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_4.ma".
16 include "basic_2/grammar/genv.ma".
17 include "basic_2/grammar/cl_shift.ma".
18 include "basic_2/relocation/ldrop_append.ma".
19 include "basic_2/relocation/lsubr.ma".
21 (* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
24 (* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *)
25 (* Note: cpr_flat: does not hold in basic_1 *)
26 inductive cpr: relation4 genv lenv term term ≝
27 | cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
28 | cpr_delta: ∀G,L,K,V,V2,W2,i.
29 ⇩[i] L ≡ K. ⓓV → cpr G K V V2 →
30 ⇧[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
31 | cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
32 cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
33 cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
34 | cpr_flat : ∀I,G,L,V1,V2,T1,T2.
35 cpr G L V1 V2 → cpr G L T1 T2 →
36 cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
37 | cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
38 ⇧[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
39 | cpr_tau : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
40 | cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
41 cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
42 cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
43 | cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
44 cpr G L V1 V → ⇧[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 →
45 cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
48 interpretation "context-sensitive parallel reduction (term)"
49 'PRed G L T1 T2 = (cpr G L T1 T2).
51 (* Basic properties *********************************************************)
53 lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
54 #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
56 | #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
57 elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -L1 *
58 /3 width=6 by cpr_delta/
59 |3,7: /4 width=1 by lsubr_bind, cpr_bind, cpr_beta/
60 |4,6: /3 width=1 by cpr_flat, cpr_tau/
61 |5,8: /4 width=3 by lsubr_bind, cpr_zeta, cpr_theta/
65 (* Basic_1: was by definition: pr2_free *)
66 lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
68 lapply (lsubr_cpr_trans … HT12 L ?) //
71 (* Basic_1: includes by definition: pr0_refl *)
72 lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
73 #G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/
76 (* Basic_1: was: pr2_head_1 *)
77 lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
78 ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
79 * /2 width=1 by cpr_bind, cpr_flat/ qed.
81 lemma cpr_delift: ∀G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓓV) →
82 ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
83 #G #K #V #T1 elim T1 -T1
84 [ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
85 #i #L #d #HLK elim (lt_or_eq_or_gt i d)
86 #Hid [1,3: /3 width=4 by cpr_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
88 elim (lift_total V 0 (i+1)) #W #HVW
89 elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
90 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
91 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
92 [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by ldrop_drop, cpr_bind, lift_bind, ex2_2_intro/
93 | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
98 lemma cpr_append: ∀G. l_appendable_sn … (cpr G).
99 #G #K #T1 #T2 #H elim H -G -K -T1 -T2
100 /2 width=3 by cpr_bind, cpr_flat, cpr_zeta, cpr_tau, cpr_beta, cpr_theta/
101 #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
102 lapply (ldrop_fwd_length_lt2 … HK0) #H
103 @(cpr_delta … (L@@K0) V1 … HVW2) //
104 @(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
107 (* Basic inversion lemmas ***************************************************)
109 fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
111 ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
112 ⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
113 #G #L #T1 #T2 * -G -L -T1 -T2
114 [ #I #G #L #J #H destruct /2 width=1 by or_introl/
115 | #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/
116 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
117 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
118 | #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
119 | #G #L #V #T1 #T2 #_ #J #H destruct
120 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
121 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
125 lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
127 ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
128 ⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
129 /2 width=3 by cpr_inv_atom1_aux/ qed-.
131 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
132 lemma cpr_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k.
134 elim (cpr_inv_atom1 … H) -H //
135 * #K #V #V2 #i #_ #_ #_ #H destruct
138 (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
139 lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
141 ∃∃K,V,V2. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
144 elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
145 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
148 lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, 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: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
155 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
156 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
159 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T &
161 #G #L #U1 #U2 * -L -U1 -U2
162 [ #I #G #L #b #J #W1 #U1 #H destruct
163 | #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
164 | #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
165 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
166 | #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
167 | #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
168 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
169 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
173 lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
174 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
177 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T &
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,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
183 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 &
186 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
187 #a #G #L #V1 #T1 #U2 #H
188 elim (cpr_inv_bind1 … H) -H *
189 /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
192 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
193 lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
194 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 &
196 #a #G #L #V1 #T1 #U2 #H
197 elim (cpr_inv_bind1 … H) -H *
198 [ /3 width=5 by ex3_2_intro/
199 | #T #_ #_ #_ #H destruct
203 fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
204 ∀I,V1,U1. U = ⓕ{I}V1.U1 →
205 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
207 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
208 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
209 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
210 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
211 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
212 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
214 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
215 #G #L #U #U2 * -L -U -U2
216 [ #I #G #L #J #W1 #U1 #H destruct
217 | #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
218 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
219 | #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/
220 | #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
221 | #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/
222 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or4_intro2, ex6_6_intro/
223 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or4_intro3, ex7_7_intro/
227 lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
228 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
230 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
231 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
232 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
233 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
234 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
235 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, 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: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 →
242 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
244 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
245 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 &
246 U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
247 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
248 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
249 U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
250 #G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
251 [ /3 width=5 by or3_intro0, ex3_2_intro/
253 | /3 width=11 by or3_intro1, ex5_6_intro/
254 | /3 width=13 by or3_intro2, ex6_7_intro/
258 (* Note: the main property of simple terms *)
259 lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
260 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
262 #G #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: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
274 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
276 ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
277 #G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
278 [ /3 width=5 by ex3_2_intro, or_introl/
279 | /2 width=1 by or_intror/
280 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
281 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
285 (* Basic forward lemmas *****************************************************)
287 lemma cpr_fwd_bind1_minus: ∀I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
288 ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
290 #I #G #L #V1 #T1 #T #H #b
291 elim (cpr_inv_bind1 … H) -H *
292 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/
293 | #T2 #_ #_ #H destruct
297 lemma cpr_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T →
298 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
299 #G #L1 @(lenv_ind_dx … L1) -L1 normalize
301 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
302 | #I #L1 #V1 #IH #L #T1 #X
303 >shift_append_assoc normalize #H
304 elim (cpr_inv_bind1 … H) -H *
305 [ #V0 #T0 #_ #HT10 #H destruct
306 elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
307 >append_length >HL12 -HL12
308 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
309 | #T #_ #_ #H destruct
314 (* Basic_1: removed theorems 11:
315 pr0_subst0_back pr0_subst0_fwd pr0_subst0
316 pr2_head_2 pr2_cflat clear_pr2_trans
317 pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
318 pr2_gen_ctail pr2_ctail
320 (* Basic_1: removed local theorems 4:
321 pr0_delta_tau pr0_cong_delta
322 pr2_free_free pr2_free_delta