Theorem: LRAT.DefaultFormula.deleteOne_preserves_strong_assignments_invariant
Proof state before:
case neg.h_3.inl
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
  ∀ (x : PosFin n),
    ¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
      ¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
  ∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
    (nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
deleteOne_f_rw :
  f.deleteOne id =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
hb : hasAssignment b f.assignments[i.val] = true
hf : some (unit (i, b)) ∈ f.clauses.data
⊢ some (unit (i, b)) ∈ (f.clauses.set! id none).data ∨
    (∃ a,
        (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = unit (i, b)) ∨
      ∃ a,
        (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = unit (i, b)

case neg.h_3.inr
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
  ∀ (x : PosFin n),
    ¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
      ¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
  ∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
    (nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
deleteOne_f_rw :
  f.deleteOne id =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
hb : hasAssignment b f.assignments[i.val] = true
hf :
  (∃ a,
      (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = unit (i, b) ∨
        (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = unit (i, b)) ∨
    ∃ a,
      (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = unit (i, b) ∨
        (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = unit (i, b)
⊢ some (unit (i, b)) ∈ (f.clauses.set! id none).data ∨
    (∃ a,
        (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = unit (i, b)) ∨
      ∃ a,
        (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = unit (i, b)
Tactic:
. apply Or.inl
  simp only [Array.set!, Array.setD]
  split
  . rcases List.get_of_mem hf with ⟨idx, hidx⟩
    simp only [← hidx, Array.data_set]
    rw [List.mem_iff_get]
    have idx_in_bounds : idx.1 < List.length (List.set f.clauses.data id none) := by
      simp only [List.length_set]
      exact idx.2
    apply Exists.intro ⟨idx.1, idx_in_bounds⟩
    by_cases id = idx
    . next id_eq_idx =>
      exfalso
      have idx_in_bounds2 : idx.1 < f.clauses.size := by
        have f_clauses_rw : f.clauses = { data := f.clauses.data } := rfl
        conv => rhs; rw [f_clauses_rw, Array.size_mk]
        exact idx.2
      simp only [getElem!, id_eq_idx, Array.data_length, idx_in_bounds2, ↓reduceDite,
        Fin.eta, Array.get_eq_getElem, Array.getElem_eq_data_get] at heq
      rw [hidx] at heq
      simp only [Option.some.injEq] at heq
      rw [← heq] at hl
      specialize hl i
      simp only [unit, DefaultClause.mk.injEq, List.cons.injEq, Prod.mk.injEq, true_and, and_true,
        Bool.not_eq_false, Bool.not_eq_true] at hl
      by_cases b_val : b
      . simp only [b_val, and_false] at hl
      . simp only [b_val, false_and] at hl
    . next id_ne_idx =>
      simp only [ne_eq, id_ne_idx, not_false_eq_true, List.get_set_ne]
  . exact hf
Proof state after:
case neg.h_3.inr
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
hl :
  ∀ (x : PosFin n),
    ¬c = { clause := [(x, false)], nodupkey := ⋯, nodup := ⋯ } ∧
      ¬c = { clause := [(x, true)], nodupkey := ⋯, nodup := ⋯ }
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
  ∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
    (nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup := nodup } → False
heq✝ : some c = some val✝
deleteOne_f_rw :
  f.deleteOne id =
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits, assignments := f.assignments }
hb : hasAssignment b f.assignments[i.val] = true
hf :
  (∃ a,
      (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = unit (i, b) ∨
        (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = unit (i, b)) ∨
    ∃ a,
      (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = unit (i, b) ∨
        (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = unit (i, b)
⊢ some (unit (i, b)) ∈ (f.clauses.set! id none).data ∨
    (∃ a,
        (a, false) ∈ f.rupUnits.data ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.rupUnits.data ∧ unit (a, true) = unit (i, b)) ∨
      ∃ a,
        (a, false) ∈ f.ratUnits.data ∧ unit (a, false) = unit (i, b) ∨
          (a, true) ∈ f.ratUnits.data ∧ unit (a, true) = unit (i, b)