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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/btsnalt_5.ma".
include "basic_2/computation/fpbs_fpbs.ma".
include "basic_2/computation/fsb.ma".

(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)

(* Note: alternative definition of fsb *)
inductive fsba (h) (g): relation3 genv lenv term ≝
| fsba_intro: ∀G1,L1,T1. (
                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
                 (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
              ) → fsba h g G1 L1 T1.

interpretation
   "'big tree' strong normalization (closure) alternative"
   'BTSNAlt h g G L T = (fsba h g G L T).

(* Basic eliminators ********************************************************)

theorem fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
                         ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h,g] T1 → (
                            ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
                            (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → R G2 L2 T2
                         ) → R G1 L1 T1
                      ) →
                      ∀G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → R G L T.
#h #g #R #IH #G #L #T #H elim H -G -L -T
/5 width=1 by fsba_intro/
qed-.

(* Basic_properties *********************************************************)

fact fsba_intro_aux: ∀h,g,G1,L1,T1. (
                        ∀G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
                        ⦃G1, L1, T1⦄ ⋕ ⦃G, L, T⦄ →
                        (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
                     ) → fsba h g G1 L1 T1.
/4 width=5 by fsba_intro/ qed-.

lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1 →
                       ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2.
#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H0 #IH0 #G #L #T #H1 @fsba_intro
#G2 #L2 #T2 #H2 #_ lapply (fpbs_trans … H1 … H2) -G -L -T
#H12 elim (bteq_dec G1 G2 L1 L2 T1 T2) /3 width=6 by fpb_fpbs/
-IH0 #H212 
 

 -H0 -H #H @(IH0 … H) -IH0 -H // @(fpbs_trans … H1 … H2)    

lemma fsba_intro_fpb: ∀h,g,G1,L1,T1. (
                         ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
                         (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2
                      ) → ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1.
#h #g #G1 #L1 #T1 #IH1 @fsba_intro_aux
#G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T
[ #H1 #H2 -IH1 elim H2 -H2 //
| #G0 #G #L0 #L #T0 #T #H10 #H12 #IH2 #H210 #H212 elim (bteq_dec G1 G L1 L T1 T) 
  [ -IH1 -H210 -H10 -H12 /3 width=1 by/
  | -IH2 -H212 #H21 lapply (IH1 … H21) -IH1 -H21
    [
    | -H10 -H210 #H 
(*
(* Main inversion lemmas ****************************************************)

theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
#h #g #G #L #T #H elim H -G -L -T
/5 width=12 by fsb_intro, fpb_fpbs, fpbc_fwd_fpb, fpbc_fwd_gen/
qed-.

(* Main properties **********************************************************)

theorem fsb_fsba: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥⦥[h, g] T.
#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
/4 width=1 by fsba_intro_fpb/
qed.
(*
| fsba_intro: ∀G1,L1,T1. (
                ∀G2,L2,T2.  ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → fsb h g G2 L2 T2
              ) → fsb h g G1 L1 T1
.



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include "basic_2/substitution/fqup.ma".

lemma fsb_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
#h #g #G #L #T #H @(csx_ind … H) -T
*)*)