Theorem: LRAT.DefaultFormula.insertRupUnits_preserves_assignments_invariant
Proof state before:
case pos
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.readyForRupAdd
units : List (Literal (PosFin n))
h :
  let assignments := (f.insertRupUnits units).fst.assignments;
  let_fun hsize := ⋯;
  let rupUnits := (f.insertRupUnits units).fst.rupUnits;
  insertUnit_invariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.insertRupUnits units).fst.assignments.size = n
i : PosFin n
b : Bool
p : PosFin n → Bool
hp :
  ∀ (x : DefaultClause n),
    (some x ∈ (f.insertRupUnits units).fst.clauses.data ∨
        (∃ a,
            (a, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (a, false) = x ∨
              (a, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (a, true) = x) ∨
          ∃ a,
            (a, false) ∈ (f.insertRupUnits units).fst.ratUnits.data ∧ unit (a, false) = x ∨
              (a, true) ∈ (f.insertRupUnits units).fst.ratUnits.data ∧ unit (a, true) = x) →
      ∃ a, (a, false) ∈ Clause.toList x ∧ decide (p a = false) = true ∨ (a, true) ∈ Clause.toList x ∧ p a = true
pf : p ⊨ f
j : Fin (f.insertRupUnits units).fst.rupUnits.size
b' : Bool
hb : hasAssignment b (addAssignment b' f.assignments[↑⟨i.val, ⋯⟩]) = true
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRupUnits units).fst.rupUnits[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b')
h2 : (f.insertRupUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = addAssignment b' f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b' f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (f.insertRupUnits units).fst.rupUnits.size),
    k ≠ j → (f.insertRupUnits units).fst.rupUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
b_eq_b' : b = b'
j_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j]
j_unit_def : j_unit = unit (f.insertRupUnits units).fst.rupUnits[j]
hb' : b' = true
⊢ (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = unit (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b') ∨
    (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = unit (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b')

case neg
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.readyForRupAdd
units : List (Literal (PosFin n))
h :
  let assignments := (f.insertRupUnits units).fst.assignments;
  let_fun hsize := ⋯;
  let rupUnits := (f.insertRupUnits units).fst.rupUnits;
  insertUnit_invariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.insertRupUnits units).fst.assignments.size = n
i : PosFin n
b : Bool
p : PosFin n → Bool
hp :
  ∀ (x : DefaultClause n),
    (some x ∈ (f.insertRupUnits units).fst.clauses.data ∨
        (∃ a,
            (a, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (a, false) = x ∨
              (a, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (a, true) = x) ∨
          ∃ a,
            (a, false) ∈ (f.insertRupUnits units).fst.ratUnits.data ∧ unit (a, false) = x ∨
              (a, true) ∈ (f.insertRupUnits units).fst.ratUnits.data ∧ unit (a, true) = x) →
      ∃ a, (a, false) ∈ Clause.toList x ∧ decide (p a = false) = true ∨ (a, true) ∈ Clause.toList x ∧ p a = true
pf : p ⊨ f
j : Fin (f.insertRupUnits units).fst.rupUnits.size
b' : Bool
hb : hasAssignment b (addAssignment b' f.assignments[↑⟨i.val, ⋯⟩]) = true
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRupUnits units).fst.rupUnits[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b')
h2 : (f.insertRupUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = addAssignment b' f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b' f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (f.insertRupUnits units).fst.rupUnits.size),
    k ≠ j → (f.insertRupUnits units).fst.rupUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
b_eq_b' : b = b'
j_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j]
j_unit_def : j_unit = unit (f.insertRupUnits units).fst.rupUnits[j]
hb' : ¬b' = true
⊢ (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = unit (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b') ∨
    (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = unit (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b')
Tactic:
. rw [hb']
  apply Or.inr
  constructor
  . have h1 : (insertRupUnits f units).fst.rupUnits[j] = (i, true) := by
      rw [hb'] at h1
      rw [h1]
      simp only [Prod.mk.injEq, and_true]
      apply Subtype.ext
      rfl
    rw [← h1]
    apply Array.getElem_mem_data
  . rfl
Proof state after:
case neg
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.readyForRupAdd
units : List (Literal (PosFin n))
h :
  let assignments := (f.insertRupUnits units).fst.assignments;
  let_fun hsize := ⋯;
  let rupUnits := (f.insertRupUnits units).fst.rupUnits;
  insertUnit_invariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.insertRupUnits units).fst.assignments.size = n
i : PosFin n
b : Bool
p : PosFin n → Bool
hp :
  ∀ (x : DefaultClause n),
    (some x ∈ (f.insertRupUnits units).fst.clauses.data ∨
        (∃ a,
            (a, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (a, false) = x ∨
              (a, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (a, true) = x) ∨
          ∃ a,
            (a, false) ∈ (f.insertRupUnits units).fst.ratUnits.data ∧ unit (a, false) = x ∨
              (a, true) ∈ (f.insertRupUnits units).fst.ratUnits.data ∧ unit (a, true) = x) →
      ∃ a, (a, false) ∈ Clause.toList x ∧ decide (p a = false) = true ∨ (a, true) ∈ Clause.toList x ∧ p a = true
pf : p ⊨ f
j : Fin (f.insertRupUnits units).fst.rupUnits.size
b' : Bool
hb : hasAssignment b (addAssignment b' f.assignments[↑⟨i.val, ⋯⟩]) = true
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRupUnits units).fst.rupUnits[j] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b')
h2 : (f.insertRupUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = addAssignment b' f.assignments[↑⟨i.val, ⋯⟩]
h3 : ¬hasAssignment b' f.assignments[↑⟨i.val, ⋯⟩] = true
h4 :
  ∀ (k : Fin (f.insertRupUnits units).fst.rupUnits.size),
    k ≠ j → (f.insertRupUnits units).fst.rupUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
b_eq_b' : b = b'
j_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j]
j_unit_def : j_unit = unit (f.insertRupUnits units).fst.rupUnits[j]
hb' : ¬b' = true
⊢ (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = unit (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b') ∨
    (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = unit (⟨↑⟨i.val, ⋯⟩, ⋯⟩, b')