+lemma cpms_tdneq_fwd_fpbg (h) (o) (n): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ (T1 ≛[h,o] T2 → ⊥) → ⦃G,L,T1⦄ >[h,o] ⦃G,L,T2⦄.
+/3 width=2 by cpms_fwd_cpxs, cpxs_tdneq_fpbg/ qed-.
+
+lemma fpbg_cpms_trans (h) (o) (n): ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T⦄ →
+ ∀T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fpbs_trans, cpms_fwd_fpbs/ qed-.
+
+lemma cpms_fpbg_trans (h) (o) (n): ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ➡*[n,h] T →
+ ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h,o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fpbg_trans, cpms_fwd_fpbs/ qed-.
+