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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/unfold/lpqs_cpqs.ma".
17 (* SN RESTRICTED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *****************)
19 (* Main properties on context-sensitive rest parallel computation for terms *)
21 fact cpqs_conf_lpqs_atom_atom:
22 ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T.
25 fact cpqs_conf_lpqs_atom_delta:
27 ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
28 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
29 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
30 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
32 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
33 ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
34 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
35 ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T.
36 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
37 elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
38 elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
39 elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
40 elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
41 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
42 lapply (fsupp_lref … HLK0) -HLK0 #HLK0
43 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
44 elim (lift_total V 0 (i+1)) #T #HVT
45 lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
48 fact cpqs_conf_lpqs_delta_delta:
50 ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
51 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
52 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
53 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
55 ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
56 ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
57 ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
58 ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
59 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
60 ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
61 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
62 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
63 lapply (ldrop_mono … H … HLK0) -H #H destruct
64 elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
65 elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
66 lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
67 elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
68 elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
69 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
70 lapply (fsupp_lref … HLK0) -HLK0 #HLK0
71 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
72 elim (lift_total V 0 (i+1)) #T #HVT
73 lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
74 lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
77 fact cpqs_conf_lpqs_bind_bind:
79 ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
80 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
81 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
82 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
84 ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 →
85 ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 →
86 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
87 ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T.
88 #a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
89 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
90 elim (IH … HV01 … HV02 … HL01 … HL02) //
91 elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
94 fact cpqs_conf_lpqs_bind_zeta:
96 ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ →
97 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
98 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
99 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
101 ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 →
102 ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
103 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
104 ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T.
105 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
106 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
107 elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
108 elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
111 fact cpqs_conf_lpqs_zeta_zeta:
113 ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L,T⦄ →
114 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
115 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
116 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
118 ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
119 ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
120 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
121 ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T.
122 #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
123 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
124 elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
125 elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
126 elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2
127 lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
130 fact cpqs_conf_lpqs_flat_flat:
132 ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
133 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
134 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
135 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
137 ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 →
138 ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 →
139 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
140 ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T.
141 #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
142 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
143 elim (IH … HV01 … HV02 … HL01 … HL02) //
144 elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
147 fact cpqs_conf_lpqs_flat_tau:
149 ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ →
150 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
151 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
152 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
154 ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
155 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
156 ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T.
157 #L0 #V0 #T0 #IH #V1 #T1 #HT01
158 #T2 #HT02 #L1 #HL01 #L2 #HL02
159 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
162 fact cpqs_conf_lpqs_tau_tau:
164 ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L,T⦄ →
165 ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 →
166 ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 →
167 ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0
169 ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 →
170 ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 →
171 ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T.
172 #L0 #V0 #T0 #IH #T1 #HT01
173 #T2 #HT02 #L1 #HL01 #L2 #HL02
174 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
177 theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs.
178 #L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
179 [ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
180 elim (cpqs_inv_atom1 … H1) -H1
181 elim (cpqs_inv_atom1 … H2) -H2
183 /2 width=1 by cpqs_conf_lpqs_atom_atom/
184 | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
185 /3 width=10 by cpqs_conf_lpqs_atom_delta/
186 | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
187 /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/
188 | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
189 * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
190 /3 width=17 by cpqs_conf_lpqs_delta_delta/
192 | #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
193 elim (cpqs_inv_bind1 … H1) -H1 *
194 [ #V1 #T1 #HV01 #HT01 #H1
195 | #T1 #HT01 #HXT1 #H11 #H12
197 elim (cpqs_inv_bind1 … H2) -H2 *
198 [1,3: #V2 #T2 #HV02 #HT02 #H2
199 |2,4: #T2 #HT02 #HXT2 #H21 #H22
201 [ /3 width=10 by cpqs_conf_lpqs_bind_bind/
202 | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/
203 | /3 width=11 by cpqs_conf_lpqs_bind_zeta/
204 | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/
206 | #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
207 elim (cpqs_inv_flat1 … H1) -H1 *
208 [ #V1 #T1 #HV01 #HT01 #H1
211 elim (cpqs_inv_flat1 … H2) -H2 *
212 [1,3: #V2 #T2 #HV02 #HT02 #H2
215 [ /3 width=10 by cpqs_conf_lpqs_flat_flat/
216 | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/
217 | /3 width=8 by cpqs_conf_lpqs_flat_tau/
218 | /3 width=7 by cpqs_conf_lpqs_tau_tau/
223 theorem cpqs_conf: ∀L. confluent … (cpqs L).
224 /2 width=6 by cpqs_conf_lpqs/ qed-.
226 (* Properties on context-sensitive rest. parallel computation for terms *****)
228 lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
229 ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T.
230 #L0 #T0 #T1 #HT01 #L1 #HL01
231 elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
234 lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 →
235 ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T.
236 #L0 #T0 #T1 #HT01 #L1 #HL01
237 elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
240 (* Main properties **********************************************************)
242 theorem lpqs_conf: confluent … lpqs.
243 /3 width=6 by lpx_sn_conf, cpqs_conf_lpqs/
246 theorem lpqs_trans: Transitive … lpqs.
247 /3 width=5 by lpx_sn_trans, cpqs_trans_lpqs/
250 (* Advanced forward lemmas **************************************************)
252 lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
253 ∃∃L2,T2. L @@ L1 ⊢ ➤* L @@ L2 & L @@ L1 ⊢ T1 ➤* T2 &
255 #L1 @(lenv_ind_dx … L1) -L1
257 @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
258 | #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
259 elim (cpqs_inv_bind1 … H) -H *
260 [ #V2 #T2 #HV12 #HT12 #H destruct
261 elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
262 lapply (lpqs_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
263 @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
264 | #T #_ #_ #H destruct