include "basic_2/grammar/cl_shift.ma".
include "basic_2/relocation/ldrop_append.ma".

lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
#G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
/2 width=1 by cpy_atom, cpy_bind, cpy_flat/
#I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
lapply (ldrop_fwd_length_lt2 … HK0) #H
@(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
qed-.

lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶[d, e] T →
                      ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
#G #L1 @(lenv_ind_dx … L1) -L1 normalize
[ #L #T1 #T #d #e #HT1
  @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
| #I #L1 #V1 #IH #L #T1 #X #d #e
  >shift_append_assoc normalize #H
  elim (cpy_inv_bind1 … H) -H
  #V0 #T0 #_ #HT10 #H destruct
  elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
  >append_length >HL12 -HL12
  @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
  /2 width=3 by trans_eq/
]
qed-.