next c rupHints =>
by_cases performRupAdd_success : (performRupAdd f c rupHints).2
. let f' := (performRupAdd f c rupHints).1
  have heq : performRupAdd f c rupHints = (f', true) := by rw [← performRupAdd_success]
  have f_liff_f' := rupAdd_sound f c rupHints f' f_readyForRupAdd heq
  unfold incrementalLRATChecker at incrementalChecker_res_unsat
  simp only [performRupAdd_success, ite_true] at incrementalChecker_res_unsat
  rw [liff_unsat f f' f_liff_f']
  exact incrementalChecker_res_unsat
. simp only [Bool.not_eq_true] at performRupAdd_success
  simp only [performRupAdd_success, ite_false] at incrementalChecker_res
case right.h_2
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : DecidableEq α
inst✝¹ : Clause α β
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : readyForRupAdd f
f_readyForRatAdd : readyForRatAdd f
action✝ : Action β α
id✝ : Nat
c✝ : β
rupHints✝ : Array Nat
aWellFormed : wellFormedAction (Action.addRup id✝ c✝ rupHints✝)
incrementalChecker_res_unsat : unsatisfiable α (incrementalLRATChecker f (Action.addRup id✝ c✝ rupHints✝)).fst
incrementalChecker_res :
  (if (performRupAdd f c✝ rupHints✝).snd = true then ((performRupAdd f c✝ rupHints✝).fst, out_of_proof)
      else ((performRupAdd f c✝ rupHints✝).fst, rup_failure)).snd =
    out_of_proof
⊢ unsatisfiable α f
no goals