-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean
theorem insert_iff {n : Nat} (f : DefaultFormula n) (c1 : DefaultClause n) (c2 : DefaultClause n) :
    c2 ∈ toList (insert f c1) ↔ c2 = c1 ∨ c2 ∈ toList f

Proof state:
case mpr.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_eq_c1 : c2 = c1
a✝ :
  c2 = c1 ∨
    some c2 ∈ f.clauses.toList ∨
      (∃ a,
          (a, false) ∈ f.rupUnits.toList ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.toList ∧ unit (a, true) = c2) ∨
        ∃ a, (a, false) ∈ f.ratUnits.toList ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.toList ∧ unit (a, true) = c2
⊢ some c2 ∈ (f.insert c1).clauses.toList

Tactic:
simp only [c2_eq_c1, insert]

Result:
case mpr.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_eq_c1 : c2 = c1
a✝ :
  c2 = c1 ∨
    some c2 ∈ f.clauses.toList ∨
      (∃ a,
          (a, false) ∈ f.rupUnits.toList ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.rupUnits.toList ∧ unit (a, true) = c2) ∨
        ∃ a, (a, false) ∈ f.ratUnits.toList ∧ unit (a, false) = c2 ∨ (a, true) ∈ f.ratUnits.toList ∧ unit (a, true) = c2
⊢ some c1 ∈
    (match c1.isUnit with
        | none =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments }
        | some (l, true) =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addPosAssignment }
        | some (l, false) =>
          { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
            assignments := f.assignments.modify l.val addNegAssignment }).clauses.toList

Proof state:
case mpr.inr.inl
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ f.clauses.toList
⊢ some c2 ∈ (f.insert c1).clauses.toList ∨
    (∃ a,
        (a, false) ∈ (f.insert c1).rupUnits.toList ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).rupUnits.toList ∧ unit (a, true) = c2) ∨
      ∃ a,
        (a, false) ∈ (f.insert c1).ratUnits.toList ∧ unit (a, false) = c2 ∨
          (a, true) ∈ (f.insert c1).ratUnits.toList ∧ unit (a, true) = c2

Tactic:
apply Or.inl

Result:
case mpr.inr.inl.h
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
h : some c2 ∈ f.clauses.toList
⊢ some c2 ∈ (f.insert c1).clauses.toList

Proof state:
case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h :
  some c2 ∈
    { clauses := f.clauses.push (some c1), rupUnits := f.rupUnits, ratUnits := f.ratUnits,
          assignments := f.assignments.modify l✝.val addNegAssignment }.clauses.toList
⊢ some c2 ∈ f.clauses.toList

Tactic:
simp only [Array.push_toList, List.mem_append, List.mem_singleton, Option.some.injEq] at h

Result:
case mp.h.inl.h.h_3
n : Nat
f : DefaultFormula n
c1 c2 : DefaultClause n
c2_ne_c1 : ¬c2 = c1
x✝ : Option (Literal (PosFin n))
l✝ : PosFin n
heq✝ : c1.isUnit = some (l✝, false)
h : some c2 ∈ f.clauses.toList ∨ c2 = c1
⊢ some c2 ∈ f.clauses.toList