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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/notation/relations/predevalstar_6.ma".
include "basic_2/rt_transition/cnu.ma".
include "basic_2/rt_computation/cpms.ma".

(* T-UNBOUND EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS ******************)

definition cpmue (h) (n) (G) (L): relation2 term term ≝
           λT1,T2. ∧∧ ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.

interpretation "t-unbound evaluation for t-bound context-sensitive parallel rt-transition (term)"
   'PRedEvalStar h n G L T1 T2 = (cpmue h n G L T1 T2).

definition R_cpmue (h) (G) (L) (T): predicate nat ≝
           λn. ∃U. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍*⦃U⦄.

(* Basic properties *********************************************************)

lemma cpmue_intro (h) (n) (G) (L):
      ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍*⦃T2⦄.
/2 width=1 by conj/ qed.

(* Basic forward lemmas *****************************************************)

lemma cpmue_fwd_cpms (h) (n) (G) (L):
      ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍*⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
#h #n #G #L #T1 #T2 * #HT12 #_ //
qed-.