1 lemma lpx_append: ∀h,g,G,K1,K2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
2 ⦃G, L1 @@ K1⦄ ⊢ ➡[h, g] L2 @@ K2.
3 /3 width=1 by lpx_sn_append, cpx_append/ qed.
5 (* Advanced forward lemmas **************************************************)
7 lemma lpx_fwd_append1: ∀h,g,G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ➡[h, g] L →
8 ∃∃K2,L2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K2 @@ L2.
9 /2 width=2 by lpx_sn_fwd_append1/ qed-.
11 lemma lpx_fwd_append2: ∀h,g,G,L,K2,L2. ⦃G, L⦄ ⊢ ➡[h, g] K2 @@ L2 →
12 ∃∃K1,L1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & L = K1 @@ L1.
13 /2 width=2 by lpx_sn_fwd_append2/ qed-.