-lemma cpx_fwd_cpys_tpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+axiom cpys_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T2 →
+ ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*×[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶*×[d, i-d] T2.
+
+lemma cpys_tpxs_trans: ∀h,g,G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶*×[d, e] T →
+ ∀T2. ⦃G, ⋆⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T #d #e #HT1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by cpxs_strap1, cpys_cpx, lsubr_cpx_trans, cpx_cpxs/
+qed-.
+
+axiom cpx_fwd_cpys_tpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →