-- Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
theorem clear_insertRup {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
    (units : CNF.Clause (PosFin n)) :
    clearRupUnits (f.insertRupUnits units).1 = f

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

Tactic:
intro i hi1 hi2

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

Proof state:
case ratUnits
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
⊢ { clauses := (f.insertRupUnits units).1.clauses, rupUnits := #[], ratUnits := (f.insertRupUnits units).1.ratUnits,
        assignments :=
          Array.foldl clearUnit (f.insertRupUnits units).1.assignments (f.insertRupUnits units).1.rupUnits }.ratUnits =
    f.ratUnits

Tactic:
simp only [insertRupUnits]

Result:
no goals

Proof state:
case assignments
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
motive : Nat → Array Assignment → Prop := f.ClearInsertInductionMotive ⋯ (f.insertRupUnits units).fst.rupUnits
h_base : motive 0 (f.insertRupUnits units).fst.assignments
⊢ Array.foldl clearUnit (f.insertRupUnits units).1.assignments (f.insertRupUnits units).1.rupUnits = f.assignments

Tactic:
have h_inductive (idx : Fin (insertRupUnits f units).1.rupUnits.size) (assignments : Array Assignment)
  (ih : motive idx.val assignments) : motive (idx.val + 1) (clearUnit assignments (insertRupUnits f units).1.rupUnits[idx]) :=
  clear_insert_inductive_case f f_readyForRupAdd.2.1 (insertRupUnits f units).1.rupUnits
    (nodup_insertRupUnits f f_readyForRupAdd units) idx assignments ih

Result:
case assignments
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
motive : Nat → Array Assignment → Prop := f.ClearInsertInductionMotive ⋯ (f.insertRupUnits units).fst.rupUnits
h_base : motive 0 (f.insertRupUnits units).fst.assignments
h_inductive :
  ∀ (idx : Fin (f.insertRupUnits units).fst.rupUnits.size) (assignments : Array Assignment),
    motive (↑idx) assignments → motive (↑idx + 1) (clearUnit assignments (f.insertRupUnits units).fst.rupUnits[idx])
⊢ Array.foldl clearUnit (f.insertRupUnits units).1.assignments (f.insertRupUnits units).1.rupUnits = f.assignments