-lemma fpbg_fpbs_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⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fpbs_ind … H) -G2 -L2 -T2
-/2 width=5 by fpbg_strap1/
+(* Note: this is used in the closure proof *)
+lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/4 width=1 by fpbc_fpbg, fpbu_fpbc, fpbu_fqup/ qed.
+
+(* Basic eliminators ********************************************************)
+
+lemma fpbg_ind: ∀h,g,G1,L1,T1. ∀R:relation3 ….
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻≡[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#h #g #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)