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/grammar/lpx_sn_lpx_sn.ma".
16 include "basic_2/relocation/fsup.ma".
17 include "basic_2/reduction/lpr_ldrop.ma".
19 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
21 (* Main properties on context-sensitive parallel reduction for terms ********)
23 fact cpr_conf_lpr_atom_atom:
24 ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➡ T & L2 ⊢ ⓪{I} ➡ T.
27 fact cpr_conf_lpr_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 (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
42 elim (lpr_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 (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
51 fact cpr_conf_lpr_delta_delta:
53 ∀L,T.♯{L, T} < ♯{L0, #i} →
54 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
55 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
56 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
58 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
59 ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
60 ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
61 ∀V2. KX ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
62 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
63 ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T.
64 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
65 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
66 lapply (ldrop_mono … H … HLK0) -H #H destruct
67 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
68 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
69 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
70 elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
71 elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
72 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
73 lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
74 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
75 elim (lift_total V 0 (i+1)) #T #HVT
76 lapply (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
77 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
80 fact cpr_conf_lpr_bind_bind:
82 ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
83 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
84 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
85 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
87 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
88 ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➡ T2 →
89 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
90 ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➡ T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
91 #a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
92 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
93 elim (IH … HV01 … HV02 … HL01 … HL02) //
94 elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
97 fact cpr_conf_lpr_bind_zeta:
99 ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
100 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
101 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
102 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
104 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 →
105 ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
106 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
107 ∃∃T. L1 ⊢ +ⓓV1.T1 ➡ T & L2 ⊢ X2 ➡ T.
108 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
109 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
110 elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
111 elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
114 fact cpr_conf_lpr_zeta_zeta:
116 ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
117 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
118 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
119 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
121 ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
122 ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
123 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
124 ∃∃T. L1 ⊢ X1 ➡ T & L2 ⊢ X2 ➡ T.
125 #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
126 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
127 elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
128 elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
129 elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
130 lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
133 fact cpr_conf_lpr_flat_flat:
135 ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
136 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
137 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
138 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
140 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
141 ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0 ⊢ T0 ➡ T2 →
142 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
143 ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➡ T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
144 #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
145 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
146 elim (IH … HV01 … HV02 … HL01 … HL02) //
147 elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
150 fact cpr_conf_lpr_flat_tau:
152 ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
153 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
154 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
155 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
157 ∀V1,T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
158 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
159 ∃∃T. L1 ⊢ ⓝV1.T1 ➡ T & L2 ⊢ T2 ➡ T.
160 #L0 #V0 #T0 #IH #V1 #T1 #HT01
161 #T2 #HT02 #L1 #HL01 #L2 #HL02
162 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
165 fact cpr_conf_lpr_tau_tau:
167 ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
168 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
169 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
170 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
172 ∀T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
173 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
174 ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T.
175 #L0 #V0 #T0 #IH #T1 #HT01
176 #T2 #HT02 #L1 #HL01 #L2 #HL02
177 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
180 fact cpr_conf_lpr_flat_beta:
182 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
183 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
184 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
185 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
187 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓛ{a}W0.T0 ➡ T1 →
188 ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 →
189 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
190 ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T.
191 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
192 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
193 elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
194 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
195 elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW1)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
196 lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/
199 (* Basic-1: includes:
200 pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
201 pr0_cong_upsilon_cong pr0_cong_upsilon_delta
203 fact cpr_conf_lpr_flat_theta:
205 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
206 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
207 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
208 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
210 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓓ{a}W0.T0 ➡ T1 →
211 ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
212 ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 →
213 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
214 ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
215 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
216 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
217 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
218 elim (lift_total V 0 1) #U #HVU
219 lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ #HU2
220 elim (cpr_inv_abbr1 … H) -H *
221 [ #W1 #T1 #HW01 #HT01 #H destruct
222 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
223 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 /4 width=7/
224 | #T1 #HT01 #HXT1 #H destruct
225 elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
226 elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /2 width=1/ #Y #HYT #HXY
227 @(ex2_intro … (ⓐV.Y)) /2 width=1/ /3 width=5/ (**) (* auto /4 width=9/ is too slow *)
231 fact cpr_conf_lpr_beta_beta:
233 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
234 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
235 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
236 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
238 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
239 ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 →
240 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
241 ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T.
242 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
243 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
244 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
245 elim (IH … HT01 … HT02 (L1.ⓛW0) … (L2.ⓛW0)) /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
246 lapply (cpr_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/
247 lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/
250 (* Basic_1: was: pr0_upsilon_upsilon *)
251 fact cpr_conf_lpr_theta_theta:
253 ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
254 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
255 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
256 ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
258 ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
259 ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
260 ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
261 ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 →
262 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
263 ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
264 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
265 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
266 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
267 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
268 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
269 elim (lift_total V 0 1) #U #HVU
270 lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/
271 lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ /4 width=7/
274 theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr.
275 #L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
276 [ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
277 elim (cpr_inv_atom1 … H1) -H1
278 elim (cpr_inv_atom1 … H2) -H2
280 /2 width=1 by cpr_conf_lpr_atom_atom/
281 | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
282 /3 width=10 by cpr_conf_lpr_atom_delta/
283 | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
284 /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
285 | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
286 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
287 /3 width=17 by cpr_conf_lpr_delta_delta/
289 | #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
290 elim (cpr_inv_bind1 … H1) -H1 *
291 [ #V1 #T1 #HV01 #HT01 #H1
292 | #T1 #HT01 #HXT1 #H11 #H12
294 elim (cpr_inv_bind1 … H2) -H2 *
295 [1,3: #V2 #T2 #HV02 #HT02 #H2
296 |2,4: #T2 #HT02 #HXT2 #H21 #H22
298 [ /3 width=10 by cpr_conf_lpr_bind_bind/
299 | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
300 | /3 width=11 by cpr_conf_lpr_bind_zeta/
301 | /3 width=12 by cpr_conf_lpr_zeta_zeta/
303 | #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
304 elim (cpr_inv_flat1 … H1) -H1 *
305 [ #V1 #T1 #HV01 #HT01 #H1
307 | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
308 | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
310 elim (cpr_inv_flat1 … H2) -H2 *
311 [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
313 |3,7,11,15: #a2 #V2 #Y2 #Z2 #T2 #HV02 #HZT2 #H21 #H22 #H23
314 |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
316 [ /3 width=10 by cpr_conf_lpr_flat_flat/
317 | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/
318 | /4 width=11 by ex2_commute, cpr_conf_lpr_flat_beta/
319 | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
320 | /3 width=8 by cpr_conf_lpr_flat_tau/
321 | /3 width=7 by cpr_conf_lpr_tau_tau/
322 | /3 width=11 by cpr_conf_lpr_flat_beta/
323 | /3 width=11 by cpr_conf_lpr_beta_beta/
324 | /3 width=14 by cpr_conf_lpr_flat_theta/
325 | /3 width=17 by cpr_conf_lpr_theta_theta/
330 (* Basic_1: includes: pr0_confluence pr2_confluence *)
331 theorem cpr_conf: ∀L. confluent … (cpr L).
332 /2 width=6 by cpr_conf_lpr/ qed-.
334 (* Properties on context-sensitive parallel reduction for terms *************)
336 lemma lpr_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 →
337 ∃∃T. L1 ⊢ T0 ➡ T & L1 ⊢ T1 ➡ T.
338 #L0 #T0 #T1 #HT01 #L1 #HL01
339 elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
342 lemma lpr_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 →
343 ∃∃T. L1 ⊢ T0 ➡ T & L0 ⊢ T1 ➡ T.
344 #L0 #T0 #T1 #HT01 #L1 #HL01
345 elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
348 lemma fsup_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
349 ∃∃L,U1. L1 ⊢ ➡ L & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
350 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
351 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
352 elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
353 elim (lift_total T d e) #U #HTU
354 elim (ldrop_lpr_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K
355 lapply (cpr_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/