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, ⋯⟩
h✝ : b = b'
⊢ p ⊨ (i, 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, ⋯⟩
h✝ : ¬b = b'
⊢ p ⊨ (i, b)
Tactic:
. next b_eq_b' =>
  let j_unit := unit (insertRupUnits f units).1.rupUnits[j]
  have j_unit_def : j_unit = unit (insertRupUnits f units).1.rupUnits[j] := rfl
  have j_unit_in_insertRupUnits_res :
    ∃ i : PosFin n,
      (i, false) ∈ (insertRupUnits f units).1.rupUnits.data ∧ unit (i, false) = j_unit ∨
      (i, true) ∈ (insertRupUnits f units).1.rupUnits.data ∧ unit (i, true) = j_unit := by
    apply Exists.intro i
    rw [j_unit_def, h1]
    by_cases hb' : b'
    . 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
    . simp only [Bool.not_eq_true] at hb'
      rw [hb']
      apply Or.inl
      constructor
      . have h1 : (insertRupUnits f units).fst.rupUnits[j] = (i, false) := 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
  specialize hp j_unit ((Or.inr ∘ Or.inl) j_unit_in_insertRupUnits_res)
  simp only [List.any_eq_true, Prod.exists, Bool.exists_bool, Bool.decide_coe, Fin.getElem_fin, List.find?, j_unit] at hp
  simp only [Fin.getElem_fin] at h1
  rcases hp with ⟨i', hp⟩
  simp only [h1, Clause.toList, unit_eq, List.mem_singleton,
    Prod.mk.injEq] at hp
  rcases hp with ⟨hp1, hp2⟩ | ⟨hp1, hp2⟩
  . simp only [b_eq_b', ← hp1.2, HSat.eval]
    rw [hp1.1] at hp2
    exact of_decide_eq_true hp2
  . simp only [b_eq_b', ← hp1.2, HSat.eval]
    rw [hp1.1] at hp2
    exact hp2
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, ⋯⟩
h✝ : ¬b = b'
⊢ p ⊨ (i, b)