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(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
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include "basic_2/substitution/lift.ma".
include "apps_2/models/model_lower.ma".
include "apps_2/models/model_sound.ma".

(* MODEL ********************************************************************)

(* Forward lemmas on basic relocation ***************************************)

lemma sound_fwd_lift: ∀M. sound M → extensional M →
                      ∀sv,gv,T1,T2,l,m. ⬆[l, m] T1 ≡ T2 →
                      ∀lv. 〚T1〛{sv, gv, ↓[l, m]⦋M⦌ lv} = 〚T2〛{sv, gv, lv}.
#M #H1M #H2M #sv #gv #T1 #T2 #l #m #H elim H -T1 -T2 -l -m
[ #k #l #m #lv >(m1 … H1M) >(m1 … H1M) -H1M //
| #i #l #m #Hil #lv >(m2 … H1M) >(m2 … H1M) -H1M >lower_lt //
| #i #l #m #Hli #lv >(m2 … H1M) >(m2 … H1M) -H1M >lower_ge //
| #p #l #m #lv >(m3 … H1M) >(m3 … H1M) -H1M //
| #a * #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #lv
  [ >(m4 … H1M) >(m4 … H1M) /3 width=1 by raise_lower_lt, sound_ti_eq_l/
  | @(mx … H2M) /3 width=1 by raise_lower_lt, sound_ti_eq_l/
  ]
| * #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #lv
  [ >(m6 … H1M) >(m6 … H1M) /3 width=1 by eq_f2/
  | >(m7 … H1M) >(m7 … H1M) /2 width=1 by/
  ]
]
qed-.