Theorem: LRAT.DefaultFormula.insertUnit_preserves_invariant
Proof state before:
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
j1 j2 : Fin units.size
i_gt_zero : ↑i > 0
h1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
h2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
h3 : assignments[↑i] = both
h4 : assignments0[↑i] = unassigned
h5 : ∀ (k : Fin units.size), ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
j1_lt_updatedUnits_size : ↑j1 < (insertUnit (units, assignments, foundContradiction) l).fst.size
j2_lt_updatedUnits_size : ↑j2 < (insertUnit (units, assignments, foundContradiction) l).fst.size
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_ne_j1 : ¬k = ⟨↑j1, j1_lt_updatedUnits_size⟩
k_ne_j2 : ¬k = ⟨↑j2, j2_lt_updatedUnits_size⟩
k_not_lt_units_size : ¬↑k < units.size
⊢ ¬(if hasAssignment l.snd assignments[l.fst.val]! = true then (units, assignments, foundContradiction)
              else
                (units.push l, assignments.modify l.fst.val (addAssignment l.snd),
                  foundContradiction || assignments[l.fst.val]! != unassigned)).fst[↑k].fst.val =
      ↑i
Tactic:
split
Proof state after:
case isTrue
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
j1 j2 : Fin units.size
i_gt_zero : ↑i > 0
h1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
h2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
h3 : assignments[↑i] = both
h4 : assignments0[↑i] = unassigned
h5 : ∀ (k : Fin units.size), ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
j1_lt_updatedUnits_size : ↑j1 < (insertUnit (units, assignments, foundContradiction) l).fst.size
j2_lt_updatedUnits_size : ↑j2 < (insertUnit (units, assignments, foundContradiction) l).fst.size
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_ne_j1 : ¬k = ⟨↑j1, j1_lt_updatedUnits_size⟩
k_ne_j2 : ¬k = ⟨↑j2, j2_lt_updatedUnits_size⟩
k_not_lt_units_size : ¬↑k < units.size
h✝ : hasAssignment l.snd assignments[l.fst.val]! = true
⊢ ¬(units, assignments, foundContradiction).fst[↑k].fst.val = ↑i

case isFalse
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
j1 j2 : Fin units.size
i_gt_zero : ↑i > 0
h1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
h2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
h3 : assignments[↑i] = both
h4 : assignments0[↑i] = unassigned
h5 : ∀ (k : Fin units.size), ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
j1_lt_updatedUnits_size : ↑j1 < (insertUnit (units, assignments, foundContradiction) l).fst.size
j2_lt_updatedUnits_size : ↑j2 < (insertUnit (units, assignments, foundContradiction) l).fst.size
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_ne_j1 : ¬k = ⟨↑j1, j1_lt_updatedUnits_size⟩
k_ne_j2 : ¬k = ⟨↑j2, j2_lt_updatedUnits_size⟩
k_not_lt_units_size : ¬↑k < units.size
h✝ : ¬hasAssignment l.snd assignments[l.fst.val]! = true
⊢ ¬(units.push l, assignments.modify l.fst.val (addAssignment l.snd),
                foundContradiction || assignments[l.fst.val]! != unassigned).fst[↑k].fst.val =
      ↑i