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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/relocation/lleq_lleq.ma".
include "basic_2/computation/lpxs.ma".
include "basic_2/computation/fpns.ma".

(* COMPUTATION FOR "BIG TREE" NORMAL FORMS **********************************)

(* Advanced inversion lemmas ************************************************)

lemma fpns_inv_gen: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ →
                    ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 & L1 ⋕[T1] L2 & T1 = T2.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpns_ind … H) -G2 -L2 -T2 /2 width=1 by and4_intro/
#G #G2 #L #L2 #T #T2 #_ #H2 * #HG #HL1 #HT1 #HT destruct
elim (fpn_inv_gen … H2) -H2 #HG #HL2 #HT #HT2 destruct
/3 width=3 by lpxs_strap1, lleq_trans, and4_intro/
qed-.

(* Advanced properties ******************************************************)

lemma lpxs_lleq_fpns: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → L1 ⋕[T] L2 →
                      ⦃G, L1, T⦄ ⊢ ⋕➡*[h, g] ⦃G, L2, T⦄.
#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2 //
#L #L2 #HL1 #HL2 #IHL1 #HL12 elim (lleq_dec T L1 L) #HT
[ -HL1 @fpns_strap1 [4: @IHL1 // |1,2,3: skip ]
  /3 width=3 by fpn_intro, lleq_canc_sn/
| -IHL1

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

theorem fpns_trans: tri_transitive … fqus.
/2 width=5 by tri_TC_transitive/ qed-.


(*
lemma fpns_refl: ∀h,g. tri_reflexive … (fpns h g).
/2 width=1 by and3_intro/ qed.

lemma fpn_fpns: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄ →
                ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=1 by lpx_lpxs, and3_intro/
qed.

lemma fpns_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G, L, T⦄ →
                   ⦃G, L, T⦄ ⊢ ➡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * #H1G #H1L #G1T *
/3 width=3 by lpxs_strap1, and3_intro/
qed-.

lemma fpns_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ➡[h, g] ⦃G, L, T⦄ →
                   ⦃G, L, T⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * #H1G #H1L #G1T *
/3 width=3 by lpxs_strap2, and3_intro/
qed-.

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

lemma fpns_fwd_bteq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊢ ➡*[h, g] ⦃G2, L2, T2⦄ →
                     ⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=4 by lpxs_fwd_length, and3_intro/
qed-.
*)