Theorem: LRAT.DefaultFormula.performRatCheck_preserves_formula
Proof state before:
case h_1
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
p : Literal (PosFin n)
ratHint : Nat × Array Nat
x✝ : Option (DefaultClause n)
c✝ : DefaultClause n
heq✝ : f.clauses[ratHint.fst]! = some c✝
⊢ (if
          ({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                    assignments := f.assignments }.insertRatUnits
                (c✝.delete p).negate).snd =
            true then
        (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                    assignments := f.assignments }.insertRatUnits
                (c✝.delete p).negate).fst.clearRatUnits,
          true)
      else
        if
            (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).2.2.snd =
                true ∨
              (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).2.2.fst =
                false then
          ({
                clauses :=
                  (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).1.clauses,
                rupUnits :=
                  (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).1.rupUnits,
                ratUnits :=
                  (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).1.ratUnits,
                assignments :=
                  restoreAssignments
                    (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                    assignments := f.assignments }.insertRatUnits
                                (c✝.delete p).negate).fst.performRupCheck
                          ratHint.snd).1.assignments
                    (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                    assignments := f.assignments }.insertRatUnits
                                (c✝.delete p).negate).fst.performRupCheck
                          ratHint.snd).2.fst }.clearRatUnits,
            false)
        else
          ({
                clauses :=
                  (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).1.clauses,
                rupUnits :=
                  (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).1.rupUnits,
                ratUnits :=
                  (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                  assignments := f.assignments }.insertRatUnits
                              (c✝.delete p).negate).fst.performRupCheck
                        ratHint.snd).1.ratUnits,
                assignments :=
                  restoreAssignments
                    (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                    assignments := f.assignments }.insertRatUnits
                                (c✝.delete p).negate).fst.performRupCheck
                          ratHint.snd).1.assignments
                    (({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
                                    assignments := f.assignments }.insertRatUnits
                                (c✝.delete p).negate).fst.performRupCheck
                          ratHint.snd).2.fst }.clearRatUnits,
            true)).fst =
    f

case h_2
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
p : Literal (PosFin n)
ratHint : Nat × Array Nat
x✝ : Option (DefaultClause n)
heq✝ : f.clauses[ratHint.fst]! = none
⊢ ({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }, false).fst =
    f
Tactic:
. next c _ =>
  split
  . rw [clear_insertRat f hf]
  . let fc := (insertRatUnits f (negate (DefaultClause.delete c p))).1
    have fc_assignments_size : fc.assignments.size = n := by rw [insertRatUnits_preserves_assignments_size, hf.2]
    have insertRatUnits_rw : (insertRatUnits f (negate (DefaultClause.delete c p))).1 =
      ⟨(insertRatUnits f (negate (DefaultClause.delete c p))).1.clauses,
       (insertRatUnits f (negate (DefaultClause.delete c p))).1.rupUnits,
       (insertRatUnits f (negate (DefaultClause.delete c p))).1.ratUnits,
       (insertRatUnits f (negate (DefaultClause.delete c p))).1.assignments⟩ := rfl
    simp only [performRupCheck_preserves_clauses, performRupCheck_preserves_rupUnits, performRupCheck_preserves_ratUnits]
    rw [restoreAssignments_performRupCheck fc fc_assignments_size ratHint.2, ← insertRatUnits_rw,
      clear_insertRat f hf (negate (DefaultClause.delete c p))]
    split <;> rfl
Proof state after:
case h_2
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
p : Literal (PosFin n)
ratHint : Nat × Array Nat
x✝ : Option (DefaultClause n)
heq✝ : f.clauses[ratHint.fst]! = none
⊢ ({ clauses := f.clauses, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }, false).fst =
    f