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
h :
  (assignments[↑i] = assignments0[↑i] ∧ ∀ (j : Fin units.size), ¬units[↑j].fst.val = ↑i) ∨
    (∃ j b i_gt_zero,
        units[↑j] = (⟨↑i, ⋯⟩, b) ∧
          assignments[↑i] = addAssignment b assignments0[↑i] ∧
            hasAssignment b assignments0[↑i] = false ∧ ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        units[↑j1] = (⟨↑i, ⋯⟩, true) ∧
          units[↑j2] = (⟨↑i, ⋯⟩, false) ∧
            assignments[↑i] = both ∧
              assignments0[↑i] = unassigned ∧ ∀ (k : Fin units.size), ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments.size
⊢ ((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:
rcases h with ⟨h1, h2⟩ | ⟨j, b, i_gt_zero, h1, h2, h3, h4⟩ | ⟨j1, j2, i_gt_zero, h1, h2, h3, h4, h5⟩
Proof state after:
case inl.intro
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
⊢ ((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

case inr.inl.intro.intro.intro.intro.intro.intro
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
h1 : units[↑j] = (⟨↑i, ⋯⟩, b)
h2 : assignments[↑i] = addAssignment b assignments0[↑i]
h3 : hasAssignment b assignments0[↑i] = false
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
⊢ ((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

case inr.inr.intro.intro.intro.intro.intro.intro.intro
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
⊢ ((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