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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 "ground_2/relocation/rtmap_basic.ma".
16 include "static_2/relocation/lifts.ma".
17 include "apps_2/models/veq.ma".
19 (* EVALUATION EQUIVALENCE **************************************************)
21 (* Forward lemmas with generic relocation ***********************************)
23 fact lifts_fwd_vlift_aux (M) (gv): is_model M → is_extensional M →
24 ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀m. 𝐁❴m,1❵ = f →
25 ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[m←d]lv].
26 #M #gv #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
27 [ /4 width=3 by seq_trans, seq_sym, ms/
28 | #f #i1 #i2 #Hi12 #m #Hm #lv #d destruct
29 @(mr … H1M) [4,5: @(seq_sym … H1M) @(ml … H1M) |1,2: skip ]
30 elim (lt_or_ge i1 m) #Hi1
31 [ lapply (at_basic_inv_lt … Hi12) -Hi12 // #H destruct
32 >vlift_lt /2 width=1 by mq/
33 | lapply (at_basic_inv_ge … Hi12) -Hi12 // #H destruct
34 >vlift_gt /2 width=1 by mq, le_S_S/
36 | /4 width=3 by seq_trans, seq_sym, mg/
37 | #f #p * #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #m #Hm #lv #d destruct
38 [ @(mr … H1M) [4,5: @(seq_sym … H1M) @(md … H1M) |1,2: skip ]
40 [2: @(ti_comp_l … H1M) | skip ]
41 [2: @(vlift_comp … lv lv) | skip ]
42 [3: /2 width=1 by veq_refl/ ]
43 [2: @(IHV … d) // | skip ]
44 @(seq_trans … H1M) [2: @(IHT (↑m) … d) // | skip ]
45 /4 width=1 by seq_sym, ti_ext_l, vlift_swap/
46 | @mx /2 width=1 by/ #d0 @(seq_trans … H1M)
47 [3: @(seq_sym … H1M) @(ti_ext_l … H1M) | skip ]
48 [2: @vlift_swap // | skip ]
51 | #f * #V1 #v2 #T1 #T2 #_ #_ #IHV #IHT #m #Hm #lv #d
52 [ /4 width=5 by seq_sym, ma, mc, mr/
53 | /4 width=5 by seq_sym, me, mr/
58 lemma lifts_SO_fwd_vlift (M) (gv): is_model M → is_extensional M →
59 ∀T1,T2. ⬆*[1] T1 ≘ T2 →
60 ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[d]lv].
61 /2 width=3 by lifts_fwd_vlift_aux/ qed-.