-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddResult.lean
theorem clear_insertRat {n : Nat} (f : DefaultFormula n)
    (hf : f.ratUnits = #[] ∧ f.assignments.size = n) (units : CNF.Clause (PosFin n)) :
    clearRatUnits (f.insertRatUnits units).1 = f

Proof state:
case assignments.intro.h₂
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
units : CNF.Clause (PosFin n)
motive : Nat → Array Assignment → Prop := f.ClearInsertInductionMotive ⋯ (f.insertRatUnits units).fst.ratUnits
h_base : motive 0 (f.insertRatUnits units).fst.assignments
h_inductive :
  ∀ (idx : Fin (f.insertRatUnits units).fst.ratUnits.size) (assignments : Array Assignment),
    motive (↑idx) assignments → motive (↑idx + 1) (clearUnit assignments (f.insertRatUnits units).fst.ratUnits[idx])
h_size : (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments (f.insertRatUnits units).fst.ratUnits).size = n
i : Nat
hi1 : i < (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits).size
hi2 : i < f.assignments.size
i_lt_n : i < n
h :
  let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i :=
    (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments
        (f.insertRatUnits units).fst.ratUnits)[↑⟨i, i_lt_n⟩];
  let fassignments_i := f.assignments[↑⟨i, i_lt_n⟩];
  (assignments_i = fassignments_i ∧
      ∀ (j : Fin (f.insertRatUnits units).fst.ratUnits.size),
        ↑j ≥ (f.insertRatUnits units).fst.ratUnits.size →
          (f.insertRatUnits units).fst.ratUnits[j].fst.val ≠ ↑⟨i, i_lt_n⟩) ∨
    (∃ j b i_gt_zero,
        ↑j ≥ (f.insertRatUnits units).fst.ratUnits.size ∧
          (f.insertRatUnits units).fst.ratUnits[j] = (⟨↑⟨i, i_lt_n⟩, ⋯⟩, b) ∧
            assignments_i = addAssignment b fassignments_i ∧
              ¬hasAssignment b fassignments_i = true ∧
                ∀ (k : Fin (f.insertRatUnits units).fst.ratUnits.size),
                  ↑k ≥ (f.insertRatUnits units).fst.ratUnits.size →
                    k ≠ j → (f.insertRatUnits units).fst.ratUnits[k].fst.val ≠ ↑⟨i, i_lt_n⟩) ∨
      ∃ j1 j2 i_gt_zero,
        ↑j1 ≥ (f.insertRatUnits units).fst.ratUnits.size ∧
          ↑j2 ≥ (f.insertRatUnits units).fst.ratUnits.size ∧
            (f.insertRatUnits units).fst.ratUnits[j1] = (⟨↑⟨i, i_lt_n⟩, ⋯⟩, true) ∧
              (f.insertRatUnits units).fst.ratUnits[j2] = (⟨↑⟨i, i_lt_n⟩, ⋯⟩, false) ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin (f.insertRatUnits units).fst.ratUnits.size),
                      ↑k ≥ (f.insertRatUnits units).fst.ratUnits.size →
                        k ≠ j1 → k ≠ j2 → (f.insertRatUnits units).fst.ratUnits[k].fst.val ≠ ↑⟨i, i_lt_n⟩
⊢ (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits)[i] =
    f.assignments[i]

Tactic:
rcases h with h | h | h

Result:
case assignments.intro.h₂.inl
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
units : CNF.Clause (PosFin n)
motive : Nat → Array Assignment → Prop := f.ClearInsertInductionMotive ⋯ (f.insertRatUnits units).fst.ratUnits
h_base : motive 0 (f.insertRatUnits units).fst.assignments
h_inductive :
  ∀ (idx : Fin (f.insertRatUnits units).fst.ratUnits.size) (assignments : Array Assignment),
    motive (↑idx) assignments → motive (↑idx + 1) (clearUnit assignments (f.insertRatUnits units).fst.ratUnits[idx])
h_size : (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments (f.insertRatUnits units).fst.ratUnits).size = n
i : Nat
hi1 : i < (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits).size
hi2 : i < f.assignments.size
i_lt_n : i < n
h :
  (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments (f.insertRatUnits units).fst.ratUnits)[↑⟨i, i_lt_n⟩] =
      f.assignments[↑⟨i, i_lt_n⟩] ∧
    ∀ (j : Fin (f.insertRatUnits units).fst.ratUnits.size),
      ↑j ≥ (f.insertRatUnits units).fst.ratUnits.size → (f.insertRatUnits units).fst.ratUnits[j].fst.val ≠ ↑⟨i, i_lt_n⟩
⊢ (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits)[i] =
    f.assignments[i]

case assignments.intro.h₂.inr.inl
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
units : CNF.Clause (PosFin n)
motive : Nat → Array Assignment → Prop := f.ClearInsertInductionMotive ⋯ (f.insertRatUnits units).fst.ratUnits
h_base : motive 0 (f.insertRatUnits units).fst.assignments
h_inductive :
  ∀ (idx : Fin (f.insertRatUnits units).fst.ratUnits.size) (assignments : Array Assignment),
    motive (↑idx) assignments → motive (↑idx + 1) (clearUnit assignments (f.insertRatUnits units).fst.ratUnits[idx])
h_size : (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments (f.insertRatUnits units).fst.ratUnits).size = n
i : Nat
hi1 : i < (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits).size
hi2 : i < f.assignments.size
i_lt_n : i < n
h :
  ∃ j b i_gt_zero,
    ↑j ≥ (f.insertRatUnits units).fst.ratUnits.size ∧
      (f.insertRatUnits units).fst.ratUnits[j] = (⟨↑⟨i, i_lt_n⟩, ⋯⟩, b) ∧
        (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments
                (f.insertRatUnits units).fst.ratUnits)[↑⟨i, i_lt_n⟩] =
            addAssignment b f.assignments[↑⟨i, i_lt_n⟩] ∧
          ¬hasAssignment b f.assignments[↑⟨i, i_lt_n⟩] = true ∧
            ∀ (k : Fin (f.insertRatUnits units).fst.ratUnits.size),
              ↑k ≥ (f.insertRatUnits units).fst.ratUnits.size →
                k ≠ j → (f.insertRatUnits units).fst.ratUnits[k].fst.val ≠ ↑⟨i, i_lt_n⟩
⊢ (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits)[i] =
    f.assignments[i]

case assignments.intro.h₂.inr.inr
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
units : CNF.Clause (PosFin n)
motive : Nat → Array Assignment → Prop := f.ClearInsertInductionMotive ⋯ (f.insertRatUnits units).fst.ratUnits
h_base : motive 0 (f.insertRatUnits units).fst.assignments
h_inductive :
  ∀ (idx : Fin (f.insertRatUnits units).fst.ratUnits.size) (assignments : Array Assignment),
    motive (↑idx) assignments → motive (↑idx + 1) (clearUnit assignments (f.insertRatUnits units).fst.ratUnits[idx])
h_size : (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments (f.insertRatUnits units).fst.ratUnits).size = n
i : Nat
hi1 : i < (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits).size
hi2 : i < f.assignments.size
i_lt_n : i < n
h :
  ∃ j1 j2 i_gt_zero,
    ↑j1 ≥ (f.insertRatUnits units).fst.ratUnits.size ∧
      ↑j2 ≥ (f.insertRatUnits units).fst.ratUnits.size ∧
        (f.insertRatUnits units).fst.ratUnits[j1] = (⟨↑⟨i, i_lt_n⟩, ⋯⟩, true) ∧
          (f.insertRatUnits units).fst.ratUnits[j2] = (⟨↑⟨i, i_lt_n⟩, ⋯⟩, false) ∧
            (Array.foldl clearUnit (f.insertRatUnits units).fst.assignments
                    (f.insertRatUnits units).fst.ratUnits)[↑⟨i, i_lt_n⟩] =
                both ∧
              f.assignments[↑⟨i, i_lt_n⟩] = unassigned ∧
                ∀ (k : Fin (f.insertRatUnits units).fst.ratUnits.size),
                  ↑k ≥ (f.insertRatUnits units).fst.ratUnits.size →
                    k ≠ j1 → k ≠ j2 → (f.insertRatUnits units).fst.ratUnits[k].fst.val ≠ ↑⟨i, i_lt_n⟩
⊢ (Array.foldl clearUnit (f.insertRatUnits units).1.assignments (f.insertRatUnits units).1.ratUnits)[i] =
    f.assignments[i]