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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/static/lsubd_da.ma".
16 include "basic_2/unfold/lsstas_alt.ma".
17 include "basic_2/equivalence/cpcs_cpcs.ma".
18 include "basic_2/dynamic/lsubsv_ldrop.ma".
19 include "basic_2/dynamic/lsubsv_lsubd.ma".
21 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
23 (* Properties on nat-iterated stratified static type assignment *************)
25 lemma lsubsv_lsstas_trans: ∀h,g,G,L2,T,U2,l1. ⦃G, L2⦄ ⊢ T •*[h, g, l1] U2 →
26 ∀l2. l1 ≤ l2 → ⦃G, L2⦄ ⊢ T ▪[h, g] l2 →
27 ∀L1. G ⊢ L1 ¡⫃[h, g] L2 →
28 ∃∃U1. ⦃G, L1⦄ ⊢ T •*[h, g, l1] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
29 #h #g #G #L2 #T #U #l1 #H @(lsstas_ind_alt … H) -G -L2 -T -U -l1
30 [1,2: /2 width=3 by lstar_O, ex2_intro/
31 | #G #L2 #K2 #X #Y #U #i #l1 #HLK2 #_ #HYU #IHXY #l2 #Hl12 #Hl2 #L1 #HL12
32 elim (da_inv_lref … Hl2) -Hl2 * #K0 #V0 [| #l0 ] #HK0 #HV0
33 lapply (ldrop_mono … HK0 … HLK2) -HK0 #H destruct
34 elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
35 elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -HYU -IHXY -HLK1 ]
37 elim (IHXY … Hl12 HV0 … HK12) -K2 -l2 #T #HXT #HTY
38 lapply (ldrop_fwd_drop2 … HLK1) #H
39 elim (lift_total T 0 (i+1))
40 /3 width=12 by lsstas_ldef, cpcs_lift, ex2_intro/
41 | #V #l0 #_ #_ #_ #_ #_ #_ #_ #H destruct
43 | #G #L2 #K2 #X #Y #U #i #l1 #l #HLK2 #_ #_ #HYU #IHXY #l2 #Hl12 #Hl2 #L1 #HL12 -l
44 elim (da_inv_lref … Hl2) -Hl2 * #K0 #V0 [| #l0 ] #HK0 #HV0 [| #H1 ]
45 lapply (ldrop_mono … HK0 … HLK2) -HK0 #H2 destruct
46 lapply (le_plus_to_le_r … Hl12) -Hl12 #Hl12
47 elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
48 elim (lsubsv_inv_pair2 … H) -H * #K1 [| ]
50 lapply (lsubsv_fwd_lsubd … HK12) #H
51 lapply (lsubd_da_trans … HV0 … H) -H
52 elim (IHXY … Hl12 HV0 … HK12) -K2 -Hl12 #Y0
53 lapply (ldrop_fwd_drop2 … HLK1)
54 elim (lift_total Y0 0 (i+1))
55 /3 width=12 by lsstas_ldec, cpcs_lift, ex2_intro/
56 | #V #l #_ #_ #HVX #_ #HV #HX #HK12 #_ #H destruct
57 lapply (da_mono … HX … HV0) -HX #H destruct
58 elim (IHXY … Hl12 HV0 … HK12) -K2 #Y0 #HXY0 #HY0
59 elim (da_ssta … HV) -HV #W #HVW
60 elim (lsstas_total … HVW (l1+1)) -W #W #HVW
61 lapply (HVX … Hl12 HVW HXY0) -HVX -Hl12 -HXY0 #HWY0
62 lapply (cpcs_trans … HWY0 … HY0) -Y0
63 lapply (ldrop_fwd_drop2 … HLK1)
64 elim (lift_total W 0 (i+1))
65 /4 width=12 by lsstas_ldef, lsstas_cast, cpcs_lift, ex2_intro/
67 | #a #I #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
68 lapply (da_inv_bind … Hl2) -Hl2 #Hl2
69 elim (IHTU2 … Hl2 (L1.ⓑ{I}V2) …) // [2: /2 width=1/ ] -L2
70 /3 width=3 by lsstas_bind, cpcs_bind_dx, ex2_intro/
71 | #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
72 lapply (da_inv_flat … Hl2) -Hl2 #Hl2
73 elim (IHTU2 … Hl2 … HL12) -L2 //
74 /3 width=5 by lsstas_appl, cpcs_flat, ex2_intro/
75 | #G #L2 #W2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
76 lapply (da_inv_flat … Hl2) -Hl2 #Hl2
77 elim (IHTU2 … Hl2 … HL12) -L2 //
78 /3 width=3 by lsstas_cast, ex2_intro/
82 lemma lsubsv_ssta_trans: ∀h,g,G,L2,T,U2. ⦃G, L2⦄ ⊢ T •[h, g] U2 →
83 ∀l. ⦃G, L2⦄ ⊢ T ▪[h, g] l+1 →
84 ∀L1. G ⊢ L1 ¡⫃[h, g] L2 →
85 ∃∃U1. ⦃G, L1⦄ ⊢ T •[h, g] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
86 #h #g #G #L2 #T #U2 #H #l #HTl #L1 #HL12
87 elim ( lsubsv_lsstas_trans … U2 1 … HTl … HL12)
88 /3 width=3 by lsstas_inv_SO, ssta_lsstas, ex2_intro/