+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/fpbs_fleq.ma".
+include "basic_2/computation/fpbg_fpbg.ma".
+include "basic_2/computation/fsb.ma".
+
+(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
+
+axiom fleq_fpbc_trans: ∀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⦄.
+
+axiom fleq_fpbg_trans: ∀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⦄.
+
+(* Properties ******************************************************)
+
+lemma fsb_fleq_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 @(fsb_ind_alt … H) -G1 -L1 -T1
+/4 width=5 by fsb_intro, fleq_fpbc_trans/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fsb_ind_fpbg_fpbs: ∀h,g. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#h #g #R #IH #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+#G1 #L1 #T1 #H1 #IH1 #G2 #L2 #T2 #H12 elim (fpbs_fwd_fpb_sn … H12) -H12
+[ #H12 @IH -IH /2 width=5 by fsb_fleq_trans/
+ -H1 #G0 #L0 #T0 #H20 lapply (fleq_fpbg_trans … H12 H20) -G2 -L2 -T2
+ #H10 elim (fpbg_fwd_fpb_sn … H10) -H10 /2 width=5 by/
+| -H1 -IH * /2 width=5 by/
+]
+qed-.