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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 *)
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15 include "basic_2A/substitution/drop_drop.ma".
16 include "basic_2A/unfold/lstas.ma".
18 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
20 (* Properties on relocation *************************************************)
22 lemma lstas_lift: ∀h,G,d. d_liftable (lstas h G d).
23 #h #G #d #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1 -d
24 [ #G #L1 #d #k #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
25 >(lift_inv_sort1 … H1) -X1
26 >(lift_inv_sort1 … H2) -X2 //
27 | #G #L1 #K1 #V1 #W1 #W #i #d #HLK1 #_ #HW1 #IHVW1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
28 elim (lift_inv_lref1 … H) * #Hil #H destruct
29 [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
30 elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
31 elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
32 /3 width=9 by lstas_ldef/
33 | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
34 lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/
36 | #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
37 >(lift_mono … HWU2 … H) -U2
38 elim (lift_inv_lref1 … H) * #Hil #H destruct
39 [ elim (lift_total W1 (l-i-1) m) #W2 #HW12
40 elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
41 elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
42 /3 width=10 by lstas_zero/
43 | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
44 /3 width=10 by lstas_zero, drop_inv_gen/
46 | #G #L1 #K1 #W1 #V1 #W #i #d #HLK1 #_ #HW1 #IHWV1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
47 elim (lift_inv_lref1 … H) * #Hil #H destruct
48 [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
49 elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
50 elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
51 /3 width=9 by lstas_succ/
52 | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
53 lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/
55 | #a #I #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
56 elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
57 elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
58 lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_bind, drop_skip/
59 | #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
60 elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
61 elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
62 lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_appl/
63 | #G #L1 #W1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X #H #U2 #HU12
64 elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by lstas_cast/
68 (* Inversion lemmas on relocation *******************************************)
70 lemma lstas_inv_lift1: ∀h,G,d. d_deliftable_sn (lstas h G d).
71 #h #G #d #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -d
72 [ #G #L2 #d #k #L1 #s #l #m #_ #X #H
73 >(lift_inv_sort2 … H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/
74 | #G #L2 #K2 #V2 #W2 #W #i #d #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #l #m #HL21 #X #H
75 elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HVW2 | -IHVW2 ]
76 [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
77 elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
78 elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_ldef, ex2_intro/
79 | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
80 elim (le_inv_plus_l … Hil) -Hil #Hlim #mi
81 elim (lift_split … HW2 l (i-m+1)) -HW2 /2 width=1 by le_S_S, le_S/
82 #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_ldef, le_S, ex2_intro/
84 | #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #s #l #m #HL21 #X #H
85 elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ]
86 [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
87 elim (IHWV2 … HK21 … HW12) -K2
88 /3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/
89 | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2
90 /3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/
92 | #G #L2 #K2 #W2 #V2 #W #i #d #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #l #m #HL21 #X #H
93 elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HWV2 | -IHWV2 ]
94 [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
95 elim (IHWV2 … HK21 … HW12) -K2 #V1 #HV12 #HWV1
96 elim (lift_trans_le … HV12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_succ, ex2_intro/
97 | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
98 elim (le_inv_plus_l … Hil) -Hil #Hlim #mi
99 elim (lift_split … HW2 l (i-m+1)) -HW2 /2 width=1 by le_S_S, le_S/
100 #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_succ, le_S, ex2_intro/
102 | #a #I #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
103 elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
104 elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/
105 | #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
106 elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
107 elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by lstas_appl, lift_flat, ex2_intro/
108 | #G #L2 #W2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
109 elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
110 elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by lstas_cast, ex2_intro/
114 (* Advanced inversion lemmas ************************************************)
116 lemma lstas_split_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 → ∀d1,d2. d = d1 + d2 →
117 ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T & ⦃G, L⦄ ⊢ T •*[h, d2] T2.
118 #h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
119 [ #G #L #d #k #d1 #d2 #H destruct
120 >commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/
121 | #G #L #K #V1 #V2 #U2 #i #d #HLK #_ #VU2 #IHV12 #d1 #d2 #H destruct
122 elim (IHV12 d1 d2) -IHV12 // #V
123 elim (lift_total V 0 (i+1))
124 lapply (drop_fwd_drop2 … HLK)
125 /3 width=12 by lstas_lift, lstas_ldef, ex2_intro/
126 | #G #L #K #W1 #W2 #i #HLK #HW12 #_ #d1 #d2 #H
127 elim (zero_eq_plus … H) -H #H1 #H2 destruct
128 /3 width=5 by lstas_zero, ex2_intro/
129 | #G #L #K #W1 #W2 #U2 #i #d #HLK #HW12 #HWU2 #IHW12 #d1 @(nat_ind_plus … d1) -d1
130 [ #d2 normalize #H destruct
131 elim (IHW12 0 d) -IHW12 //
132 lapply (drop_fwd_drop2 … HLK)
133 /3 width=8 by lstas_succ, lstas_zero, ex2_intro/
134 | #d1 #_ #d2 <plus_plus_comm_23 #H lapply (injective_plus_l … H) -H #H
135 elim (IHW12 … H) -d #W
136 elim (lift_total W 0 (i+1))
137 lapply (drop_fwd_drop2 … HLK)
138 /3 width=12 by lstas_lift, lstas_succ, ex2_intro/
140 | #a #I #G #L #V #T #U #d #_ #IHTU #d1 #d2 #H
141 elim (IHTU … H) -d /3 width=3 by lstas_bind, ex2_intro/
142 | #G #L #V #T #U #d #_ #IHTU #d1 #d2 #H
143 elim (IHTU … H) -d /3 width=3 by lstas_appl, ex2_intro/
144 | #G #L #W #T #U #d #_ #IHTU #d1 #d2 #H
145 elim (IHTU … H) -d /3 width=3 by lstas_cast, ex2_intro/
149 lemma lstas_split: ∀h,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*[h, d1 + d2] T2 →
150 ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T & ⦃G, L⦄ ⊢ T •*[h, d2] T2.
151 /2 width=3 by lstas_split_aux/ qed-.
153 (* Advanced properties ******************************************************)
155 lemma lstas_lstas: ∀h,G,L,T,T1,d1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 →
156 ∀d2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, d2] T2.
157 #h #G #L #T #T1 #d1 #H elim H -G -L -T -T1 -d1
158 [ /2 width=2 by lstas_sort, ex_intro/
159 | #G #L #K #V #V1 #U1 #i #d1 #HLK #_ #HVU1 #IHV1 #d2
160 elim (IHV1 d2) -IHV1 #V2
161 elim (lift_total V2 0 (i+1))
162 /3 width=7 by ex_intro, lstas_ldef/
163 | #G #L #K #W #W1 #i #HLK #HW1 #IHW1 #d2
164 @(nat_ind_plus … d2) -d2 /3 width=5 by lstas_zero, ex_intro/
165 #d2 #_ elim (IHW1 d2) -IHW1 #W2
166 elim (lift_total W2 0 (i+1))
167 /3 width=7 by lstas_succ, ex_intro/
168 | #G #L #K #W #W1 #U1 #i #d #HLK #_ #_ #IHW1 #d2
169 @(nat_ind_plus … d2) -d2
170 [ elim (IHW1 0) -IHW1 /3 width=5 by lstas_zero, ex_intro/
171 | #d2 #_ elim (IHW1 d2) -IHW1
172 #W2 elim (lift_total W2 0 (i+1)) /3 width=7 by ex_intro, lstas_succ/
174 | #a #I #G #L #V #T #U #d #_ #IHTU #d2
175 elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
176 | #G #L #V #T #U #d #_ #IHTU #d2
177 elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
178 | #G #L #W #T #U #d #_ #IHTU #d2
179 elim (IHTU d2) -IHTU /3 width=2 by lstas_cast, ex_intro/