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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_sfr.ma".
16 include "basic_2/unfold/delift_lift.ma".
17 include "basic_2/unfold/thin_ldrop.ma".
18 include "basic_2/equivalence/cpcs_delift.ma".
19 include "basic_2/dynamic/nta_lift.ma".
21 (* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************)
23 (* Properties on basic local environment thibbing ***************************)
25 (* Note: this is known as the substitution lemma *)
26 (* Basic_1: was only: ty3_gen_cabbr *)
27 lemma nta_thin_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 : U1 →
28 ∀L2,d,e. ≽ [d, e] L1 → L1 ▼*[d, e] ≡ L2 →
29 ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 : U2 &
30 L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2.
31 #h #L1 #T1 #U1 #H elim H -L1 -T1 -U1
33 | #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL1 #HL12
34 elim (lt_or_ge i d) #Hdi [ -HVW1 ]
35 [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H
36 lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1
37 elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2
38 elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
39 elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #X2 #W2 #HVW2 #H #HW12
40 lapply (delift_mono … H … HV12) -H -HV12 #H destruct
41 elim (lift_total W2 0 (i+1)) #U2 #HWU2
42 lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #HLK1
43 lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 //
44 >minus_plus <plus_minus_m_m // /3 width=6/
45 | elim (lt_or_ge i (d+e)) #Hide [ -HVW1 | -Hdi -IHVW1 -HL1 ]
46 [ lapply (sfr_ldrop_trans_be_up … HLK1 … HL1 ? ?) -HL1 // /2 width=2/ <minus_n_O #H
47 elim (sfr_inv_bind … H ?) -H /2 width=1/ #HK1 #_
48 elim (thin_ldrop_conf_be … HL12 … HLK1 ? ?) -HL12 /2 width=2/ #K2 #H #HLK2
49 lapply (thin_inv_thin1 … H ?) -H /2 width=1/ #HK12
50 elim (IHVW1 … HK1 HK12) -IHVW1 -HK1 -HK12 #V2 #W2 #HVW2 #HV12 #HW12
51 elim (lift_total V2 0 d) #T2 #HVT2
52 elim (lift_total W2 0 d) #U2 #HWU2
53 elim (lift_total W2 0 (i+1)) #U #HW2U
54 lapply (nta_lift … HVW2 … HLK2 … HVT2 … HWU2) -HVW2 -HLK2 #HTU2
55 lapply (ldrop_fwd_ldrop2 … HLK1) #HLK0
56 lapply (delift_lift_ge … HW12 … HLK0 HWU1 … HW2U) -HW12 -HLK0 -HWU1 // >minus_plus #HU1
57 lapply (lift_conf_be … HWU2 … HW2U ?) -W2 /2 width=1/ #HU2
58 lapply (delift_lift_div_be … HU1 … HU2 ? ?) -U // /2 width=1/ /3 width=8/
59 | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei
60 lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1
61 elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W
62 <plus_minus in ⊢ (??%??→?); /2 width=2/ #HW1
63 <minus_minus // /2 width=2/ -Hdei >commutative_plus <minus_n_n /3 width=6/
66 | #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL1 #HL12
67 elim (lt_or_ge i d) #Hdi [ -HWV1 | -IHWV1 ]
68 [ lapply (sfr_ldrop_trans_ge … HLK1 … HL1 ?) -HL1 /2 width=2/ #H
69 lapply (sfr_inv_skip … H ?) -H /2 width=1/ #HK1
70 elim (thin_ldrop_conf_le … HL12 … HLK1 ?) -HL12 /2 width=2/ #X #H #HLK2
71 elim (thin_inv_delift1 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
72 elim (IHWV1 … HK1 HK12) -IHWV1 -HK1 -HK12 #X2 #V2 #HWV2 #H #_
73 lapply (delift_mono … H … HW12) -H #H destruct
74 elim (lift_total W2 0 (i+1)) #U2 #HWU2
75 lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
76 lapply (delift_lift_ge … HW12 … HLK1 HWU1 … HWU2) -HW12 -HLK1 -HWU1 //
77 >minus_plus <plus_minus_m_m // /3 width=6/
78 | elim (lt_or_ge i (d+e)) #Hide [ -HWV1 -HWU1 -HL12 | -Hdi -HL1 ]
79 [ lapply (sfr_inv_ldrop … HLK1 … HL1 ? ?) -HLK1 -HL1 // -Hdi -Hide #H destruct
80 | lapply (transitive_le … (i+1) Hide ?) /2 width=1/ #Hdei
81 lapply (thin_ldrop_conf_ge … HL12 … HLK1 ?) -HL12 -HLK1 // #HL2K1
82 elim (lift_split … HWU1 d (i+1-e) ? ? ?) -HWU1 // /2 width=1/ #W
83 <plus_minus in ⊢ (??%??→?); /2 width=2/ #HW1
84 <minus_minus // /2 width=2/ -Hdei >commutative_plus <minus_n_n /3 width=6/
87 | #I #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL1 #HL12
88 elim (IHVW1 … HL1 HL12) -IHVW1 #V2 #W2 #HVW2 #HV12 #_
89 elim (IHTU1 (L2.ⓑ{I}V2) (d+1) e ? ?) -IHTU1 /2 width=1/ -HL1 -HL12 #T2 #U2 #HTU2 #HT12 #HU12
90 lapply (delift_lsubs_trans … HT12 (L1.ⓑ{I}V2) ?) -HT12 /2 width=1/
91 lapply (delift_lsubs_trans … HU12 (L1.ⓑ{I}V2) ?) -HU12 /2 width=1/ /3 width=7/
92 | #L1 #V1 #W1 #T1 #U1 #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL1 #HL12
93 elim (IHVW1 … HL1 HL12) -IHVW1 #V2 #W2 #HVW2 #HV12 #HW12
94 elim (IHTU1 … HL1 HL12) -IHTU1 -HL1 -HL12 #X2 #Y2 #HXY2 #HX2 #HY2
95 elim (delift_inv_bind1 … HX2) -HX2 #Z21 #T2 #HZ21 #HT12 #H destruct
96 elim (delift_inv_bind1 … HY2) -HY2 #Z22 #U2 #HZ22 #HU12 #H destruct
97 lapply (delift_mono … HZ21 … HW12) -HZ21 #H destruct
98 lapply (delift_mono … HZ22 … HW12) -HZ22 #H destruct
99 @(ex3_2_intro … (ⓐV2.ⓛW2.T2) (ⓐV2.ⓛW2.U2)) /3 width=1/ (**) (* explict constructor, /4 depth=?/ is too slow *)
100 | #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL1 #HL12
101 elim (IHTU1 … HL1 HL12) -IHTU1 #T2 #U2 #HTU2 #HT12 #HU12
102 elim (IHUW1 … HL1 HL12) -IHUW1 -HL1 -HL12 #X2 #W2 #HXW2 #H #HW12
103 elim (delift_inv_flat1 … H) -H #V2 #Y2 #HV12 #HY2 #H destruct
104 lapply (delift_mono … HY2 … HU12) -HY2 #H destruct /3 width=7/
105 | #L1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL1 #HL12
106 elim (IHTU1 … HL1 HL12) -IHTU1 -HL1 -HL12 /3 width=5/
107 | #L1 #T1 #U11 #U12 #V1 #_ #HU112 #_ #IHT1 #IHU12 #L2 #d #e #HL1 #HL12
108 elim (IHT1 … HL1 HL12) -IHT1 #T2 #U21 #HT2 #HT12 #HU121
109 elim (IHU12 … HL1 HL12) -IHU12 -HL1 #U22 #V2 #HU22 #HU122 #_
110 lapply (thin_cpcs_delift_mono … HU112 … HL12 … HU121 … HU122) -HU112 -HL12 -HU121 /3 width=5/
114 lemma nta_ldrop_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 : U1 →
115 ∀L2,d,e. ≽ [d, e] L1 → ⇩[d, e] L1 ≡ L2 →
116 ∃∃T2,U2. ⦃h, L2⦄ ⊢ T2 : U2 &
117 L1 ⊢ T1 ▼*[d, e] ≡ T2 & L1 ⊢ U1 ▼*[d, e] ≡ U2.