Theorem: LRAT.DefaultFormula.insertRupUnits_preserves_assignments_invariant
Proof state before:
case inr.inr.intro.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, ⋯⟩
j1_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j1]
j1_unit_def : j1_unit = unit (f.insertRupUnits units).fst.rupUnits[j1]
j1_unit_in_insertRupUnits_res :
  ∃ i,
    (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = j1_unit ∨
      (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = j1_unit
j2_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j2]
j2_unit_def : j2_unit = unit (f.insertRupUnits units).fst.rupUnits[j2]
j2_unit_in_insertRupUnits_res :
  ∃ i,
    (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = j2_unit ∨
      (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = j2_unit
hp2 :
  ∃ a,
    (a, false) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j2]) ∧ decide (p a = false) = true ∨
      (a, true) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j2]) ∧ p a = true
i1 : PosFin n
hp1 :
  (i1, false) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j1]) ∧ decide (p i1 = false) = true ∨
    (i1, true) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j1]) ∧ p i1 = true
⊢ p ⊨ (i, b)
Tactic:
rcases hp2 with ⟨i2, hp2⟩
Proof state after:
case inr.inr.intro.intro.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, ⋯⟩
j1_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j1]
j1_unit_def : j1_unit = unit (f.insertRupUnits units).fst.rupUnits[j1]
j1_unit_in_insertRupUnits_res :
  ∃ i,
    (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = j1_unit ∨
      (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = j1_unit
j2_unit : DefaultClause n := unit (f.insertRupUnits units).fst.rupUnits[j2]
j2_unit_def : j2_unit = unit (f.insertRupUnits units).fst.rupUnits[j2]
j2_unit_in_insertRupUnits_res :
  ∃ i,
    (i, false) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, false) = j2_unit ∨
      (i, true) ∈ (f.insertRupUnits units).fst.rupUnits.data ∧ unit (i, true) = j2_unit
i1 : PosFin n
hp1 :
  (i1, false) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j1]) ∧ decide (p i1 = false) = true ∨
    (i1, true) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j1]) ∧ p i1 = true
i2 : PosFin n
hp2 :
  (i2, false) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j2]) ∧ decide (p i2 = false) = true ∨
    (i2, true) ∈ Clause.toList (unit (f.insertRupUnits units).fst.rupUnits[↑j2]) ∧ p i2 = true
⊢ p ⊨ (i, b)