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7 (* ||T|| The HELM team. *)
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15 include "basic_2/substitution/cpss.ma".
17 (* CONTEXT-SENSITIVE RESTRICTED PARALLEL COMPUTATION FOR TERMS **************)
19 inductive cpqs: lenv → relation term ≝
20 | cpqs_atom : ∀I,L. cpqs L (⓪{I}) (⓪{I})
21 | cpqs_delta: ∀L,K,V,V2,W2,i.
22 ⇩[0, i] L ≡ K. ⓓV → cpqs K V V2 →
23 ⇧[0, i + 1] V2 ≡ W2 → cpqs L (#i) W2
24 | cpqs_bind : ∀a,I,L,V1,V2,T1,T2.
25 cpqs L V1 V2 → cpqs (L. ⓑ{I} V1) T1 T2 →
26 cpqs L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
27 | cpqs_flat : ∀I,L,V1,V2,T1,T2.
28 cpqs L V1 V2 → cpqs L T1 T2 →
29 cpqs L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
30 | cpqs_zeta : ∀L,V,T1,T,T2. cpqs (L.ⓓV) T1 T →
31 ⇧[0, 1] T2 ≡ T → cpqs L (+ⓓV. T1) T2
32 | cpqs_tau : ∀L,V,T1,T2. cpqs L T1 T2 → cpqs L (ⓝV. T1) T2
35 interpretation "context-sensitive restricted parallel computation (term)"
36 'PRestStar L T1 T2 = (cpqs L T1 T2).
38 (* Basic properties *********************************************************)
40 lemma cpqs_lsubr_trans: lsub_trans … cpqs lsubr.
41 #L1 #T1 #T2 #H elim H -L1 -T1 -T2
43 | #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
44 elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
51 lemma cpss_cpqs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➤* T2.
52 #L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=6/
55 lemma cpqs_refl: ∀T,L. L ⊢ T ➤* T.
58 lemma cpqs_delift: ∀L,K,V,T1,d. ⇩[0, d] L ≡ (K. ⓓV) →
59 ∃∃T2,T. L ⊢ T1 ➤* T2 & ⇧[d, 1] T ≡ T2.
61 elim (cpss_delift … T1 … HLK) -HLK /3 width=4/
64 lemma cpqs_append: l_appendable_sn … cpqs.
65 #K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/
66 #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
67 lapply (ldrop_fwd_length_lt2 … HK0) #H
68 @(cpqs_delta … (L@@K0) V1 … HVW2) //
69 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
72 (* Basic inversion lemmas ***************************************************)
74 fact cpqs_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➤* T2 → ∀I. T1 = ⓪{I} →
76 ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
80 #L #T1 #T2 * -L -T1 -T2
81 [ #I #L #J #H destruct /2 width=1/
82 | #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/
83 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
84 | #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
85 | #L #V #T1 #T #T2 #_ #_ #J #H destruct
86 | #L #V #T1 #T2 #_ #J #H destruct
90 lemma cpqs_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➤* T2 →
92 ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
96 /2 width=3 by cpqs_inv_atom1_aux/ qed-.
98 lemma cpqs_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➤* T2 → T2 = ⋆k.
100 elim (cpqs_inv_atom1 … H) -H //
101 * #K #V #V2 #i #_ #_ #_ #H destruct
104 lemma cpqs_inv_lref1: ∀L,T2,i. L ⊢ #i ➤* T2 →
106 ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
110 elim (cpqs_inv_atom1 … H) -H /2 width=1/
111 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
114 lemma cpqs_inv_gref1: ∀L,T2,p. L ⊢ §p ➤* T2 → T2 = §p.
116 elim (cpqs_inv_atom1 … H) -H //
117 * #K #V #V2 #i #_ #_ #_ #H destruct
120 fact cpqs_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
121 ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 → (
122 ∃∃V2,T2. L ⊢ V1 ➤* V2 &
123 L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
126 ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
127 #L #U1 #U2 * -L -U1 -U2
128 [ #I #L #b #J #W1 #U1 #H destruct
129 | #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
130 | #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /3 width=5/
131 | #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
132 | #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W1 #U1 #H destruct /3 width=3/
133 | #L #V #T1 #T2 #_ #b #J #W1 #U1 #H destruct
137 lemma cpqs_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ➤* U2 → (
138 ∃∃V2,T2. L ⊢ V1 ➤* V2 &
139 L. ⓑ{I} V1 ⊢ T1 ➤* T2 &
142 ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
143 /2 width=3 by cpqs_inv_bind1_aux/ qed-.
145 lemma cpqs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a} V1. T1 ➤* U2 → (
146 ∃∃V2,T2. L ⊢ V1 ➤* V2 &
150 ∃∃T. L.ⓓV1 ⊢ T1 ➤* T & ⇧[0, 1] U2 ≡ T & a = true.
152 elim (cpqs_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
155 lemma cpqs_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a} V1. T1 ➤* U2 →
156 ∃∃V2,T2. L ⊢ V1 ➤* V2 &
160 elim (cpqs_inv_bind1 … H) -H *
162 | #T #_ #_ #_ #H destruct
166 fact cpqs_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ➤* U2 →
167 ∀I,V1,T1. U1 = ⓕ{I} V1. T1 → (
168 ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
171 (L ⊢ T1 ➤* U2 ∧ I = Cast).
172 #L #U1 #U2 * -L -U1 -U2
173 [ #I #L #J #W1 #U1 #H destruct
174 | #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
175 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
176 | #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /3 width=5/
177 | #L #V #T1 #T #T2 #_ #_ #J #W1 #U1 #H destruct
178 | #L #V #T1 #T2 #HT12 #J #W1 #U1 #H destruct /3 width=1/
182 lemma cpqs_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ➤* U2 → (
183 ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
186 (L ⊢ T1 ➤* U2 ∧ I = Cast).
187 /2 width=3 by cpqs_inv_flat1_aux/ qed-.
189 lemma cpqs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐ V1. T1 ➤* U2 →
190 ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
192 #L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H *
198 lemma cpqs_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓝ V1. T1 ➤* U2 → (
199 ∃∃V2,T2. L ⊢ V1 ➤* V2 & L ⊢ T1 ➤* T2 &
203 #L #V1 #T1 #U2 #H elim (cpqs_inv_flat1 … H) -H * /2 width=1/ /3 width=5/
206 (* Basic forward lemmas *****************************************************)
208 lemma cpqs_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➤* T →
209 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
210 #L1 @(lenv_ind_dx … L1) -L1 normalize
212 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
213 | #I #L1 #V1 #IH #L #T1 #X
214 >shift_append_assoc normalize #H
215 elim (cpqs_inv_bind1 … H) -H *
216 [ #V0 #T0 #_ #HT10 #H destruct
217 elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
218 >append_length >HL12 -HL12
219 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
220 | #T #_ #_ #H destruct