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 "basic_2/notation/relations/pred_3.ma".
16 include "basic_2/grammar/cl_shift.ma".
17 include "basic_2/relocation/ldrop_append.ma".
18 include "basic_2/substitution/lsubr.ma".
20 (* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
22 (* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *)
23 (* Note: cpr_flat: does not hold in basic_1 *)
24 inductive cpr: lenv → relation term ≝
25 | cpr_atom : ∀I,L. cpr L (⓪{I}) (⓪{I})
26 | cpr_delta: ∀L,K,V,V2,W2,i.
27 ⇩[0, i] L ≡ K. ⓓV → cpr K V V2 →
28 ⇧[0, i + 1] V2 ≡ W2 → cpr L (#i) W2
29 | cpr_bind : ∀a,I,L,V1,V2,T1,T2.
30 cpr L V1 V2 → cpr (L.ⓑ{I}V1) T1 T2 →
31 cpr L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
32 | cpr_flat : ∀I,L,V1,V2,T1,T2.
33 cpr L V1 V2 → cpr L T1 T2 →
34 cpr L (ⓕ{I} V1. T1) (ⓕ{I}V2.T2)
35 | cpr_zeta : ∀L,V,T1,T,T2. cpr (L.ⓓV) T1 T →
36 ⇧[0, 1] T2 ≡ T → cpr L (+ⓓV.T1) T2
37 | cpr_tau : ∀L,V,T1,T2. cpr L T1 T2 → cpr L (ⓝV.T1) T2
38 | cpr_beta : ∀a,L,V1,V2,W1,W2,T1,T2.
39 cpr L V1 V2 → cpr L W1 W2 → cpr (L.ⓛW1) T1 T2 →
40 cpr L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
41 | cpr_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
42 cpr L V1 V → ⇧[0, 1] V ≡ V2 → cpr L W1 W2 → cpr (L.ⓓW1) T1 T2 →
43 cpr L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
46 interpretation "context-sensitive parallel reduction (term)"
47 'PRed L T1 T2 = (cpr L T1 T2).
49 (* Basic properties *********************************************************)
51 lemma lsubr_cpr_trans: lsub_trans … cpr lsubr.
52 #L1 #T1 #T2 #H elim H -L1 -T1 -T2
54 | #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
55 elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -L1 * /3 width=6/
62 (* Basic_1: was by definition: pr2_free *)
63 lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
65 lapply (lsubr_cpr_trans … HT12 L ?) //
68 (* Basic_1: includes by definition: pr0_refl *)
69 lemma cpr_refl: ∀T,L. L ⊢ T ➡ T.
70 #T elim T -T // * /2 width=1/
73 (* Basic_1: was: pr2_head_1 *)
74 lemma cpr_pair_sn: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 →
75 ∀T. L ⊢ ②{I}V1.T ➡ ②{I}V2.T.
78 lemma cpr_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓓV) →
79 ∃∃T2,T. L ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
81 [ * #i #L #d #HLK /2 width=4/
82 elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
84 elim (lift_total V 0 (i+1)) #W #HVW
85 elim (lift_split … HVW i i) // /3 width=6/
86 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
87 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
88 [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
89 | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
94 lemma cpr_append: l_appendable_sn … cpr.
95 #K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
96 #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
97 lapply (ldrop_fwd_length_lt2 … HK0) #H
98 @(cpr_delta … (L@@K0) V1 … HVW2) //
99 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
102 (* Basic inversion lemmas ***************************************************)
104 fact cpr_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
106 ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
108 ⇧[O, i + 1] V2 ≡ T2 &
110 #L #T1 #T2 * -L -T1 -T2
111 [ #I #L #J #H destruct /2 width=1/
112 | #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
113 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
114 | #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
115 | #L #V #T1 #T #T2 #_ #_ #J #H destruct
116 | #L #V #T1 #T2 #_ #J #H destruct
117 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
118 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
122 lemma cpr_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➡ T2 →
124 ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
126 ⇧[O, i + 1] V2 ≡ T2 &
128 /2 width=3 by cpr_inv_atom1_aux/ qed-.
130 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
131 lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
133 elim (cpr_inv_atom1 … H) -H //
134 * #K #V #V2 #i #_ #_ #_ #H destruct
137 (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
138 lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
140 ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
144 elim (cpr_inv_atom1 … H) -H /2 width=1/
145 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
148 lemma cpr_inv_gref1: ∀L,T2,p. L ⊢ §p ➡ T2 → T2 = §p.
150 elim (cpr_inv_atom1 … H) -H //
151 * #K #V #V2 #i #_ #_ #_ #H destruct
154 fact cpr_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
155 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
156 ∃∃V2,T2. L ⊢ V1 ➡ V2 &
157 L. ⓑ{I}V1 ⊢ T1 ➡ T2 &
160 ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
161 #L #U1 #U2 * -L -U1 -U2
162 [ #I #L #b #J #W1 #U1 #H destruct
163 | #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
164 | #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/
165 | #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
166 | #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/
167 | #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
168 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
169 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
173 lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
174 ∃∃V2,T2. L ⊢ V1 ➡ V2 &
175 L. ⓑ{I}V1 ⊢ T1 ➡ T2 &
178 ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
179 /2 width=3 by cpr_inv_bind1_aux/ qed-.
181 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
182 lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1.T1 ➡ U2 → (
183 ∃∃V2,T2. L ⊢ V1 ➡ V2 &
187 ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
189 elim (cpr_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
192 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
193 lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 →
194 ∃∃V2,T2. L ⊢ V1 ➡ V2 & L.ⓛV1 ⊢ T1 ➡ T2 &
197 elim (cpr_inv_bind1 … H) -H *
199 | #T #_ #_ #_ #H destruct
203 fact cpr_inv_flat1_aux: ∀L,U,U2. L ⊢ U ➡ U2 →
204 ∀I,V1,U1. U = ⓕ{I}V1.U1 →
205 ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
207 | (L ⊢ U1 ➡ U2 ∧ I = Cast)
208 | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
209 L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
210 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
211 | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
212 L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
214 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
215 #L #U #U2 * -L -U -U2
216 [ #I #L #J #W1 #U1 #H destruct
217 | #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
218 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
219 | #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/
220 | #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
221 | #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/
222 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/
223 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/
227 lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. L ⊢ ⓕ{I}V1.U1 ➡ U2 →
228 ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
230 | (L ⊢ U1 ➡ U2 ∧ I = Cast)
231 | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
232 L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
233 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
234 | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
235 L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
237 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
238 /2 width=3 by cpr_inv_flat1_aux/ qed-.
240 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
241 lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐV1.U1 ➡ U2 →
242 ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
244 | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
246 U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
247 | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
248 L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
249 U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
250 #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
258 (* Note: the main property of simple terms *)
259 lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
260 ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
262 #L #V1 #T1 #U #H #HT1
263 elim (cpr_inv_appl1 … H) -H *
265 | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
266 elim (simple_inv_bind … HT1)
267 | #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
268 elim (simple_inv_bind … HT1)
272 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
273 lemma cpr_inv_cast1: ∀L,V1,U1,U2. L ⊢ ⓝ V1. U1 ➡ U2 → (
274 ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
278 #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
281 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
282 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
286 (* Basic forward lemmas *****************************************************)
288 lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
289 ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
291 #I #L #V1 #T1 #T #H #b
292 elim (cpr_inv_bind1 … H) -H *
293 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
294 | #T2 #_ #_ #H destruct
298 lemma cpr_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➡ T →
299 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
300 #L1 @(lenv_ind_dx … L1) -L1 normalize
302 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
303 | #I #L1 #V1 #IH #L #T1 #X
304 >shift_append_assoc normalize #H
305 elim (cpr_inv_bind1 … H) -H *
306 [ #V0 #T0 #_ #HT10 #H destruct
307 elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
308 >append_length >HL12 -HL12
309 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
310 | #T #_ #_ #H destruct
315 (* Basic_1: removed theorems 11:
316 pr0_subst0_back pr0_subst0_fwd pr0_subst0
317 pr2_head_2 pr2_cflat clear_pr2_trans
318 pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
319 pr2_gen_ctail pr2_ctail
321 (* Basic_1: removed local theorems 4:
322 pr0_delta_tau pr0_cong_delta
323 pr2_free_free pr2_free_delta