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 "ground_2/ynat/ynat_lt.ma".
16 include "basic_2/notation/relations/exclaim_5.ma".
17 include "basic_2/notation/relations/exclaim_4.ma".
18 include "basic_2/notation/relations/exclaimstar_4.ma".
19 include "basic_2/rt_computation/cpms.ma".
21 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
24 (* Basic_2A1: uses: snv *)
25 inductive cnv (a:ynat) (h): relation3 genv lenv term ≝
26 | cnv_sort: ∀G,L,s. cnv a h G L (⋆s)
27 | cnv_zero: ∀I,G,K,V. cnv a h G K V → cnv a h G (K.ⓑ{I}V) (#0)
28 | cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
29 | 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)
30 | cnv_appl: ∀n,p,G,L,V,W0,T,U0. yinj n < a → cnv a h G L V → cnv a h G L T →
31 ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
32 | cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
33 ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv a h G L (ⓝU.T)
36 interpretation "context-sensitive native validity (term)"
37 'Exclaim a h G L T = (cnv a h G L T).
39 interpretation "context-sensitive restricted native validity (term)"
40 'Exclaim h G L T = (cnv (yinj (S (S O))) h G L T).
42 interpretation "context-sensitive extended native validity (term)"
43 'ExclaimStar h G L T = (cnv Y h G L T).
45 (* Basic inversion lemmas ***************************************************)
47 fact cnv_inv_zero_aux (a) (h):
48 ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → X = #0 →
49 ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
50 #a #h #G #L #X * -G -L -X
51 [ #G #L #s #H destruct
52 | #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
53 | #I #G #K #i #_ #H destruct
54 | #p #I #G #L #V #T #_ #_ #H destruct
55 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #H destruct
56 | #G #L #U #T #U0 #_ #_ #_ #_ #H destruct
60 lemma cnv_inv_zero (a) (h):
61 ∀G,L. ⦃G,L⦄ ⊢ #0 ![a,h] →
62 ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
63 /2 width=3 by cnv_inv_zero_aux/ qed-.
65 fact cnv_inv_lref_aux (a) (h):
66 ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀i. X = #(↑i) →
67 ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
68 #a #h #G #L #X * -G -L -X
69 [ #G #L #s #j #H destruct
70 | #I #G #K #V #_ #j #H destruct
71 | #I #G #L #i #Hi #j #H destruct /2 width=4 by ex2_2_intro/
72 | #p #I #G #L #V #T #_ #_ #j #H destruct
73 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #j #H destruct
74 | #G #L #U #T #U0 #_ #_ #_ #_ #j #H destruct
78 lemma cnv_inv_lref (a) (h):
79 ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![a,h] →
80 ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
81 /2 width=3 by cnv_inv_lref_aux/ qed-.
83 fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀l. X = §l → ⊥.
84 #a #h #G #L #X * -G -L -X
85 [ #G #L #s #l #H destruct
86 | #I #G #K #V #_ #l #H destruct
87 | #I #G #K #i #_ #l #H destruct
88 | #p #I #G #L #V #T #_ #_ #l #H destruct
89 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #l #H destruct
90 | #G #L #U #T #U0 #_ #_ #_ #_ #l #H destruct
94 (* Basic_2A1: uses: snv_inv_gref *)
95 lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![a,h] → ⊥.
96 /2 width=8 by cnv_inv_gref_aux/ qed-.
98 fact cnv_inv_bind_aux (a) (h):
99 ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] →
100 ∀p,I,V,T. X = ⓑ{p,I}V.T →
101 ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
102 #a #h #G #L #X * -G -L -X
103 [ #G #L #s #q #Z #X1 #X2 #H destruct
104 | #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
105 | #I #G #K #i #_ #q #Z #X1 #X2 #H destruct
106 | #p #I #G #L #V #T #HV #HT #q #Z #X1 #X2 #H destruct /2 width=1 by conj/
107 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
108 | #G #L #U #T #U0 #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
112 (* Basic_2A1: uses: snv_inv_bind *)
113 lemma cnv_inv_bind (a) (h):
114 ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
115 ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
116 /2 width=4 by cnv_inv_bind_aux/ qed-.
118 fact cnv_inv_appl_aux (a) (h):
119 ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀V,T. X = ⓐV.T →
120 ∃∃n,p,W0,U0. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
121 ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
122 #a #h #G #L #X * -L -X
123 [ #G #L #s #X1 #X2 #H destruct
124 | #I #G #K #V #_ #X1 #X2 #H destruct
125 | #I #G #K #i #_ #X1 #X2 #H destruct
126 | #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
127 | #n #p #G #L #V #W0 #T #U0 #Ha #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /3 width=7 by ex5_4_intro/
128 | #G #L #U #T #U0 #_ #_ #_ #_ #X1 #X2 #H destruct
132 (* Basic_2A1: uses: snv_inv_appl *)
133 lemma cnv_inv_appl (a) (h):
134 ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
135 ∃∃n,p,W0,U0. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
136 ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
137 /2 width=3 by cnv_inv_appl_aux/ qed-.
139 fact cnv_inv_cast_aux (a) (h):
140 ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀U,T. X = ⓝU.T →
141 ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
142 ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
143 #a #h #G #L #X * -G -L -X
144 [ #G #L #s #X1 #X2 #H destruct
145 | #I #G #K #V #_ #X1 #X2 #H destruct
146 | #I #G #K #i #_ #X1 #X2 #H destruct
147 | #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
148 | #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #X1 #X2 #H destruct
149 | #G #L #U #T #U0 #HV #HT #HU0 #HTU0 #X1 #X2 #H destruct /2 width=3 by ex4_intro/
153 (* Basic_2A1: uses: snv_inv_cast *)
154 lemma cnv_inv_cast (a) (h):
155 ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] →
156 ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
157 ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
158 /2 width=3 by cnv_inv_cast_aux/ qed-.
160 (* Basic forward lemmas *****************************************************)
162 lemma cnv_fwd_flat (a) (h) (I) (G) (L):
163 ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![a,h] →
164 ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h].
165 #a #h * #G #L #V #T #H
166 [ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_
167 | elim (cnv_inv_cast … H) #U #HV #HT #_ #_
168 ] -H /2 width=1 by conj/
171 lemma cnv_fwd_pair_sn (a) (h) (I) (G) (L):
172 ∀V,T. ⦃G,L⦄ ⊢ ②{I}V.T ![a,h] → ⦃G,L⦄ ⊢ V ![a,h].
173 #a #h * [ #p ] #I #G #L #V #T #H
174 [ elim (cnv_inv_bind … H) -H #HV #_
175 | elim (cnv_fwd_flat … H) -H #HV #_
179 (* Basic_2A1: removed theorems 3:
180 shnv_cast shnv_inv_cast snv_shnv_cast