-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
theorem derivedLitsInvariant_confirmRupHint {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
    (rupHints : Array Nat) (i : Fin rupHints.size)
    (acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool)
    (ih : ∃ hsize : acc.1.size = n, DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.1) :
    let rupHint_res := (confirmRupHint f.clauses) acc rupHints[i]
    ∃ hsize : rupHint_res.1.size = n, DerivedLitsInvariant f f_assignments_size rupHint_res.1 hsize rupHint_res.2.1

Proof state:
case intro.isFalse.inr.inr.intro.inr.inl
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
c : DefaultClause n
rupHint_clause_eq_c : f.clauses[rupHints[↑i]]? = some (some c)
hreducedToEmpty : c.reduce acc.fst = ReduceResult.reducedToEmpty
⊢ f.DerivedLitsInvariant f_assignments_size
    (match c.reduce acc.fst with
      | ReduceResult.encounteredBoth => (acc.fst, acc.2.fst, true, false)
      | ReduceResult.reducedToEmpty => (acc.fst, acc.2.fst, true, false)
      | ReduceResult.reducedToUnit (l, b) =>
        if hasAssignment b acc.fst[l.val]! = true then (acc.fst, acc.2.fst, false, false)
        else (acc.fst.modify l.val (addAssignment b), (l, b) :: acc.2.fst, false, false)
      | ReduceResult.reducedToNonunit => (acc.fst, acc.2.fst, false, true)).fst
    ⋯
    (match c.reduce acc.fst with
        | ReduceResult.encounteredBoth => (acc.fst, acc.2.fst, true, false)
        | ReduceResult.reducedToEmpty => (acc.fst, acc.2.fst, true, false)
        | ReduceResult.reducedToUnit (l, b) =>
          if hasAssignment b acc.fst[l.val]! = true then (acc.fst, acc.2.fst, false, false)
          else (acc.fst.modify l.val (addAssignment b), (l, b) :: acc.2.fst, false, false)
        | ReduceResult.reducedToNonunit => (acc.fst, acc.2.fst, false, true)).snd.fst

Tactic:
simp only [hreducedToEmpty]

Result:
case intro.isFalse.inr.inr.intro.inr.inl
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
c : DefaultClause n
rupHint_clause_eq_c : f.clauses[rupHints[↑i]]? = some (some c)
hreducedToEmpty : c.reduce acc.fst = ReduceResult.reducedToEmpty
⊢ f.DerivedLitsInvariant f_assignments_size acc.fst ⋯ acc.2.fst

Proof state:
case intro.isFalse.inr.inr.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
c : DefaultClause n
rupHint_clause_eq_c : f.clauses[rupHints[↑i]]? = some (some c)
⊢ f.DerivedLitsInvariant f_assignments_size
    (match c.reduce acc.fst with
      | ReduceResult.encounteredBoth => (acc.fst, acc.2.fst, true, false)
      | ReduceResult.reducedToEmpty => (acc.fst, acc.2.fst, true, false)
      | ReduceResult.reducedToUnit (l, b) =>
        if hasAssignment b acc.fst[l.val]! = true then (acc.fst, acc.2.fst, false, false)
        else (acc.fst.modify l.val (addAssignment b), (l, b) :: acc.2.fst, false, false)
      | ReduceResult.reducedToNonunit => (acc.fst, acc.2.fst, false, true)).fst
    ⋯
    (match c.reduce acc.fst with
        | ReduceResult.encounteredBoth => (acc.fst, acc.2.fst, true, false)
        | ReduceResult.reducedToEmpty => (acc.fst, acc.2.fst, true, false)
        | ReduceResult.reducedToUnit (l, b) =>
          if hasAssignment b acc.fst[l.val]! = true then (acc.fst, acc.2.fst, false, false)
          else (acc.fst.modify l.val (addAssignment b), (l, b) :: acc.2.fst, false, false)
        | ReduceResult.reducedToNonunit => (acc.fst, acc.2.fst, false, true)).snd.fst

Tactic:
have reduce_c_options : reduce c acc.1 = ReduceResult.encounteredBoth ∨ reduce c acc.1 = ReduceResult.reducedToEmpty ∨
  (∃ l : Literal (PosFin n), reduce c acc.1 = ReduceResult.reducedToUnit l) ∨ reduce c acc.1 = ReduceResult.reducedToNonunit := by
  match reduce c acc.fst with
  | ReduceResult.encounteredBoth => exact Or.inl rfl
  | ReduceResult.reducedToEmpty => exact (Or.inr ∘ Or.inl) rfl
  | ReduceResult.reducedToUnit l => exact (Or.inr ∘ Or.inr ∘ Or.inl ∘ Exists.intro l) rfl
  | ReduceResult.reducedToNonunit => exact (Or.inr ∘ Or.inr ∘ Or.inr) rfl

Result:
case intro.isFalse.inr.inr.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
c : DefaultClause n
rupHint_clause_eq_c : f.clauses[rupHints[↑i]]? = some (some c)
reduce_c_options :
  c.reduce acc.fst = ReduceResult.encounteredBoth ∨
    c.reduce acc.fst = ReduceResult.reducedToEmpty ∨
      (∃ l, c.reduce acc.fst = ReduceResult.reducedToUnit l) ∨ c.reduce acc.fst = ReduceResult.reducedToNonunit
⊢ f.DerivedLitsInvariant f_assignments_size
    (match c.reduce acc.fst with
      | ReduceResult.encounteredBoth => (acc.fst, acc.2.fst, true, false)
      | ReduceResult.reducedToEmpty => (acc.fst, acc.2.fst, true, false)
      | ReduceResult.reducedToUnit (l, b) =>
        if hasAssignment b acc.fst[l.val]! = true then (acc.fst, acc.2.fst, false, false)
        else (acc.fst.modify l.val (addAssignment b), (l, b) :: acc.2.fst, false, false)
      | ReduceResult.reducedToNonunit => (acc.fst, acc.2.fst, false, true)).fst
    ⋯
    (match c.reduce acc.fst with
        | ReduceResult.encounteredBoth => (acc.fst, acc.2.fst, true, false)
        | ReduceResult.reducedToEmpty => (acc.fst, acc.2.fst, true, false)
        | ReduceResult.reducedToUnit (l, b) =>
          if hasAssignment b acc.fst[l.val]! = true then (acc.fst, acc.2.fst, false, false)
          else (acc.fst.modify l.val (addAssignment b), (l, b) :: acc.2.fst, false, false)
        | ReduceResult.reducedToNonunit => (acc.fst, acc.2.fst, false, true)).snd.fst

Proof state:
case intro.isFalse.inl
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
rupHint_clause_eq_none : f.clauses[rupHints[↑i]]? = none
⊢ f.DerivedLitsInvariant f_assignments_size acc.fst ⋯ acc.2.fst

Tactic:
exact ih

Result:
no goals