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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 *)
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15 include "basic_2/substitution/ldrop_lbotr.ma".
16 include "basic_2/unfold/tpss_lift.ma".
17 include "basic_2/unfold/delift.ma".
19 (* INVERSE BASIC TERM RELOCATION *******************************************)
21 (* Advanced properties ******************************************************)
23 lemma delift_lref_be: ∀L,K,V1,V2,U2,i,d,e. d ≤ i → i < d + e →
24 ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 →
25 ⇧[0, d] V2 ≡ U2 → L ⊢ ▼*[d, e] #i ≡ U2.
26 #L #K #V1 #V2 #U2 #i #d #e #Hdi #Hide #HLK * #V #HV1 #HV2 #HVU2
27 elim (lift_total V 0 (i+1)) #U #HVU
28 lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus <plus_minus_m_m /2 width=1/ #HV2U
29 lapply (lift_conf_be … HVU2 … HV2U ?) //
30 >commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
33 lemma lbotr_delift: ∀L,T1,d,e. d + e ≤ |L| → ⊒[d, e] L →
34 ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
35 #L #T1 @(f2_ind … fw … L T1) -L -T1
36 #n #IH #L * * /2 width=2/
37 [ #i #H #d #e #Hde #HL destruct
38 elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
39 elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
40 lapply (lt_to_le_to_lt … Hide Hde) #Hi
41 elim (ldrop_O1_lt … Hi) -Hi #I #K #V1 #HLK
42 lapply (lbotr_inv_ldrop … HLK … HL ? ?) // #H destruct
43 lapply (ldrop_pair2_fwd_fw … HLK (#i)) #HKL
44 lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
45 lapply (ldrop_fwd_O1_length … HLK0) #H
46 lapply (lbotr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
47 [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
48 elim (IH … HKL … HK) -IH -HKL -HK
49 [2: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
50 elim (lift_total V2 0 d) /3 width=7/
51 | #a #I #V1 #T1 #H #d #e #Hde #HL destruct
52 elim (IH … V1 … Hde HL) // #V2 #HV12
53 elim (IH (L.ⓑ{I}V1) T1 … (d+1) e ??) -IH // [2,3: /2 width=1/ ] -Hde -HL #T2 #HT12
54 lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
55 | #I #V1 #T1 #H #d #e #Hde #HL destruct
56 elim (IH … V1 … Hde HL) // #V2 #HV12
57 elim (IH … T1 … Hde HL) -IH -Hde -HL // /3 width=2/
61 (* Advanced inversion lemmas ************************************************)
63 lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 → i < d → U2 = #i.
64 #L #U2 #i #d #e * #U #HU #HU2 #Hid
65 elim (tpss_inv_lref1 … HU) -HU
66 [ #H destruct >(lift_inv_lref2_lt … HU2) //
68 lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
69 elim (lt_refl_false … Hi)
73 lemma delift_inv_lref1_be: ∀L,U2,d,e,i. L ⊢ ▼*[d, e] #i ≡ U2 →
75 ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
76 K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
78 #L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
79 elim (tpss_inv_lref1 … HU) -HU
80 [ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
81 | * #K #V1 #V #_ #_ #HLK #HV1 #HVU
82 elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
86 lemma delift_inv_lref1_ge: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
87 d + e ≤ i → U2 = #(i - e).
88 #L #U2 #i #d #e * #U #HU #HU2 #Hdei
89 elim (tpss_inv_lref1 … HU) -HU
90 [ #H destruct >(lift_inv_lref2_ge … HU2) //
91 | * #K #V1 #V2 #_ #Hide
92 lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
93 elim (lt_refl_false … Hi)
97 lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ ▼*[d, e] #i ≡ U2 →
99 | (∃∃K,V1,V2. d ≤ i & i < d + e &
101 K ⊢ ▼*[0, d + e - i - 1] V1 ≡ V2 &
104 | (d + e ≤ i ∧ U2 = #(i - e)).
106 elim (lt_or_ge i d) #Hdi
107 [ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
108 | elim (lt_or_ge i (d+e)) #Hide
109 [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
110 | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
115 (* Properties on basic term relocation **************************************)
117 lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
118 ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
119 ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
120 L ⊢ ▼*[dt, et] U1 ≡ U2.
121 #K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
122 elim (lift_total T d e) #U #HTU
123 lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
124 elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
125 >(lift_mono … HTU2 … HT2) -T2 /2 width=3/
128 lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
129 ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
130 ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
131 L ⊢ ▼*[dt, et + e] U1 ≡ T2.
132 #K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
133 elim (lift_total T d e) #U #HTU
134 lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
135 lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
138 lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ ▼*[dt, et] T1 ≡ T2 →
139 ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
140 ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
141 L ⊢ ▼*[dt + e, et] U1 ≡ U2.
142 #K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
143 elim (lift_total T d e) #U #HTU
144 lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
145 elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
146 >(lift_mono … HTU2 … HT2) -T2 /2 width=3/
149 lemma delift_inv_lift1_eq: ∀L,U1,T2,d,e. L ⊢ ▼*[d, e] U1 ≡ T2 →
150 ∀K. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 → T1 = T2.
151 #L #U1 #T2 #d #e * #U2 #HU12 #HTU2 #K #HLK #T1 #HTU1
152 lapply (tpss_inv_lift1_eq … HU12 … HTU1) -L -K #H destruct
153 lapply (lift_inj … HTU1 … HTU2) -U2 //
156 lemma delift_lift_div_be: ∀L,T1,T,d,e,i. L ⊢ ▼*[i, d + e - i] T1 ≡ T →
157 ∀T2. ⇧[d, i - d] T2 ≡ T → d ≤ i → i ≤ d + e →
158 L ⊢ ▼*[d, e] T1 ≡ T2.
159 #L #T1 #T #d #e #i * #T0 #HT10 #HT0 #T2 #HT2 #Hdi #Hide
160 lapply (tpss_weak … HT10 d e ? ?) -HT10 // [ >commutative_plus /2 width=1/ ] #HT10
161 lapply (lift_trans_be … HT2 … HT0 ? ?) -T //
162 >commutative_plus >commutative_plus in ⊢ (? ? (? % ?) ? ? → ?);
163 <minus_le_minus_minus_comm // <plus_minus_m_m [ /2 width=3/ | /2 width=1/ ]