Theorem: LRAT.DefaultFormula.insertUnit_preserves_invariant
Proof state before:
case pos
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
h1 : assignments[↑i] = assignments0[↑i]
h2 : ∀ (j : Fin units.size), ¬units[↑j].fst.val = ↑i
h3 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
h✝ : ↑i = l.fst.val
⊢ ((insertUnit (units, assignments, foundContradiction) l).snd.fst[↑i] = assignments0[↑i] ∧
      ∀ (j : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size),
        ¬(insertUnit (units, assignments, foundContradiction) l).fst[↑j].fst.val = ↑i) ∨
    (∃ j b i_gt_zero,
        (insertUnit (units, assignments, foundContradiction) l).fst[↑j] = (⟨↑i, ⋯⟩, b) ∧
          (insertUnit (units, assignments, foundContradiction) l).snd.fst[↑i] = addAssignment b assignments0[↑i] ∧
            hasAssignment b assignments0[↑i] = false ∧
              ∀ (k : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size),
                ¬k = j → ¬(insertUnit (units, assignments, foundContradiction) l).fst[↑k].fst.val = ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        (insertUnit (units, assignments, foundContradiction) l).fst[↑j1] = (⟨↑i, ⋯⟩, true) ∧
          (insertUnit (units, assignments, foundContradiction) l).fst[↑j2] = (⟨↑i, ⋯⟩, false) ∧
            (insertUnit (units, assignments, foundContradiction) l).snd.fst[↑i] = both ∧
              assignments0[↑i] = unassigned ∧
                ∀ (k : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size),
                  ¬k = j1 → ¬k = j2 → ¬(insertUnit (units, assignments, foundContradiction) l).fst[↑k].fst.val = ↑i
Tactic:
next i_eq_l =>
apply Or.inr ∘ Or.inl
have units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).1.size := by
  simp only [insertUnit, Prod.mk.eta]
  split
  . contradiction
  . simp only [Array.size_push, Nat.lt_succ_self]
let mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).1.size :=
  ⟨units.size, units_size_lt_updatedUnits_size⟩
have i_gt_zero : i.1 > 0 := by rw [i_eq_l]; exact l.1.2.1
refine ⟨mostRecentUnitIdx, l.2, i_gt_zero, ?_⟩
simp only [insertUnit, h3, Prod.mk.eta, ite_false, Array.get_push_eq, i_eq_l]
constructor
. rfl
. constructor
  . rw [Array.get_modify_at_idx l_in_bounds]
    simp only [← i_eq_l, h1]
  . constructor
    . simp only [getElem!, l_in_bounds, ↓reduceDite, Array.get_eq_getElem,
      Bool.not_eq_true] at h3
      simp only [← i_eq_l, ← h1]
      simp only [i_eq_l, h3]
    . intro k hk
      have k_in_bounds : k.1 < units.size := by
        apply Nat.lt_of_le_of_ne
        . apply Nat.le_of_lt_succ
          have k_property := k.2
          simp only [insertUnit, h3, Prod.mk.eta, ite_false, Array.size_push] at k_property
          exact k_property
        . intro h
          simp only [← h, not_true, mostRecentUnitIdx] at hk
          exact hk rfl
      rw [Array.get_push_lt _ _ _ k_in_bounds]
      rw [i_eq_l] at h2
      exact h2 ⟨k.1, k_in_bounds⟩
Proof state after:
no goals