-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
theorem insertUnitInvariant_insertUnit {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)) :
    InsertUnitInvariant assignments0 assignments0_size units assignments assignments_size →
    let update_res := insertUnit (units, assignments, foundContradiction) l
    have update_res_size : update_res.snd.fst.size = n

Proof state:
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
h5 : hasAssignment l.snd assignments[l.fst.val]! = true
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
⊢ ∃ j b h,
    (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

Tactic:
refine ⟨⟨j.1, j_lt_updatedUnits_size⟩, b,i_gt_zero, ?_⟩

Result:
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
h5 : hasAssignment l.snd assignments[l.fst.val]! = true
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
⊢ (insertUnit (units, assignments, foundContradiction) l).fst[↑⟨↑j, j_lt_updatedUnits_size⟩] = (⟨↑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, j_lt_updatedUnits_size⟩ →
            ¬(insertUnit (units, assignments, foundContradiction) l).fst[↑k].fst.val = ↑i

Proof state:
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
h✝ : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
⊢ (assignments.modify l.fst.val (addAssignment l.snd))[l.fst.val] = both

Tactic:
rw [Array.getElem_modify_self]

Result:
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
h✝ : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
⊢ addAssignment l.snd assignments[l.fst.val] = both

Proof state:
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, ⋯⟩, true)
h2 : assignments[↑i] = addAssignment true assignments0[↑i]
h3 : hasAssignment true assignments0[↑i] = false
hb : b = true
hl : l.snd = false
h : assignments0[↑i] = neg
⊢ neg = unassigned

Tactic:
simp only [addAssignment, addPosAssignment, h, ite_true] at h2

Result:
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, ⋯⟩, true)
h3 : hasAssignment true assignments0[↑i] = false
hb : b = true
hl : l.snd = false
h : assignments0[↑i] = neg
h2 : assignments[↑i] = both
⊢ neg = unassigned