Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case intro.inl.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
ih1 : assignments[↑i] = f.assignments[↑i]
ih2 : ∀ (j : Fin units.size), ↑j ≥ ↑idx → ¬units[↑j].fst.val = ↑i
⊢ let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i := (clearUnit assignments units[idx])[↑i];
  let fassignments_i := f.assignments[↑i];
  (assignments_i = fassignments_i ∧ ∀ (j : Fin units.size), ↑j ≥ ↑idx + 1 → units[j].fst.val ≠ ↑i) ∨
    (∃ j b i_gt_zero,
        ↑j ≥ ↑idx + 1 ∧
          units[j] = (⟨↑i, ⋯⟩, b) ∧
            assignments_i = addAssignment b fassignments_i ∧
              ¬hasAssignment b fassignments_i = true ∧
                ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j → units[k].fst.val ≠ ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        ↑j1 ≥ ↑idx + 1 ∧
          ↑j2 ≥ ↑idx + 1 ∧
            units[j1] = (⟨↑i, ⋯⟩, true) ∧
              units[j2] = (⟨↑i, ⋯⟩, false) ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j1 → k ≠ j2 → units[k].fst.val ≠ ↑i

case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j : Fin units.size
j_ge_idx : ↑j ≥ ↑idx
b : Bool
i_gt_zero : ↑i > 0
ih1 : units[↑j] = (⟨↑i, ⋯⟩, b)
ih2 : assignments[↑i] = addAssignment b f.assignments[↑i]
ih3 : ¬hasAssignment b f.assignments[↑i] = true
ih4 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j → ¬units[↑k].fst.val = ↑i
⊢ let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i := (clearUnit assignments units[idx])[↑i];
  let fassignments_i := f.assignments[↑i];
  (assignments_i = fassignments_i ∧ ∀ (j : Fin units.size), ↑j ≥ ↑idx + 1 → units[j].fst.val ≠ ↑i) ∨
    (∃ j b i_gt_zero,
        ↑j ≥ ↑idx + 1 ∧
          units[j] = (⟨↑i, ⋯⟩, b) ∧
            assignments_i = addAssignment b fassignments_i ∧
              ¬hasAssignment b fassignments_i = true ∧
                ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j → units[k].fst.val ≠ ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        ↑j1 ≥ ↑idx + 1 ∧
          ↑j2 ≥ ↑idx + 1 ∧
            units[j1] = (⟨↑i, ⋯⟩, true) ∧
              units[j2] = (⟨↑i, ⋯⟩, false) ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j1 → k ≠ j2 → units[k].fst.val ≠ ↑i

case intro.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
⊢ let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i := (clearUnit assignments units[idx])[↑i];
  let fassignments_i := f.assignments[↑i];
  (assignments_i = fassignments_i ∧ ∀ (j : Fin units.size), ↑j ≥ ↑idx + 1 → units[j].fst.val ≠ ↑i) ∨
    (∃ j b i_gt_zero,
        ↑j ≥ ↑idx + 1 ∧
          units[j] = (⟨↑i, ⋯⟩, b) ∧
            assignments_i = addAssignment b fassignments_i ∧
              ¬hasAssignment b fassignments_i = true ∧
                ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j → units[k].fst.val ≠ ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        ↑j1 ≥ ↑idx + 1 ∧
          ↑j2 ≥ ↑idx + 1 ∧
            units[j1] = (⟨↑i, ⋯⟩, true) ∧
              units[j2] = (⟨↑i, ⋯⟩, false) ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Tactic:
. apply Or.inl
  constructor
  . simp only [clearUnit, Fin.getElem_fin, Array.get_eq_getElem]
    specialize ih2 idx (Nat.le_refl idx.val)
    have idx_unit_in_bounds : units[idx].1.1 < assignments.size := by
      rw [hsize]; exact units[idx].1.2.2
    have i_in_bounds : i.1 < assignments.size := by
      rw [hsize]
      exact i.2
    have h := Array.get_modify_unchanged idx_unit_in_bounds i_in_bounds (removeAssignment units[idx.val].2) ih2
    simp only [Fin.getElem_fin] at h
    rw [h]
    exact ih1
  . intro j j_ge_idx_add_one
    exact ih2 j (Nat.le_of_succ_le j_ge_idx_add_one)
Proof state after:
case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j : Fin units.size
j_ge_idx : ↑j ≥ ↑idx
b : Bool
i_gt_zero : ↑i > 0
ih1 : units[↑j] = (⟨↑i, ⋯⟩, b)
ih2 : assignments[↑i] = addAssignment b f.assignments[↑i]
ih3 : ¬hasAssignment b f.assignments[↑i] = true
ih4 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j → ¬units[↑k].fst.val = ↑i
⊢ let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i := (clearUnit assignments units[idx])[↑i];
  let fassignments_i := f.assignments[↑i];
  (assignments_i = fassignments_i ∧ ∀ (j : Fin units.size), ↑j ≥ ↑idx + 1 → units[j].fst.val ≠ ↑i) ∨
    (∃ j b i_gt_zero,
        ↑j ≥ ↑idx + 1 ∧
          units[j] = (⟨↑i, ⋯⟩, b) ∧
            assignments_i = addAssignment b fassignments_i ∧
              ¬hasAssignment b fassignments_i = true ∧
                ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j → units[k].fst.val ≠ ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        ↑j1 ≥ ↑idx + 1 ∧
          ↑j2 ≥ ↑idx + 1 ∧
            units[j1] = (⟨↑i, ⋯⟩, true) ∧
              units[j2] = (⟨↑i, ⋯⟩, false) ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j1 → k ≠ j2 → units[k].fst.val ≠ ↑i

case intro.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
⊢ let_fun i_lt_assignments_size := ⋯;
  let_fun i_lt_f_assignments_size := ⋯;
  let assignments_i := (clearUnit assignments units[idx])[↑i];
  let fassignments_i := f.assignments[↑i];
  (assignments_i = fassignments_i ∧ ∀ (j : Fin units.size), ↑j ≥ ↑idx + 1 → units[j].fst.val ≠ ↑i) ∨
    (∃ j b i_gt_zero,
        ↑j ≥ ↑idx + 1 ∧
          units[j] = (⟨↑i, ⋯⟩, b) ∧
            assignments_i = addAssignment b fassignments_i ∧
              ¬hasAssignment b fassignments_i = true ∧
                ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j → units[k].fst.val ≠ ↑i) ∨
      ∃ j1 j2 i_gt_zero,
        ↑j1 ≥ ↑idx + 1 ∧
          ↑j2 ≥ ↑idx + 1 ∧
            units[j1] = (⟨↑i, ⋯⟩, true) ∧
              units[j2] = (⟨↑i, ⋯⟩, false) ∧
                assignments_i = both ∧
                  fassignments_i = unassigned ∧
                    ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j1 → k ≠ j2 → units[k].fst.val ≠ ↑i