Theorem: LRAT.DefaultFormula.deleteOne_preserves_strong_assignments_invariant
Proof state before:
case pos
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
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
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
hb : hasAssignment b (f.assignments.modify l.fst.val (removeAssignment l.snd))[i.val] = true
h✝ : l.fst.val = i.val
⊢ unit (i, b) ∈
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList

case neg
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
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
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
hb : hasAssignment b (f.assignments.modify l.fst.val (removeAssignment l.snd))[i.val] = true
h✝ : ¬l.fst.val = i.val
⊢ unit (i, b) ∈
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList
Tactic:
. next l_eq_i =>
  simp only [l_eq_i, Array.get_modify_at_idx i_in_bounds] at hb
  have l_ne_b : l.2 ≠ b := by
    intro l_eq_b
    rw [← l_eq_b] at hb
    have hb' := not_has_of_remove f.assignments[i.1] l.2
    simp [hb] at hb'
  rw [← Bool.eq_not_iff] at l_ne_b
  rw [l_ne_b] at hb
  have hb := has_of_remove_irrelevant f.assignments[i.1] b hb
  specialize hf i b hb
  simp only [toList, Array.toList_eq, List.append_assoc, List.mem_append, List.mem_filterMap, id_eq,
    exists_eq_right, List.mem_map, Prod.exists, Bool.exists_bool] at hf
  simp only [toList, Array.toList_eq, List.append_assoc, List.mem_append, List.mem_filterMap, id_eq,
    exists_eq_right, List.mem_map, Prod.exists, Bool.exists_bool]
  rcases hf with hf | hf
  . 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, hl] at heq
        simp only [unit, Option.some.injEq, DefaultClause.mk.injEq, List.cons.injEq, and_true] at heq
        simp only [← heq, not] at l_ne_b
        split at l_ne_b
        . simp only at l_ne_b
        . simp only at l_ne_b
      . next id_ne_idx =>
        simp only [ne_eq, id_ne_idx, not_false_eq_true, List.get_set_ne]
    . exact hf
  . exact Or.inr hf
Proof state after:
case neg
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hf : ∀ (i : PosFin n) (b : Bool), hasAssignment b f.assignments[i.val] = true → unit (i, b) ∈ f.toList
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
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
hb : hasAssignment b (f.assignments.modify l.fst.val (removeAssignment l.snd))[i.val] = true
h✝ : ¬l.fst.val = i.val
⊢ unit (i, b) ∈
    { clauses := f.clauses.set! id none, rupUnits := f.rupUnits, ratUnits := f.ratUnits,
        assignments := f.assignments.modify l.fst.val (removeAssignment l.snd) }.toList