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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/grammar/lpx_sn_lpx_sn.ma".
16 include "basic_2/substitution/lpss_ldrop.ma".
17 include "basic_2/reduction/lpr_ldrop.ma".
19 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
21 (* Properties on context-sensitive parallel substitution for terms **********)
23 fact cpr_cpss_conf_lpr_lpss_atom_atom:
24 ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T.
27 fact cpr_cpss_conf_lpr_lpss_atom_delta:
29 ∀L,T.♯{L, T} < ♯{L0, #i} →
30 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
31 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
32 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
34 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
35 ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
36 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
37 ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T.
38 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
39 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
40 elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
41 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
42 elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
43 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
44 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
45 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
46 elim (lift_total V 0 (i+1)) #T #HVT
47 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
50 fact cpr_cpss_conf_lpr_lpss_delta_atom:
52 ∀L,T.♯{L, T} < ♯{L0, #i} →
53 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
54 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
55 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
57 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
58 ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
59 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
60 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T.
61 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02
62 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
63 elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
64 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
65 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
66 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
67 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
68 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
69 elim (lift_total V 0 (i+1)) #T #HVT
70 lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/
73 fact cpr_cpss_conf_lpr_lpss_delta_delta:
75 ∀L,T.♯{L, T} < ♯{L0, #i} →
76 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
77 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
78 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
80 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
81 ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
82 ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
83 ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
84 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
85 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T.
86 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
87 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
88 lapply (ldrop_mono … H … HLK0) -H #H destruct
89 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
90 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
91 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
92 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
93 elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
94 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
95 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
96 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
97 elim (lift_total V 0 (i+1)) #T #HVT
98 lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
99 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
102 fact cpr_cpss_conf_lpr_lpss_bind_bind:
104 ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
105 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
106 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
107 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
109 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
110 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
111 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
112 ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
113 #a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
114 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
115 elim (IH … HV01 … HV02 … HL01 … HL02) //
116 elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
119 fact cpr_cpss_conf_lpr_lpss_bind_zeta:
121 ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
122 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
123 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
124 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
126 ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
127 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 →
128 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
129 ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T.
130 #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
131 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
132 elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
133 elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/
136 fact cpr_cpss_conf_lpr_lpss_flat_flat:
138 ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
139 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
140 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
141 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
143 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
144 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
145 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
146 ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
147 #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
148 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
149 elim (IH … HV01 … HV02 … HL01 … HL02) //
150 elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
153 fact cpr_cpss_conf_lpr_lpss_flat_tau:
155 ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
156 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
157 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
158 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
160 ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 →
161 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
162 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T.
163 #L0 #V0 #T0 #IH #T1 #HT01
164 #V2 #T2 #HT02 #L1 #HL01 #L2 #HL02
165 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
168 fact cpr_cpss_conf_lpr_lpss_flat_beta:
170 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
171 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
172 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
173 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
175 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
176 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 →
177 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
178 ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
179 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
180 #V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
181 elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
182 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
183 elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
184 lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/
187 fact cpr_cpss_conf_lpr_lpss_flat_theta:
189 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
190 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
191 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
192 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
194 ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
195 ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
196 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 →
197 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
198 ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
199 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
200 #V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
201 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
202 elim (lift_total V 0 1) #U #HVU
203 lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1
204 elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
205 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
206 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
207 /4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *)
210 lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss.
211 #L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
212 [ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
213 elim (cpr_inv_atom1 … H1) -H1
214 elim (cpss_inv_atom1 … H2) -H2
216 /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/
217 | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
218 /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/
219 | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
220 /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/
221 | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
222 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
223 /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/
225 | #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
226 elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
227 elim (cpr_inv_bind1 … H1) -H1 *
228 [ #V1 #T1 #HV01 #HT01 #H1
229 | #T1 #HT01 #HXT1 #H11 #H12
231 [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/
232 | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/
234 | #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
235 elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
236 elim (cpr_inv_flat1 … H1) -H1 *
237 [ #V1 #T1 #HV01 #HT01 #H1
239 | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
240 | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
242 [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/
243 | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/
244 | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/
245 | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/
250 (* Basic_1: includes: pr0_subst1 *)
251 lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L).
252 /2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-.
254 lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
255 ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
256 #L0 #T0 #T1 #HT01 #L1 #HL01
257 elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/
260 lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
261 ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
262 #L0 #T0 #T1 #HT01 #L1 #HL01
263 elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
266 (* Basic_1: includes: pr0_subst1_fwd *)
267 lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 →
268 ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T.
269 #L0 #T0 #T1 #HT01 #L1 #HL01
270 elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/