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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 *)
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15 include "basic_2/notation/relations/pred_4.ma".
16 include "basic_2/static/lsubr.ma".
17 include "basic_2/unfold/lstas.ma".
19 (* 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: relation4 genv lenv term term ≝
25 | cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
26 | cpr_delta: ∀G,L,K,V,V2,W2,i.
27 ⬇[i] L ≡ K. ⓓV → cpr G K V V2 →
28 ⬆[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
29 | cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
30 cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
31 cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
32 | cpr_flat : ∀I,G,L,V1,V2,T1,T2.
33 cpr G L V1 V2 → cpr G L T1 T2 →
34 cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
35 | cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
36 ⬆[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
37 | cpr_eps : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
38 | cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
39 cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
40 cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
41 | cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
42 cpr G L V1 V → ⬆[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 →
43 cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
46 interpretation "context-sensitive parallel reduction (term)"
47 'PRed G L T1 T2 = (cpr G L T1 T2).
49 (* Basic properties *********************************************************)
51 lemma cpr_cpx: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2.
52 #h #o #G #L #T1 #T2 #H elim H -L -T1 -T2
53 /2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
56 lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
57 #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
59 | #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
60 elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 *
61 /3 width=6 by cpr_delta/
62 |3,7: /4 width=1 by lsubr_pair, cpr_bind, cpr_beta/
63 |4,6: /3 width=1 by cpr_flat, cpr_eps/
64 |5,8: /4 width=3 by lsubr_pair, cpr_zeta, cpr_theta/
68 (* Basic_1: was by definition: pr2_free *)
69 lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
71 lapply (lsubr_cpr_trans … HT12 L ?) //
74 (* Basic_1: includes by definition: pr0_refl *)
75 lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
76 #G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/
79 (* Basic_1: was: pr2_head_1 *)
80 lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
81 ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
82 * /2 width=1 by cpr_bind, cpr_flat/ qed.
84 lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) →
85 ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2.
86 #G #K #V #T1 elim T1 -T1
87 [ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
88 #i #L #l #HLK elim (lt_or_eq_or_gt i l)
89 #Hil [1,3: /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
91 elim (lift_total V 0 (i+1)) #W #HVW
92 elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
93 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
94 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
95 [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
96 | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
101 fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 →
102 d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
103 #h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
104 /3 width=1 by cpr_eps, cpr_flat, cpr_bind/
105 [ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
106 /3 width=6 by cpr_delta/
107 | #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
111 lemma lstas_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, 0] T2 → ⦃G, L⦄ ⊢ T1 ➡ T2.
112 /2 width=4 by lstas_cpr_aux/ qed.
114 (* Basic inversion lemmas ***************************************************)
116 fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
118 ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
119 ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
120 #G #L #T1 #T2 * -G -L -T1 -T2
121 [ #I #G #L #J #H destruct /2 width=1 by or_introl/
122 | #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/
123 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
124 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
125 | #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
126 | #G #L #V #T1 #T2 #_ #J #H destruct
127 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
128 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
132 lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
134 ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
135 ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
136 /2 width=3 by cpr_inv_atom1_aux/ qed-.
138 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
139 lemma cpr_inv_sort1: ∀G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡ T2 → T2 = ⋆s.
141 elim (cpr_inv_atom1 … H) -H //
142 * #K #V #V2 #i #_ #_ #_ #H destruct
145 (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
146 lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
148 ∃∃K,V,V2. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
151 elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
152 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
155 lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
157 elim (cpr_inv_atom1 … H) -H //
158 * #K #V #V2 #i #_ #_ #_ #H destruct
161 fact cpr_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
162 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
163 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
166 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T &
168 #G #L #U1 #U2 * -L -U1 -U2
169 [ #I #G #L #b #J #W1 #U1 #H destruct
170 | #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
171 | #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/
172 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
173 | #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
174 | #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
175 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
176 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
180 lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
181 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
184 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T &
186 /2 width=3 by cpr_inv_bind1_aux/ qed-.
188 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
189 lemma cpr_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
190 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 &
193 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T & a = true.
194 #a #G #L #V1 #T1 #U2 #H
195 elim (cpr_inv_bind1 … H) -H *
196 /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
199 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
200 lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
201 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 &
203 #a #G #L #V1 #T1 #U2 #H
204 elim (cpr_inv_bind1 … H) -H *
205 [ /3 width=5 by ex3_2_intro/
206 | #T #_ #_ #_ #H destruct
210 fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
211 ∀I,V1,U1. U = ⓕ{I}V1.U1 →
212 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
214 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
215 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
216 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
217 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
218 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
219 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
221 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
222 #G #L #U #U2 * -L -U -U2
223 [ #I #G #L #J #W1 #U1 #H destruct
224 | #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
225 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
226 | #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/
227 | #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
228 | #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/
229 | #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/
230 | #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/
234 lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
235 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
237 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
238 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
239 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
240 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
241 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
242 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
244 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
245 /2 width=3 by cpr_inv_flat1_aux/ qed-.
247 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
248 lemma cpr_inv_appl1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 →
249 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
251 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
252 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 &
253 U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
254 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
255 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
256 U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
257 #G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
258 [ /3 width=5 by or3_intro0, ex3_2_intro/
260 | /3 width=11 by or3_intro1, ex5_6_intro/
261 | /3 width=13 by or3_intro2, ex6_7_intro/
265 (* Note: the main property of simple terms *)
266 lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
267 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
269 #G #L #V1 #T1 #U #H #HT1
270 elim (cpr_inv_appl1 … H) -H *
271 [ /2 width=5 by ex3_2_intro/
272 | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
273 elim (simple_inv_bind … HT1)
274 | #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
275 elim (simple_inv_bind … HT1)
279 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
280 lemma cpr_inv_cast1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
281 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
283 ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
284 #G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
285 [ /3 width=5 by ex3_2_intro, or_introl/
286 | /2 width=1 by or_intror/
287 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
288 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
292 (* Basic forward lemmas *****************************************************)
294 lemma cpr_fwd_bind1_minus: ∀I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
295 ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
297 #I #G #L #V1 #T1 #T #H #b
298 elim (cpr_inv_bind1 … H) -H *
299 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/
300 | #T2 #_ #_ #H destruct
304 (* Basic_1: removed theorems 11:
305 pr0_subst0_back pr0_subst0_fwd pr0_subst0
306 pr2_head_2 pr2_cflat clear_pr2_trans
307 pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
308 pr2_gen_ctail pr2_ctail
310 (* Basic_1: removed local theorems 4:
311 pr0_delta_eps pr0_cong_delta
312 pr2_free_free pr2_free_delta