(* *)
(**************************************************************************)
-include "basic_2/computation/csx_aaa.ma".
-include "basic_2/computation/fsb.ma".
+include "basic_2/computation/fpbs_ext.ma".
+include "basic_2/computation/csx_fpbs.ma".
+include "basic_2/computation/lsx_csx.ma".
+include "basic_2/computation/fsb_alt.ma".
(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
-(* Advanced propreties ******************************************************)
+(* Advanced propreties on context-senstive extended bormalizing terms *******)
-lemma csx_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
-#h #g #G #L #T #H @(csx_ind_fpbc … H) -T /3 width=1 by fsb_intro/
+lemma csx_fsb_fpbs: ∀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 @(csx_ind_alt … H) -T1
+#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind … G2 L2 T2) -G2 -L2 -T2
+#G0 #L0 #T0 #IHu #H10 lapply (csx_fpbs_conf … H10) // -HT1
+#HT0 generalize in match IHu; -IHu generalize in match H10; -H10
+@(lsx_ind_alt … (csx_lsx … HT0 0)) -L0
+#L0 #_ #IHl #H10 #IHu @fsb_intro
+#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHl -IHc | -IHu -IHl | ]
+[ /3 width=5 by fpbs_fqup_trans/
+| #T2 #HT02 #HnT02 elim (fpbs_cpxs_trans_neq … H10 … HT02 HnT02) -T0
+ /3 width=4 by/
+| #L2 #HL02 #HnL02 @(IHl … HL02 HnL02) -IHl -HnL02 [ -IHu -IHc | ]
+ [ /2 width=3 by fpbs_lpxs_trans/
+ | #G3 #L3 #T3 #H03 #_ elim (lpxs_fqup_trans … H03 … HL02) -L2
+ #L4 #T4 elim (eq_term_dec T0 T4) [ -IHc | -IHu ]
+ [ #H destruct /4 width=5 by fsb_fpbs_trans, lpxs_fpbs, fpbs_fqup_trans/
+ | #HnT04 #HT04 #H04 elim (fpbs_cpxs_trans_neq … H10 … HT04 HnT04) -T0
+ /4 width=8 by fpbs_fqup_trans, fpbs_lpxs_trans/
+ ]
+ ]
+]
qed.
-(* Main properties **********************************************************)
+lemma csx_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
+/2 width=5 by csx_fsb_fpbs/ qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma csx_ind_fpbu: ∀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
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
-(* Note: this is the "big tree" theorem ("small step" version) *)
-theorem aaa_fsb: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⦥[h, g] T.
-/3 width=2 by aaa_csx, csx_fsb/ qed.
+lemma csx_ind_fpbg: ∀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
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.