #q #Z #X1 #X2 #H destruct
99 | #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
100 | #I #G #K #i #_ #q #Z #X1 #X2 #H destruct
101 | #p #I #G #L #V #T #HV #HT #q #Z #X1 #X2 #H destruct /2 width=1 by conj/
102 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
103 | #G #L #U #T #U0 #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
107 (* Basic_2A1: uses: snv_inv_bind *)
108 lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] →
109 ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
110 & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
111 /2 width=4 by cnv_inv_bind_aux/ qed-.
113 fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T →
114 ∃∃n,p,W0,U0. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
115 ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
116 #a #h #G #L #X * -L -X
117 [ #G #L #s #X1 #X2 #H destruct
118 | #I #G #K #V #_ #X1 #X2 #H destruct
119 | #I #G #K #i #_ #X1 #X2 #H destruct
120 | #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
121 | #n #p #G #L #V #W0 #T #U0 #Ha #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /3 width=7 by ex5_4_intro/
122 | #G #L #U #T #U0 #_ #_ #_ #_ #X1 #X2 #H destruct
126 (* Basic_2A1: uses: snv_inv_appl *)
127 lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
128 ∃∃n,p,W0,U0. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
129 ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
130 /2 width=3 by cnv_inv_appl_aux/ qed-.
132 fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T →
133 ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
134 ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
135 #a #h #G #L #X * -G -L -X
136 [ #G #L #s #X1 #X2 #H destruct
137 | #I #G #K #V #_ #X1 #X2 #H destruct
138 | #I #G #K #i #_ #X1 #X2 #H destruct
139 | #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
140 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #X1 #X2 #H destruct
141 | #G #L #U #T #U0 #HV #HT #HU0 #HTU0 #X1 #X2 #H destruct /2 width=3 by ex4_intro/
145 (* Basic_2A1: uses: snv_inv_appl *)
146 lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
147 ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
148 ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
149 /2 width=3 by cnv_inv_cast_aux/ qed-.
151 (* Basic forward lemmas *****************************************************)
153 lemma cnv_fwd_flat (a) (h) (I) (G) (L):
154 ∀V,T. ⦃G, L⦄ ⊢ ⓕ{I}V.T ![a,h] →
155 ∧∧ ⦃G, L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ T ![a,h].
156 #a #h * #G #L #V #T #H
157 [ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_
158 | elim (cnv_inv_cast … H) #U #HV #HT #_ #_
159 ] -H /2 width=1 by conj/
162 (* Basic_2A1: removed theorems 3:
163 shnv_cast shnv_inv_cast snv_shnv_cast