(**************************************************************************)
(*       ___                                                              *)
(*      ||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/static/ssta_ltpss.ma".
include "basic_2/unwind/sstas.ma".

(* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************)

(* Properties about parallel unfold *****************************************)

lemma sstas_ltpss_tpss_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
                             ∀L2,d,e. L1 ▶* [d, e] L2 →
                             ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
                             ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 &
                                   L2 ⊢ U1 ▶* [d, e] U2.
#h #g #L1 #T1 #U1 #H @(sstas_ind_alt … H) -T1
[ #T1 #HUT1 #L2 #d #e #HL12 #U2 #HU12
  elim (ssta_ltpss_tpss_conf … HUT1 … HL12 … HU12) -HUT1 -HL12 /3 width=3/
| #T0 #U0 #l0 #HTU0 #_ #IHU01 #L2 #d #e #HL12 #T #HT0
  elim (ssta_ltpss_tpss_conf … HTU0 … HL12 … HT0) -HTU0 -HT0 #U #HTU #HU0
  elim (IHU01 … HL12 … HU0) -IHU01 -HL12 -U0 /3 width=4/
]
qed.

lemma sstas_ltpss_tps_conf: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
                            ∀L2,d,e. L1 ▶* [d, e] L2 →
                            ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
                            ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
/3 width=5/ qed.

lemma sstas_ltpss_conf: ∀h,g,L1,T,U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
                        ∀L2,d,e. L1 ▶* [d, e] L2 →
                        ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L2 ⊢ U1 ▶* [d, e] U2.
/2 width=5/ qed.

lemma sstas_tpss_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
                       ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
                       ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
/2 width=5/ qed.

lemma sstas_tps_conf: ∀h,g,L,T1,U1. ⦃h, L⦄ ⊢ T1 •*[g] U1 →
                      ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
                      ∃∃U2. ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U1 ▶* [d, e] U2.
/2 width=5/ qed.