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
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
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
⊢ ∃ 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:
match hb : b, hl : l.2 with
| true, true =>
  exfalso
  have assignments_i_rw : assignments[i.1]! = assignments[i.1] := by
    simp only [getElem!, i_in_bounds, ↓reduceDite, Array.get_eq_getElem]
  rw [hl, ← i_eq_l, assignments_i_rw, h2] at h5
  exact h5 (has_of_add _ true)
| true, false =>
  refine ⟨⟨j.1, j_lt_updatedUnits_size⟩, mostRecentUnitIdx, i_gt_zero, ?_⟩
  simp only [insertUnit, h5, Prod.mk.eta, ite_false, Array.get_push_eq, ne_eq]
  constructor
  . rw [Array.get_push_lt units l j.1 j.2, h1]
  . constructor
    . simp only [i_eq_l, ← hl]
      apply Prod.mk.eta
    . constructor
      . simp only [i_eq_l]
        rw [Array.get_modify_at_idx l_in_bounds]
        simp only [addAssignment, hl, ← i_eq_l, h2, ite_true, ite_false]
        apply addNeg_of_addPos_eq_both
      . constructor
        . match h : assignments0[i.val]'_ with
          | unassigned => rfl
          | pos => simp (config := {decide := true}) [h] at h3
          | neg =>
            simp only [addAssignment, addPosAssignment, h, ite_true] at h2
            simp only [i_eq_l] at h2
            simp [hasAssignment, hl, getElem!, l_in_bounds, h2, hasNegAssignment] at h5
          | both => simp (config := {decide := true}) only [h] at h3
        . intro k k_ne_j k_ne_l
          rw [Array.get_push]
          by_cases h : k.1 < units.size
          . simp only [h, dite_true]
            have k_ne_j : ⟨k.1, h⟩ ≠ j := by
              intro k_eq_j
              simp only [← k_eq_j, not_true] at k_ne_j
            exact h4 ⟨k.1, h⟩ k_ne_j
          . exfalso
            have k_property := k.2
            simp only [insertUnit, h5, Prod.mk.eta, ite_false, Array.size_push] at k_property
            rcases Nat.lt_or_eq_of_le <| Nat.le_of_lt_succ k_property with k_lt_units_size | k_eq_units_size
            . exact h k_lt_units_size
            . simp only [← k_eq_units_size, not_true, mostRecentUnitIdx] at k_ne_l
              exact k_ne_l rfl
| false, true =>
  refine ⟨mostRecentUnitIdx, ⟨j.1, j_lt_updatedUnits_size⟩, i_gt_zero, ?_⟩
  simp only [insertUnit, h5, Prod.mk.eta, ite_false, Array.get_push_eq, ne_eq]
  constructor
  . simp only [i_eq_l, ← hl]
    apply Prod.mk.eta
  . constructor
    . rw [Array.get_push_lt units l j.1 j.2, h1]
    . constructor
      . simp only [i_eq_l]
        rw [Array.get_modify_at_idx l_in_bounds]
        simp only [addAssignment, hl, ← i_eq_l, h2, ite_true, ite_false]
        apply addPos_of_addNeg_eq_both
      . constructor
        . match h : assignments0[i.val]'_ with
          | unassigned => rfl
          | pos =>
            simp only [addAssignment, h, ite_false, addNegAssignment] at h2
            simp only [i_eq_l] at h2
            simp [hasAssignment, hl, getElem!, l_in_bounds, h2, hasPosAssignment] at h5
          | neg  => simp (config := {decide := true}) only [h] at h3
          | both => simp (config := {decide := true}) only [h] at h3
        . intro k k_ne_l k_ne_j
          rw [Array.get_push]
          by_cases h : k.1 < units.size
          . simp only [h, dite_true]
            have k_ne_j : ⟨k.1, h⟩ ≠ j := by
              intro k_eq_j
              simp only [← k_eq_j, not_true] at k_ne_j
            exact h4 ⟨k.1, h⟩ k_ne_j
          . exfalso
            have k_property := k.2
            simp only [insertUnit, h5, Prod.mk.eta, ite_false, Array.size_push] at k_property
            rcases Nat.lt_or_eq_of_le $ Nat.le_of_lt_succ k_property with k_lt_units_size | k_eq_units_size
            . exact h k_lt_units_size
            . simp only [← k_eq_units_size, not_true, mostRecentUnitIdx] at k_ne_l
              exact k_ne_l rfl
| false, false =>
  exfalso
  have assignments_i_rw : assignments[i.1]! = assignments[i.1] := by
    simp [getElem!, i_in_bounds]
  rw [hl, ← i_eq_l, assignments_i_rw, h2] at h5
  exact h5 (has_of_add _ false)
Proof state after:
no goals