induction sch with
| nil =>
  simp only [Schema.lookupType]
| cons s ss ih =>
  simp only [Schema.lookupType]
  apply ih (dropColumn t _)
η : Type u_η
dec_η : DecidableEq η
sch : Schema
τ : Type u
inst✝ : DecidableEq τ
t : Table sch
c : η
vs : List (Option τ)
⊢ Schema.lookupType (schema (addColumn t c vs))
      { fst := c, snd := Schema.colImpliesName (Schema.hasAppendedHead sch c τ []) } =
    τ
no goals