lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ →
♯❨G2,L2,T2❩ < ♯❨G1,L1,T1❩.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=3 by fqu_fwd_fw, transitive_lt/
+/3 width=3 by fqu_fwd_fw, nlt_trans/
qed-.
(* Advanced eliminators *****************************************************)
∀G1,L1,T1. (∀G2,L2,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ → Q G2 L2 T2) →
Q G1 L1 T1
) → ∀G1,L1,T1. Q G1 L1 T1.
-#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
+#b #Q #HQ @(wf3_ind_nlt … fw) #x #IHx #G1 #L1 #T1 #H destruct
/4 width=2 by fqup_fwd_fw/
qed-.
∀G1,L1,T1. (∀G2,L2,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ → Q G2 L2 T2) →
∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → Q G2 L2 T2
) → ∀G1,L1,T1. Q G1 L1 T1.
-#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
+#b #Q #HQ @(wf3_ind_nlt … fw) #x #IHx #G1 #L1 #T1 #H destruct
/4 width=7 by fqup_fwd_fw/
qed-.