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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/relocation/ldrop_append.ma".
17 (* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
19 inductive cpss: lenv → relation term ≝
20 | cpss_atom : ∀I,L. cpss L (⓪{I}) (⓪{I})
21 | cpss_delta: ∀L,K,V,V2,W2,i.
22 ⇩[0, i] L ≡ K. ⓓV → cpss K V V2 →
23 ⇧[0, i + 1] V2 ≡ W2 → cpss L (#i) W2
24 | cpss_bind : ∀a,I,L,V1,V2,T1,T2.
25 cpss L V1 V2 → cpss (L. ⓑ{I} V1) T1 T2 →
26 cpss L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
27 | cpss_flat : ∀I,L,V1,V2,T1,T2.
28 cpss L V1 V2 → cpss L T1 T2 →
29 cpss L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
32 interpretation "context-sensitive parallel substitution (term)"
33 'PSubstStar L T1 T2 = (cpss L T1 T2).
35 (* Basic properties *********************************************************)
37 (* Note: it does not hold replacing |L1| with |L2| *)
38 lemma cpss_lsubr_trans: ∀L1,T1,T2. L1 ⊢ T1 ▶* T2 →
39 ∀L2. L2 ⊑ [0, |L1|] L1 → L2 ⊢ T1 ▶* T2.
40 #L1 #T1 #T2 #H elim H -L1 -T1 -T2
42 | #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
43 lapply (ldrop_fwd_ldrop2_length … HLK1) #Hi
44 lapply (ldrop_fwd_O1_length … HLK1) #H2i
45 elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // -Hi
46 <H2i -H2i <minus_plus_m_m /3 width=6/
52 (* Basic_1: was by definition: subst1_refl *)
53 lemma cpss_refl: ∀T,L. L ⊢ T ▶* T.
55 #I elim I -I /2 width=1/
58 (* Basic_1: was only: subst1_ex *)
59 lemma cpss_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
60 ∃∃T2,T. L ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
62 [ * #i #L #d #HLK /2 width=4/
63 elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
65 elim (lift_total V 0 (i+1)) #W #HVW
66 elim (lift_split … HVW i i ? ? ?) // /3 width=6/
67 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
68 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
69 [ elim (IHU1 (L. ⓑ{I} W1) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=9/
70 | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
75 lemma cpss_append: l_appendable_sn … cpss.
76 #K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/
77 #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
78 lapply (ldrop_fwd_ldrop2_length … HK0) #H
79 @(cpss_delta … (L@@K0) V1 … HVW2) //
80 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
83 (* Basic inversion lemmas ***************************************************)
85 fact cpss_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ∀I. T1 = ⓪{I} →
87 ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
91 #L #T1 #T2 * -L -T1 -T2
92 [ #I #L #J #H destruct /2 width=1/
93 | #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #I #H destruct /3 width=8/
94 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
95 | #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
99 lemma cpss_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ▶* T2 →
101 ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
103 ⇧[O, i + 1] V2 ≡ T2 &
105 /2 width=3 by cpss_inv_atom1_aux/ qed-.
107 (* Basic_1: was only: subst1_gen_sort *)
108 lemma cpss_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ▶* T2 → T2 = ⋆k.
110 elim (cpss_inv_atom1 … H) -H //
111 * #K #V #V2 #i #_ #_ #_ #H destruct
114 (* Basic_1: was only: subst1_gen_lref *)
115 lemma cpss_inv_lref1: ∀L,T2,i. L ⊢ #i ▶* T2 →
117 ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
121 elim (cpss_inv_atom1 … H) -H /2 width=1/
122 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
125 lemma cpss_inv_gref1: ∀L,T2,p. L ⊢ §p ▶* T2 → T2 = §p.
127 elim (cpss_inv_atom1 … H) -H //
128 * #K #V #V2 #i #_ #_ #_ #H destruct
131 fact cpss_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
132 ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
133 ∃∃V2,T2. L ⊢ V1 ▶* V2 &
134 L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
136 #L #U1 #U2 * -L -U1 -U2
137 [ #I #L #b #J #W1 #U1 #H destruct
138 | #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
139 | #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5/
140 | #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
144 lemma cpss_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* U2 →
145 ∃∃V2,T2. L ⊢ V1 ▶* V2 &
146 L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
148 /2 width=3 by cpss_inv_bind1_aux/ qed-.
150 fact cpss_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
151 ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
152 ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
154 #L #U1 #U2 * -L -U1 -U2
155 [ #I #L #J #W1 #U1 #H destruct
156 | #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
157 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
158 | #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5/
162 lemma cpss_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* U2 →
163 ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
165 /2 width=3 by cpss_inv_flat1_aux/ qed-.
167 (* Basic forward lemmas *****************************************************)
169 lemma cpss_fwd_tw: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ♯{T1} ≤ ♯{T2}.
170 #L #T1 #T2 #H elim H -L -T1 -T2 normalize
171 /3 width=1 by monotonic_le_plus_l, le_plus/ (**) (* auto is too slow without trace *)
174 lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
175 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
176 #L1 @(lenv_ind_dx … L1) -L1 normalize
178 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
179 | #I #L1 #V1 #IH #L #T1 #X
180 >shift_append_assoc normalize #H
181 elim (cpss_inv_bind1 … H) -H
182 #V0 #T0 #_ #HT10 #H destruct
183 elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
184 >append_length >HL12 -HL12
185 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
189 (* Basic_1: removed theorems 27:
190 subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
191 subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
192 subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
193 subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
194 subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
195 subst0_confluence_lift subst0_tlt
196 subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
197 subst1_gen_lift_eq subst1_confluence_neq