lemma lpx_append: ∀h,g,G,K1,K2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
                  ⦃G, L1 @@ K1⦄ ⊢ ➡[h, g] L2 @@ K2.
/3 width=1 by lpx_sn_append, cpx_append/ qed.

(* Advanced forward lemmas **************************************************)

lemma lpx_fwd_append1: ∀h,g,G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡[h, g] L →
                       ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K2 @@ L2.
/2 width=2 by lpx_sn_fwd_append1/ qed-.

lemma lpx_fwd_append2: ∀h,g,G,L,K2,L2. ⦃G, L⦄ ⊢ ➡[h, g] K2 @@ L2 →
                       ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K1 @@ L1.
/2 width=2 by lpx_sn_fwd_append2/ qed-.