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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 *)
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15 include "basic_2/notation/relations/pred_5.ma".
16 include "basic_2/static/ssta.ma".
17 include "basic_2/reduction/cpr.ma".
19 (* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
21 inductive cpx (h) (g): lenv → relation term ≝
22 | cpx_atom : ∀I,L. cpx h g L (⓪{I}) (⓪{I})
23 | cpx_sort : ∀L,k,l. deg h g k (l+1) → cpx h g L (⋆k) (⋆(next h k))
24 | cpx_delta: ∀I,L,K,V,V2,W2,i.
25 ⇩[0, i] L ≡ K.ⓑ{I}V → cpx h g K V V2 →
26 ⇧[0, i + 1] V2 ≡ W2 → cpx h g L (#i) W2
27 | cpx_bind : ∀a,I,L,V1,V2,T1,T2.
28 cpx h g L V1 V2 → cpx h g (L. ⓑ{I}V1) T1 T2 →
29 cpx h g L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
30 | cpx_flat : ∀I,L,V1,V2,T1,T2.
31 cpx h g L V1 V2 → cpx h g L T1 T2 →
32 cpx h g L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
33 | cpx_zeta : ∀L,V,T1,T,T2. cpx h g (L.ⓓV) T1 T →
34 ⇧[0, 1] T2 ≡ T → cpx h g L (+ⓓV.T1) T2
35 | cpx_tau : ∀L,V,T1,T2. cpx h g L T1 T2 → cpx h g L (ⓝV.T1) T2
36 | cpx_ti : ∀L,V1,V2,T. cpx h g L V1 V2 → cpx h g L (ⓝV1.T) V2
37 | cpx_beta : ∀a,L,V1,V2,W1,W2,T1,T2.
38 cpx h g L V1 V2 → cpx h g L W1 W2 → cpx h g (L.ⓛW1) T1 T2 →
39 cpx h g L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
40 | cpx_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
41 cpx h g L V1 V → ⇧[0, 1] V ≡ V2 → cpx h g L W1 W2 →
42 cpx h g (L.ⓓW1) T1 T2 →
43 cpx h g L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
47 "context-sensitive extended parallel reduction (term)"
48 'PRed h g L T1 T2 = (cpx h g L T1 T2).
50 (* Basic properties *********************************************************)
52 lemma lsubr_cpx_trans: ∀h,g. lsub_trans … (cpx h g) lsubr.
53 #h #g #L1 #T1 #T2 #H elim H -L1 -T1 -T2
56 | #I #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
57 elim (lsubr_fwd_ldrop2_bind … HL12 … HLK1) -HL12 -HLK1 *
58 [ /3 width=7/ | /4 width=7/ ]
65 (* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
66 lemma cpx_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
67 #h #g #T elim T -T // * /2 width=1/
70 lemma cpr_cpx: ∀h,g,L,T1,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
71 #h #g #L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=3/ /2 width=7/
74 fact ssta_cpx_aux: ∀h,g,L,T1,T2,l0. ⦃h, L⦄ ⊢ T1 •[g] ⦃l0, T2⦄ →
75 ∀l. l0 = l+1 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
76 #h #g #L #T1 #T2 #l0 #H elim H -L -T1 -T2 -l0 /2 width=2/ /2 width=7/ /3 width=2/ /3 width=7/
79 lemma ssta_cpx: ∀h,g,L,T1,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
80 /2 width=4 by ssta_cpx_aux/ qed.
82 lemma cpx_pair_sn: ∀h,g,I,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
83 ∀T. ⦃h, L⦄ ⊢ ②{I}V1.T ➡[g] ②{I}V2.T.
84 #h #g * /2 width=1/ qed.
86 lemma cpx_delift: ∀h,g,I,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
87 ∃∃T2,T. ⦃h, L⦄ ⊢ T1 ➡[g] T2 & ⇧[d, 1] T ≡ T2.
88 #h #g #I #K #V #T1 elim T1 -T1
89 [ * #i #L #d #HLK /2 width=4/
90 elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
92 elim (lift_total V 0 (i+1)) #W #HVW
93 elim (lift_split … HVW i i) // /3 width=7/
94 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
95 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
96 [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
97 | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
102 lemma cpx_append: ∀h,g. l_appendable_sn … (cpx h g).
103 #h #g #K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
104 #I #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
105 lapply (ldrop_fwd_length_lt2 … HK0) #H
106 @(cpx_delta … I … (L@@K0) V1 … HVW2) //
107 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
110 (* Basic inversion lemmas ***************************************************)
112 fact cpx_inv_atom1_aux: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ∀J. T1 = ⓪{J} →
114 | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
115 | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
116 ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
117 #h #g #L #T1 #T2 * -L -T1 -T2
118 [ #I #L #J #H destruct /2 width=1/
119 | #L #k #l #Hkl #J #H destruct /3 width=5/
120 | #I #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9/
121 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
122 | #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
123 | #L #V #T1 #T #T2 #_ #_ #J #H destruct
124 | #L #V #T1 #T2 #_ #J #H destruct
125 | #L #V1 #V2 #T #_ #J #H destruct
126 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
127 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
131 lemma cpx_inv_atom1: ∀h,g,J,L,T2. ⦃h, L⦄ ⊢ ⓪{J} ➡[g] T2 →
133 | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
134 | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
135 ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
136 /2 width=3 by cpx_inv_atom1_aux/ qed-.
138 lemma cpx_inv_sort1: ∀h,g,L,T2,k. ⦃h, L⦄ ⊢ ⋆k ➡[g] T2 → T2 = ⋆k ∨
139 ∃∃l. deg h g k (l+1) & T2 = ⋆(next h k).
141 elim (cpx_inv_atom1 … H) -H /2 width=1/ *
142 [ #k0 #l0 #Hkl0 #H1 #H2 destruct /3 width=4/
143 | #I #K #V #V2 #i #_ #_ #_ #H destruct
147 lemma cpx_inv_lref1: ∀h,g,L,T2,i. ⦃h, L⦄ ⊢ #i ➡[g] T2 →
149 ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
152 elim (cpx_inv_atom1 … H) -H /2 width=1/ *
153 [ #k #l #_ #_ #H destruct
154 | #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7/
158 lemma cpx_inv_gref1: ∀h,g,L,T2,p. ⦃h, L⦄ ⊢ §p ➡[g] T2 → T2 = §p.
160 elim (cpx_inv_atom1 … H) -H // *
161 [ #k #l #_ #_ #H destruct
162 | #I #K #V #V2 #i #_ #_ #_ #H destruct
166 fact cpx_inv_bind1_aux: ∀h,g,L,U1,U2. ⦃h, L⦄ ⊢ U1 ➡[g] U2 →
167 ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 → (
168 ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓑ{J}V1⦄ ⊢ T1 ➡[g] T2 &
171 ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T &
173 #h #g #L #U1 #U2 * -L -U1 -U2
174 [ #I #L #b #J #W #U1 #H destruct
175 | #L #k #l #_ #b #J #W #U1 #H destruct
176 | #I #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
177 | #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/
178 | #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
179 | #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/
180 | #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
181 | #L #V1 #V2 #T #_ #b #J #W #U1 #H destruct
182 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
183 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
187 lemma cpx_inv_bind1: ∀h,g,a,I,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[g] U2 → (
188 ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡[g] T2 &
191 ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T &
193 /2 width=3 by cpx_inv_bind1_aux/ qed-.
195 lemma cpx_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓓ{a}V1.T1 ➡[g] U2 → (
196 ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T2 &
199 ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T & a = true.
200 #h #g #a #L #V1 #T1 #U2 #H
201 elim (cpx_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
204 lemma cpx_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 ➡[g] U2 →
205 ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡[g] T2 &
207 #h #g #a #L #V1 #T1 #U2 #H
208 elim (cpx_inv_bind1 … H) -H *
210 | #T #_ #_ #_ #H destruct
214 fact cpx_inv_flat1_aux: ∀h,g,L,U,U2. ⦃h, L⦄ ⊢ U ➡[g] U2 →
215 ∀J,V1,U1. U = ⓕ{J}V1.U1 →
216 ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
218 | (⦃h, L⦄ ⊢ U1 ➡[g] U2 ∧ J = Cast)
219 | (⦃h, L⦄ ⊢ V1 ➡[g] U2 ∧ J = Cast)
220 | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
221 ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
223 U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
224 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
225 ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
227 U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
228 #h #g #L #U #U2 * -L -U -U2
229 [ #I #L #J #W #U1 #H destruct
230 | #L #k #l #_ #J #W #U1 #H destruct
231 | #I #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
232 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
233 | #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/
234 | #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
235 | #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/
236 | #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=1/
237 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/
238 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/
242 lemma cpx_inv_flat1: ∀h,g,I,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓕ{I}V1.U1 ➡[g] U2 →
243 ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
245 | (⦃h, L⦄ ⊢ U1 ➡[g] U2 ∧ I = Cast)
246 | (⦃h, L⦄ ⊢ V1 ➡[g] U2 ∧ I = Cast)
247 | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
248 ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
250 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
251 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
252 ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
254 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
255 /2 width=3 by cpx_inv_flat1_aux/ qed-.
257 lemma cpx_inv_appl1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓐ V1.U1 ➡[g] U2 →
258 ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
260 | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
261 ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
262 U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
263 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
264 ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
265 U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
266 #h #g #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
274 (* Note: the main property of simple terms *)
275 lemma cpx_inv_appl1_simple: ∀h,g,L,V1,T1,U. ⦃h, L⦄ ⊢ ⓐV1.T1 ➡[g] U → 𝐒⦃T1⦄ →
276 ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ T1 ➡[g] T2 &
278 #h #g #L #V1 #T1 #U #H #HT1
279 elim (cpx_inv_appl1 … H) -H *
281 | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
282 elim (simple_inv_bind … HT1)
283 | #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
284 elim (simple_inv_bind … HT1)
288 lemma cpx_inv_cast1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓝV1.U1 ➡[g] U2 →
289 ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
291 | ⦃h, L⦄ ⊢ U1 ➡[g] U2
292 | ⦃h, L⦄ ⊢ V1 ➡[g] U2.
293 #h #g #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
296 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
297 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
301 (* Basic forward lemmas *****************************************************)
303 lemma cpx_fwd_bind1_minus: ∀h,g,I,L,V1,T1,T. ⦃h, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[g] T → ∀b.
304 ∃∃V2,T2. ⦃h, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[g] ⓑ{b,I}V2.T2 &
306 #h #g #I #L #V1 #T1 #T #H #b
307 elim (cpx_inv_bind1 … H) -H *
308 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
309 | #T2 #_ #_ #H destruct
313 lemma cpx_fwd_shift1: ∀h,g,L1,L,T1,T. ⦃h, L⦄ ⊢ L1 @@ T1 ➡[g] T →
314 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
315 #h #g #L1 @(lenv_ind_dx … L1) -L1 normalize
317 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
318 | #I #L1 #V1 #IH #L #T1 #X
319 >shift_append_assoc normalize #H
320 elim (cpx_inv_bind1 … H) -H *
321 [ #V0 #T0 #_ #HT10 #H destruct
322 elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
323 >append_length >HL12 -HL12
324 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
325 | #T #_ #_ #H destruct