qed.
lemma ltps_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
L1 [d1, e1] ≫ L0 → L0 ⊢ T2 [d2, e2] ≫* U2 →
∃∃T. L1 ⊢ T2 [d2, e2] ≫* T & L0 ⊢ T [d1, e1] ≫* U2.
#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
qed.
lemma ltps_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
L1 [d1, e1] ≫ L0 → L0 ⊢ T2 [d2, e2] ≫* U2 →
∃∃T. L1 ⊢ T2 [d2, e2] ≫* T & L0 ⊢ T [d1, e1] ≫* U2.
#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2