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Theorem repls-rec-addboth

⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ cren(ca) = ren(a) ∧ n = # aal → repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = lift(aσ.take(n), s) + (s, aσ[n]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα)