Theorem: LRAT.DefaultFormula.insertRupUnits_preserves_assignments_invariant
Proof state before:
case inl.intro
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
hb : hasAssignment b (f.insertRupUnits units).fst.assignments[i.val] = true
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
h1 : (f.insertRupUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = f.assignments[↑⟨i.val, ⋯⟩]
h2 :
  ∀ (j : Fin (f.insertRupUnits units).fst.rupUnits.size), (f.insertRupUnits units).fst.rupUnits[j].fst.val ≠ ↑⟨i.val, ⋯⟩
⊢ p ⊨ (i, b)

case inr.inl.intro.intro.intro.intro.intro.intro
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
hb : hasAssignment b (f.insertRupUnits units).fst.assignments[i.val] = true
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
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, ⋯⟩
⊢ p ⊨ (i, b)

case inr.inr.intro.intro.intro.intro.intro.intro.intro
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
hb : hasAssignment b (f.insertRupUnits units).fst.assignments[i.val] = true
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
j1 j2 : Fin (f.insertRupUnits units).fst.rupUnits.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRupUnits units).fst.rupUnits[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (f.insertRupUnits units).fst.rupUnits[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (f.insertRupUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
  ∀ (k : Fin (f.insertRupUnits units).fst.rupUnits.size),
    k ≠ j1 → k ≠ j2 → (f.insertRupUnits units).fst.rupUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
⊢ p ⊨ (i, b)
Tactic:
. rw [h1] at hb
  exact (assignments_invariant_of_strong_assignments_invariant f f_readyForRupAdd.2).2 i b hb p pf
Proof state after:
case inr.inl.intro.intro.intro.intro.intro.intro
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
hb : hasAssignment b (f.insertRupUnits units).fst.assignments[i.val] = true
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
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, ⋯⟩
⊢ p ⊨ (i, b)

case inr.inr.intro.intro.intro.intro.intro.intro.intro
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
hb : hasAssignment b (f.insertRupUnits units).fst.assignments[i.val] = true
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
j1 j2 : Fin (f.insertRupUnits units).fst.rupUnits.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRupUnits units).fst.rupUnits[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (f.insertRupUnits units).fst.rupUnits[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (f.insertRupUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
  ∀ (k : Fin (f.insertRupUnits units).fst.rupUnits.size),
    k ≠ j1 → k ≠ j2 → (f.insertRupUnits units).fst.rupUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
⊢ p ⊨ (i, b)