lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
                    ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
                    ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
(* ALTERNATIVE PROOF
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
[ /2 width=1 by or_introl/
| #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
  elim (fpb_fpbu … H1) -H1 #H1
  [ /3 width=1 by  
*)
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
#L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
[ -HT1 elim (fqus_inv_gen … HT2) -HT2
  [ #H elim (fqup_inv_step_sn … H) -H
    /4 width=11 by fpbs_intro_alt, fpbu_fqu, ex2_3_intro, or_intror/
  | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
    [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
    | elim (lpxs_nlleq_inv_step_sn … HL0 H) -HL0 -H
      /5 width=7 by lpxs_lleq_fpbs, fpbu_lpx, lleq_trans, ex2_3_intro, or_intror/
    ]
  ]
| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
  /5 width=11 by fpbs_intro_alt, fpbu_cpx, ex2_3_intro, or_intror/
]
qed-.

(* alternative proof that needs decidability of bteq to go in fpbs.ma 
 * or lpx_fpbc_trans to go in fpbs_lift.ma (possibly)  
*)

axiom lpx_bteq_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1⦄ ⊢➡ [h, g] L →
                      ⦃G1, L, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄.

lemma fpbs_fwd_fpb_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
                       ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ ∨
                       ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1 [ /2 width=1 by or_introl/ ] (**) (* auto fails without brackets *)
#G1 #G #L1 #L #T1 #T *
[ #G0 #L0 #T0 #H #H02 #IH1 elim (fquq_inv_gen … H) -H
  [ -IH1 /4 width=5 by fpbc_fqu, ex2_3_intro, or_intror/
  | -H02 * #HG #HL #HT destruct /2 width=1 by/
  ]
| #T0 #HT10 #H02 #IH02 elim (eq_term_dec T1 T0) #H destruct
  [ -H02 /2 width=1 by/
  | -IH02 /5 width=5 by fpbc_cpx, ex2_3_intro, or_intror/
  ]
| #L0 #HL10 #_ * [ /3 width=3 by or_introl, lpx_bteq_trans/ ]
  * #G3 #L3 #T3 #H13 #H32