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 *)
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15 include "basic_2/notation/relations/pred_4.ma".
16 include "basic_2/grammar/genv.ma".
17 include "basic_2/static/lsubr.ma".
18 include "basic_2/unfold/lstas.ma".
20 (* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
23 (* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *)
24 (* Note: cpr_flat: does not hold in basic_1 *)
25 inductive cpr: relation4 genv lenv term term ≝
26 | cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
27 | cpr_delta: ∀G,L,K,V,V2,W2,i.
28 ⇩[i] L ≡ K. ⓓV → cpr G K V V2 →
29 ⇧[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
30 | cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
31 cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
32 cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
33 | cpr_flat : ∀I,G,L,V1,V2,T1,T2.
34 cpr G L V1 V2 → cpr G L T1 T2 →
35 cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
36 | cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
37 ⇧[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
38 | cpr_eps : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
39 | cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
40 cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
41 cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
42 | cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
43 cpr G L V1 V → ⇧[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 →
44 cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
47 interpretation "context-sensitive parallel reduction (term)"
48 'PRed G L T1 T2 = (cpr G L T1 T2).
50 (* Basic properties *********************************************************)
52 lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
53 #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
55 | #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
56 elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 *
57 /3 width=6 by cpr_delta/
58 |3,7: /4 width=1 by lsubr_pair, cpr_bind, cpr_beta/
59 |4,6: /3 width=1 by cpr_flat, cpr_eps/
60 |5,8: /4 width=3 by lsubr_pair, cpr_zeta, cpr_theta/
64 (* Basic_1: was by definition: pr2_free *)
65 lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
67 lapply (lsubr_cpr_trans … HT12 L ?) //
70 (* Basic_1: includes by definition: pr0_refl *)
71 lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
72 #G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/
75 (* Basic_1: was: pr2_head_1 *)
76 lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
77 ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
78 * /2 width=1 by cpr_bind, cpr_flat/ qed.
80 lemma cpr_delift: ∀G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓓV) →
81 ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
82 #G #K #V #T1 elim T1 -T1
83 [ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
84 #i #L #d #HLK elim (lt_or_eq_or_gt i d)
85 #Hid [1,3: /3 width=4 by cpr_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
87 elim (lift_total V 0 (i+1)) #W #HVW
88 elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
89 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
90 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
91 [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
92 | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
97 fact lstas_cpr_aux: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
98 l = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
99 #h #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2 -l
100 /3 width=1 by cpr_eps, cpr_flat, cpr_bind/
101 [ #G #L #l #k #H0 destruct normalize //
102 | #G #L #K #V1 #V2 #W2 #i #l #HLK #_ #HVW2 #IHV12 #H destruct
103 /3 width=6 by cpr_delta/
104 | #G #L #K #V1 #V2 #W2 #i #l #_ #_ #_ #_ <plus_n_Sm #H destruct
108 lemma lstas_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, 0] T2 → ⦃G, L⦄ ⊢ T1 ➡ T2.
109 /2 width=4 by lstas_cpr_aux/ qed.
111 (* Basic inversion lemmas ***************************************************)
113 fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
115 ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
116 ⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
117 #G #L #T1 #T2 * -G -L -T1 -T2
118 [ #I #G #L #J #H destruct /2 width=1 by or_introl/
119 | #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/
120 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
121 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
122 | #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
123 | #G #L #V #T1 #T2 #_ #J #H destruct
124 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
125 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
129 lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
131 ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
132 ⇧[O, i + 1] V2 ≡ T2 & I = LRef i.
133 /2 width=3 by cpr_inv_atom1_aux/ qed-.
135 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
136 lemma cpr_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k.
138 elim (cpr_inv_atom1 … H) -H //
139 * #K #V #V2 #i #_ #_ #_ #H destruct
142 (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
143 lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
145 ∃∃K,V,V2. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
148 elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
149 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
152 lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
154 elim (cpr_inv_atom1 … H) -H //
155 * #K #V #V2 #i #_ #_ #_ #H destruct
158 fact cpr_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
159 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
160 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
163 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T &
165 #G #L #U1 #U2 * -L -U1 -U2
166 [ #I #G #L #b #J #W1 #U1 #H destruct
167 | #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
168 | #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/
169 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
170 | #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
171 | #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
172 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
173 | #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
177 lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
178 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
181 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T &
183 /2 width=3 by cpr_inv_bind1_aux/ qed-.
185 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
186 lemma cpr_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
187 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 &
190 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
191 #a #G #L #V1 #T1 #U2 #H
192 elim (cpr_inv_bind1 … H) -H *
193 /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
196 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
197 lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
198 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 &
200 #a #G #L #V1 #T1 #U2 #H
201 elim (cpr_inv_bind1 … H) -H *
202 [ /3 width=5 by ex3_2_intro/
203 | #T #_ #_ #_ #H destruct
207 fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
208 ∀I,V1,U1. U = ⓕ{I}V1.U1 →
209 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
211 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
212 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
213 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
214 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
215 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
216 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
218 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
219 #G #L #U #U2 * -L -U -U2
220 [ #I #G #L #J #W1 #U1 #H destruct
221 | #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
222 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
223 | #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/
224 | #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
225 | #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/
226 | #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/
227 | #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/
231 lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
232 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
234 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
235 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
236 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
237 U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
238 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
239 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
241 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
242 /2 width=3 by cpr_inv_flat1_aux/ qed-.
244 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
245 lemma cpr_inv_appl1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 →
246 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
248 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
249 ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 &
250 U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
251 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
252 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
253 U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
254 #G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
255 [ /3 width=5 by or3_intro0, ex3_2_intro/
257 | /3 width=11 by or3_intro1, ex5_6_intro/
258 | /3 width=13 by or3_intro2, ex6_7_intro/
262 (* Note: the main property of simple terms *)
263 lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
264 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
266 #G #L #V1 #T1 #U #H #HT1
267 elim (cpr_inv_appl1 … H) -H *
269 | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
270 elim (simple_inv_bind … HT1)
271 | #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
272 elim (simple_inv_bind … HT1)
276 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
277 lemma cpr_inv_cast1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
278 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
280 ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
281 #G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
282 [ /3 width=5 by ex3_2_intro, or_introl/
283 | /2 width=1 by or_intror/
284 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
285 | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
289 (* Basic forward lemmas *****************************************************)
291 lemma cpr_fwd_bind1_minus: ∀I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
292 ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
294 #I #G #L #V1 #T1 #T #H #b
295 elim (cpr_inv_bind1 … H) -H *
296 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/
297 | #T2 #_ #_ #H destruct
301 (* Basic_1: removed theorems 11:
302 pr0_subst0_back pr0_subst0_fwd pr0_subst0
303 pr2_head_2 pr2_cflat clear_pr2_trans
304 pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
305 pr2_gen_ctail pr2_ctail
307 (* Basic_1: removed local theorems 4:
308 pr0_delta_eps pr0_cong_delta
309 pr2_free_free pr2_free_delta