Theorem: LRAT.DefaultFormula.insertUnit_preserves_invariant
Proof state before:
case right.right.left
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
  ⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, false)
h2 : assignments[↑i] = addAssignment false assignments0[↑i]
h3 : hasAssignment false assignments0[↑i] = false
hb : b = false
hl : l.snd = true
⊢ (assignments.modify l.fst.val (addAssignment l.snd))[↑i] = both

case right.right.right
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
  ⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, false)
h2 : assignments[↑i] = addAssignment false assignments0[↑i]
h3 : hasAssignment false assignments0[↑i] = false
hb : b = false
hl : l.snd = true
⊢ assignments0[↑i] = unassigned ∧
    ∀
      (k :
        Fin
          (if hasAssignment l.snd assignments[l.fst.val]! = true then (units, assignments, foundContradiction)
              else
                (units.push (l.fst, l.snd), assignments.modify l.fst.val (addAssignment l.snd),
                  foundContradiction || assignments[l.fst.val]! != unassigned)).fst.size),
      ¬k = mostRecentUnitIdx → ¬k = ⟨↑j, j_lt_updatedUnits_size⟩ → ¬(units.push l)[↑k].fst.val = ↑i
Tactic:
. simp only [i_eq_l]
  rw [Array.get_modify_at_idx l_in_bounds]
  simp only [addAssignment, hl, ← i_eq_l, h2, ite_true, ite_false]
  apply addPos_of_addNeg_eq_both
Proof state after:
case right.right.right
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
  ⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, false)
h2 : assignments[↑i] = addAssignment false assignments0[↑i]
h3 : hasAssignment false assignments0[↑i] = false
hb : b = false
hl : l.snd = true
⊢ assignments0[↑i] = unassigned ∧
    ∀
      (k :
        Fin
          (if hasAssignment l.snd assignments[l.fst.val]! = true then (units, assignments, foundContradiction)
              else
                (units.push (l.fst, l.snd), assignments.modify l.fst.val (addAssignment l.snd),
                  foundContradiction || assignments[l.fst.val]! != unassigned)).fst.size),
      ¬k = mostRecentUnitIdx → ¬k = ⟨↑j, j_lt_updatedUnits_size⟩ → ¬(units.push l)[↑k].fst.val = ↑i