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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_2/static/lsubd_da.ma".
16 include "basic_2/unfold/lstas_alt.ma".
17 include "basic_2/equivalence/scpes_cpcs.ma".
18 include "basic_2/dynamic/lsubsv_lsubd.ma".
20 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
22 (* Properties on nat-iterated static type assignment ************************)
24 lemma lsubsv_lstas_trans: ∀h,g,G,L2,T,U2,l2. ⦃G, L2⦄ ⊢ T •*[h, l2] U2 →
25 ∀l1. l2 ≤ l1 → ⦃G, L2⦄ ⊢ T ▪[h, g] l1 →
26 ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
27 ∃∃U1. ⦃G, L1⦄ ⊢ T •*[h, l2] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
28 #h #g #G #L2 #T #U #l2 #H @(lstas_ind_alt … H) -G -L2 -T -U -l2
29 [1,2: /2 width=3 by ex2_intro/
30 | #G #L2 #K2 #X #Y #U #i #l2 #HLK2 #_ #HYU #IHXY #l1 #Hl21 #Hl1 #L1 #HL12
31 elim (da_inv_lref … Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0
32 lapply (drop_mono … HK0 … HLK2) -HK0 #H destruct
33 elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
34 elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -HYU -IHXY -HLK1 ]
36 elim (IHXY … Hl21 HV0 … HK12) -K2 -l1 #T #HXT #HTY
37 lapply (drop_fwd_drop2 … HLK1) #H
38 elim (lift_total T 0 (i+1))
39 /3 width=12 by lstas_ldef, cpcs_lift, ex2_intro/
40 | #V #l0 #_ #_ #_ #_ #_ #H destruct
42 | #G #L2 #K2 #X #Y #Y0 #U #i #l2 #HLK2 #HXY0 #_ #HYU #IHXY #l1 #Hl21 #Hl1 #L1 #HL12
43 elim (da_inv_lref … Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0 [| #H1 ]
44 lapply (drop_mono … HK0 … HLK2) -HK0 #H2 destruct
45 lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21
46 elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
47 elim (lsubsv_inv_pair2 … H) -H * #K1
49 lapply (lsubsv_fwd_lsubd … HK12) #H
50 lapply (lsubd_da_trans … HV0 … H) -H #H
51 elim (da_inv_sta … H) -H
52 elim (IHXY … Hl21 HV0 … HK12) -K2 -Hl21 #Y1
53 lapply (drop_fwd_drop2 … HLK1)
54 elim (lift_total Y1 0 (i+1))
55 /3 width=12 by lstas_ldec, cpcs_lift, ex2_intro/
56 | #V #l1 #HXV #_ #HV #HX #HK12 #_ #H destruct
57 lapply (da_mono … HV0 … HX) -HX #H destruct
58 elim (shnv_inv_cast … HXV) -HXV #_ #_ #H
59 lapply (H … Hl21) -H #HXV
60 elim (IHXY … Hl21 HV0 … HK12) -K2 -Hl21 #Y0 #HXY0 #HY0
61 elim (da_inv_sta … HV) -HV #W #HVW
62 elim (lstas_total … HVW (l2+1)) -W #W #HVW
63 lapply (scpes_inv_lstas_eq … HXV … HXY0 … HVW) -HXV -HXY0 #HY0W
64 lapply (cpcs_canc_sn … HY0W … HY0) -Y0 #HYW
65 elim (lift_total W 0 (i+1))
66 lapply (drop_fwd_drop2 … HLK1)
67 /4 width=12 by cpcs_lift, lstas_cast, lstas_ldef, ex2_intro/
69 | #a #I #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
70 lapply (da_inv_bind … Hl2) -Hl2 #Hl2
71 elim (IHTU2 … Hl2 (L1.ⓑ{I}V2) …)
72 /3 width=3 by lsubsv_pair, lstas_bind, cpcs_bind_dx, ex2_intro/
73 | #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
74 lapply (da_inv_flat … Hl2) -Hl2 #Hl2
75 elim (IHTU2 … Hl2 … HL12) -L2
76 /3 width=5 by lstas_appl, cpcs_flat, ex2_intro/
77 | #G #L2 #W2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
78 lapply (da_inv_flat … Hl2) -Hl2 #Hl2
79 elim (IHTU2 … Hl2 … HL12) -L2
80 /3 width=3 by lstas_cast, ex2_intro/
84 lemma lsubsv_sta_trans: ∀h,g,G,L2,T,U2. ⦃G, L2⦄ ⊢ T •[h] U2 →
85 ∀l. ⦃G, L2⦄ ⊢ T ▪[h, g] l+1 →
86 ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
87 ∃∃U1. ⦃G, L1⦄ ⊢ T •[h] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
88 #h #g #G #L2 #T #U2 #H #l #HTl #L1 #HL12
89 elim (lsubsv_lstas_trans … U2 1 … HTl … HL12)
90 /3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/