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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/substitution/lpss_ldrop.ma".
16 include "basic_2/reduction/lpr_ldrop.ma".
18 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
20 (* Properties on context-sensitive parallel substitution for terms **********)
22 fact cpr_cpss_conf_lpr_lpss_atom_atom:
23 ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T.
26 fact cpr_cpss_conf_lpr_lpss_atom_delta:
28 ∀L,T.♯{L, T} < ♯{L0, #i} →
29 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
30 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
31 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
33 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
34 ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
35 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
36 ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T.
37 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
38 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
39 elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
40 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
41 elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
42 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
43 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
44 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
45 elim (lift_total V 0 (i+1)) #T #HVT
46 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
49 fact cpr_cpss_conf_lpr_lpss_delta_atom:
51 ∀L,T.♯{L, T} < ♯{L0, #i} →
52 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
53 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
54 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
56 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
57 ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
58 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
59 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T.
60 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02
61 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
62 elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
63 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
64 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
65 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
66 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
67 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
68 elim (lift_total V 0 (i+1)) #T #HVT
69 lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/
72 fact cpr_cpss_conf_lpr_lpss_delta_delta:
74 ∀L,T.♯{L, T} < ♯{L0, #i} →
75 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
76 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
77 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
79 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
80 ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
81 ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
82 ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
83 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
84 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T.
85 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
86 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
87 lapply (ldrop_mono … H … HLK0) -H #H destruct
88 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
89 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
90 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
91 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
92 elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
93 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
94 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
95 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
96 elim (lift_total V 0 (i+1)) #T #HVT
97 lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
98 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
101 fact cpr_cpss_conf_lpr_lpss_bind_bind:
103 ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
104 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
105 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
106 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
108 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
109 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
110 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
111 ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
112 #a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
113 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
114 elim (IH … HV01 … HV02 … HL01 … HL02) //
115 elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
118 fact cpr_cpss_conf_lpr_lpss_bind_zeta:
120 ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
121 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
122 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
123 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
125 ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
126 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 →
127 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
128 ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T.
129 #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
130 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
131 elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
132 elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/
135 fact cpr_cpss_conf_lpr_lpss_flat_flat:
137 ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
138 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
139 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
140 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
142 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
143 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
144 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
145 ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
146 #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
147 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
148 elim (IH … HV01 … HV02 … HL01 … HL02) //
149 elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
152 fact cpr_cpss_conf_lpr_lpss_flat_tau:
154 ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
155 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
156 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
157 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
159 ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 →
160 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
161 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T.
162 #L0 #V0 #T0 #IH #T1 #HT01
163 #V2 #T2 #HT02 #L1 #HL01 #L2 #HL02
164 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
167 fact cpr_cpss_conf_lpr_lpss_flat_beta:
169 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
170 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
171 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
172 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
174 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
175 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 →
176 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
177 ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
178 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
179 #V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
180 elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
181 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
182 elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
183 lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/
186 fact cpr_cpss_conf_lpr_lpss_flat_theta:
188 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
189 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
190 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
191 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
193 ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
194 ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
195 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 →
196 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
197 ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
198 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
199 #V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
200 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
201 elim (lift_total V 0 1) #U #HVU
202 lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1
203 elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
204 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
205 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
206 /4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *)
209 lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss.
210 #L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
211 [ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
212 elim (cpr_inv_atom1 … H1) -H1
213 elim (cpss_inv_atom1 … H2) -H2
215 /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/
216 | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
217 /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/
218 | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
219 /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/
220 | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
221 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
222 /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/
224 | #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
225 elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
226 elim (cpr_inv_bind1 … H1) -H1 *
227 [ #V1 #T1 #HV01 #HT01 #H1
228 | #T1 #HT01 #HXT1 #H11 #H12
230 [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/
231 | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/
233 | #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
234 elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
235 elim (cpr_inv_flat1 … H1) -H1 *
236 [ #V1 #T1 #HV01 #HT01 #H1
238 | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
239 | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
241 [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/
242 | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/
243 | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/
244 | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/
249 (* Basic_1: includes: pr0_subst1 *)
250 lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L).
251 /2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-.
253 lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
254 ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
255 #L0 #T0 #T1 #HT01 #L1 #HL01
256 elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/
259 lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
260 ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
261 #L0 #T0 #T1 #HT01 #L1 #HL01
262 elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
265 (* Basic_1: includes: pr0_subst1_fwd *)
266 lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 →
267 ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T.
268 #L0 #T0 #T1 #HT01 #L1 #HL01
269 elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/