. next c pivot rupHints ratHints =>
  by_cases performRatAdd_success : (performRatAdd f c pivot rupHints ratHints).2
  . let f' := (performRatAdd f c pivot rupHints ratHints).1
    have heq : performRatAdd f c pivot rupHints ratHints = (f', true) := by rw [← performRatAdd_success]
    simp only [wellFormedAction, limplies_iff_mem] at aWellFormed
    unfold incrementalLRATChecker at incrementalChecker_res_unsat
    simp only [performRatAdd_success, ite_true] at incrementalChecker_res_unsat
    have h := ratAdd_sound f c pivot rupHints ratHints f' f_readyForRatAdd aWellFormed heq
    rw [equisat] at h
    rw [h]
    exact incrementalChecker_res_unsat
  . simp only [Bool.not_eq_true] at performRatAdd_success
    simp only [performRatAdd_success, ite_false] at incrementalChecker_res
case right.h_3
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
pivot✝ : Literal α
rupHints✝ : Array Nat
ratHints✝ : Array (Nat × Array Nat)
aWellFormed : wellFormedAction (Action.addRat id✝ c✝ pivot✝ rupHints✝ ratHints✝)
incrementalChecker_res_unsat :
  unsatisfiable α (incrementalLRATChecker f (Action.addRat id✝ c✝ pivot✝ rupHints✝ ratHints✝)).fst
incrementalChecker_res :
  (if (performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).snd = true then
        ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, out_of_proof)
      else ((performRatAdd f c✝ pivot✝ rupHints✝ ratHints✝).fst, rup_failure)).snd =
    out_of_proof
⊢ unsatisfiable α f

case right.h_4
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
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_res_unsat : unsatisfiable α (incrementalLRATChecker f (Action.del ids✝)).fst
incrementalChecker_res : (Formula.delete f ids✝, out_of_proof).snd = out_of_proof
⊢ unsatisfiable α f
case right.h_4
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
ids✝ : Array Nat
aWellFormed : wellFormedAction (Action.del ids✝)
incrementalChecker_res_unsat : unsatisfiable α (incrementalLRATChecker f (Action.del ids✝)).fst
incrementalChecker_res : (Formula.delete f ids✝, out_of_proof).snd = out_of_proof
⊢ unsatisfiable α f