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/substitution/fsupp.ma".
17 include "basic_2/substitution/lpss_ldrop.ma".
18 include "basic_2/reduction/lpr_ldrop.ma".
20 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
22 (* Properties on context-sensitive parallel substitution for terms **********)
24 fact cpr_cpss_conf_lpr_lpss_atom_atom:
25 ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T.
28 fact cpr_cpss_conf_lpr_lpss_atom_delta:
30 ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
31 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
32 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
33 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
35 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
36 ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
37 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
38 ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T.
39 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
40 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
41 elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
42 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
43 elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
44 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
45 lapply (fsupp_lref … HLK0) -HLK0 #HLK0
46 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
47 elim (lift_total V 0 (i+1)) #T #HVT
48 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
51 fact cpr_cpss_conf_lpr_lpss_delta_atom:
53 ∀L,T.⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
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 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
61 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T.
62 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02
63 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
64 elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
65 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
66 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
67 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
68 lapply (fsupp_lref … HLK0) -HLK0 #HLK0
69 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
70 elim (lift_total V 0 (i+1)) #T #HVT
71 lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/
74 fact cpr_cpss_conf_lpr_lpss_delta_delta:
76 ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
77 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
78 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
79 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
81 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
82 ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
83 ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
84 ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
85 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
86 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T.
87 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
88 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
89 lapply (ldrop_mono … H … HLK0) -H #H destruct
90 elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
91 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
92 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
93 elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
94 elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
95 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
96 lapply (fsupp_lref … HLK0) -HLK0 #HLK0
97 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
98 elim (lift_total V 0 (i+1)) #T #HVT
99 lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
100 lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
103 fact cpr_cpss_conf_lpr_lpss_bind_bind:
105 ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
106 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
107 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
108 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
110 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
111 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
112 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
113 ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
114 #a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
115 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
116 elim (IH … HV01 … HV02 … HL01 … HL02) //
117 elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
120 fact cpr_cpss_conf_lpr_lpss_bind_zeta:
122 ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L, T⦄ →
123 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
124 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
125 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
127 ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
128 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 →
129 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
130 ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T.
131 #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
132 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
133 elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
134 elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/
137 fact cpr_cpss_conf_lpr_lpss_flat_flat:
139 ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
140 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
141 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
142 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
144 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
145 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
146 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
147 ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
148 #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
149 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
150 elim (IH … HV01 … HV02 … HL01 … HL02) //
151 elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
154 fact cpr_cpss_conf_lpr_lpss_flat_tau:
156 ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L, T⦄ →
157 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
158 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
159 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
161 ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 →
162 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
163 ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T.
164 #L0 #V0 #T0 #IH #T1 #HT01
165 #V2 #T2 #HT02 #L1 #HL01 #L2 #HL02
166 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
169 fact cpr_cpss_conf_lpr_lpss_flat_beta:
171 ∀L,T. ⦃L0,ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
172 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
173 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
174 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
176 ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
177 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 →
178 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
179 ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
180 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
181 #V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
182 elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
183 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
184 elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
185 lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/
188 fact cpr_cpss_conf_lpr_lpss_flat_theta:
190 ∀L,T. ⦃L0,ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
191 ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 →
192 ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 →
193 ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0
195 ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
196 ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
197 ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 →
198 ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 →
199 ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T.
200 #a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
201 #V2 #HV02 #X #H #L1 #HL01 #L2 #HL02
202 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
203 elim (lift_total V 0 1) #U #HVU
204 lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1
205 elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct
206 elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
207 elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
208 /4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *)
211 lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss.
212 #L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
213 [ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
214 elim (cpr_inv_atom1 … H1) -H1
215 elim (cpss_inv_atom1 … H2) -H2
217 /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/
218 | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
219 /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/
220 | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
221 /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/
222 | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
223 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
224 /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/
226 | #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
227 elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
228 elim (cpr_inv_bind1 … H1) -H1 *
229 [ #V1 #T1 #HV01 #HT01 #H1
230 | #T1 #HT01 #HXT1 #H11 #H12
232 [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/
233 | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/
235 | #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
236 elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2
237 elim (cpr_inv_flat1 … H1) -H1 *
238 [ #V1 #T1 #HV01 #HT01 #H1
240 | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
241 | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
243 [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/
244 | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/
245 | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/
246 | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/
251 (* Basic_1: includes: pr0_subst1 *)
252 (* Basic_1: was: pr2_subst1 *)
253 lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L).
254 /2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-.
256 lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
257 ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
258 #L0 #T0 #T1 #HT01 #L1 #HL01
259 elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/
262 lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 →
263 ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T.
264 #L0 #T0 #T1 #HT01 #L1 #HL01
265 elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
268 (* Basic_1: includes: pr0_subst1_fwd *)
269 lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 →
270 ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T.
271 #L0 #T0 #T1 #HT01 #L1 #HL01
272 elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/
275 (* Properties on sn parallel substitution on local environments *************)
277 lemma lpr_lpss_conf: confluent2 … lpr lpss.
278 /3 width=6 by lpx_sn_conf, cpr_cpss_conf_lpr_lpss/