ash/lib/sat_solver/sat_solver.ex

defmodule Ash.SatSolver do
  @moduledoc """
  Tools for working with the satsolver that drives filter subset checking (for authorization)
  """

  alias Ash.Filter
  alias Ash.Query.{BooleanExpression, Not, Ref}

  @dialyzer {:nowarn_function, overlap?: 2}

  @doc """
  Creates tuples of a boolean statement.

  i.e `b(1 and 2) #=> {:and, 1, 2}`
  """
  defmacro b(statement) do
    value =
      Macro.prewalk(
        statement,
        fn
          {:and, _, [left, right]} ->
            quote do
              {:and, unquote(left), unquote(right)}
            end

          {:or, _, [left, right]} ->
            quote do
              {:or, unquote(left), unquote(right)}
            end

          {:not, _, [value]} ->
            quote do
              {:not, unquote(value)}
            end

          other ->
            other
        end
      )

    quote do
      unquote(value)
      |> Ash.SatSolver.balance()
    end
  end

  @doc false
  def balance({op, left, right}) do
    left = balance(left)
    right = balance(right)
    [left, right] = Enum.sort([left, right])

    {op, left, right}
  end

  def balance({:not, {:not, right}}) do
    balance(right)
  end

  def balance({:not, statement}) do
    {:not, balance(statement)}
  end

  def balance(other), do: other

  @doc "Returns true if the candidate filter returns the same or less data than the filter"
  def strict_filter_subset(filter, candidate) do
    case {filter, candidate} do
      {%{expression: nil}, %{expression: nil}} ->
        true

      {%{expression: nil}, _candidate_expr} ->
        true

      {_filter_expr, %{expression: nil}} ->
        false

      {filter, candidate} ->
        do_strict_filter_subset(filter, candidate)
    end
  end

  defp do_strict_filter_subset(filter, candidate) do
    filter =
      Filter.map(filter, fn
        %Ref{} = ref ->
          %{ref | input?: false}

        other ->
          other
      end)

    candidate =
      Filter.map(candidate, fn
        %Ref{} = ref ->
          %{ref | input?: false}

        other ->
          other
      end)

    expr = BooleanExpression.new(:and, filter.expression, candidate.expression)

    case transform_and_solve(
           filter.resource,
           expr
         ) do
      {:error, :unsatisfiable} ->
        false

      {:ok, _scenario} ->
        expr = BooleanExpression.new(:and, Not.new(filter.expression), candidate.expression)

        case transform_and_solve(
               filter.resource,
               expr
             ) do
          {:error, :unsatisfiable} ->
            true

          {:ok, _scenario} ->
            :maybe
        end
    end
  end

  defp filter_to_expr(nil), do: nil
  defp filter_to_expr(false), do: false
  defp filter_to_expr(true), do: true
  defp filter_to_expr(%Filter{expression: expression}), do: filter_to_expr(expression)
  defp filter_to_expr(%{__predicate__?: _} = op_or_func), do: op_or_func
  defp filter_to_expr(%Ash.Query.Exists{} = exists), do: exists
  defp filter_to_expr(%Ash.Query.Parent{} = parent), do: parent
  defp filter_to_expr(%Ash.CustomExpression{expression: expression}), do: expression
  defp filter_to_expr(%Not{expression: expression}), do: b(not filter_to_expr(expression))

  defp filter_to_expr(%BooleanExpression{op: op, left: left, right: right}) do
    {op, filter_to_expr(left), filter_to_expr(right)}
  end

  defp filter_to_expr(expr) do
    raise ArgumentError, message: "Invalid filter expression #{inspect(expr)}"
  end

  @doc "Prepares a filter for comparison"
  def transform(resource, expression) do
    expression
    |> consolidate_relationships(resource)
    |> upgrade_related_filters_to_join_keys(resource)
    |> build_expr_with_predicate_information()
  end

  @doc "Calls `transform/2` and solves the expression"
  def transform_and_solve(resource, expression) do
    resource
    |> transform(expression)
    |> to_cnf()
    |> elem(0)
    |> solve_expression()
  end

  defp upgrade_related_filters_to_join_keys(
         %BooleanExpression{op: op, left: left, right: right},
         resource
       ) do
    BooleanExpression.new(
      op,
      upgrade_related_filters_to_join_keys(left, resource),
      upgrade_related_filters_to_join_keys(right, resource)
    )
  end

  defp upgrade_related_filters_to_join_keys(%Not{expression: expression}, resource) do
    Not.new(upgrade_related_filters_to_join_keys(expression, resource))
  end

  defp upgrade_related_filters_to_join_keys(
         %Ash.Query.Exists{path: path, expr: expr} = exists,
         resource
       ) do
    related = Ash.Resource.Info.related(resource, path)

    %{exists | expr: upgrade_related_filters_to_join_keys(expr, related)}
  end

  defp upgrade_related_filters_to_join_keys(
         %{__operator__?: true, left: left, right: right} = op,
         resource
       ) do
    %{op | left: upgrade_ref(left, resource), right: upgrade_ref(right, resource)}
  end

  defp upgrade_related_filters_to_join_keys(
         %{__function__?: true, arguments: arguments} = function,
         resource
       ) do
    %{function | arguments: Enum.map(arguments, &upgrade_ref(&1, resource))}
  end

  defp upgrade_related_filters_to_join_keys(expr, _), do: expr

  defp upgrade_ref({key, ref}, resource) when is_atom(key) do
    {key, upgrade_ref(ref, resource)}
  end

  defp upgrade_ref(
         %Ash.Query.Ref{attribute: attribute, relationship_path: path} = ref,
         resource
       )
       when path != [] do
    with relationship when not is_nil(relationship) <-
           Ash.Resource.Info.relationship(resource, path),
         true <- attribute.name == relationship.destination_attribute,
         new_attribute when not is_nil(new_attribute) <-
           Ash.Resource.Info.attribute(relationship.source, relationship.source_attribute) do
      %{
        ref
        | relationship_path: :lists.droplast(path),
          attribute: new_attribute,
          resource: resource
      }
    else
      _ ->
        ref
    end
  end

  defp upgrade_ref(other, _), do: other

  defp consolidate_relationships(expression, resource) do
    {replacements, _all_relationship_paths} =
      expression
      |> Filter.relationship_paths(true)
      |> Enum.uniq()
      |> Enum.reduce({%{}, []}, fn path, {replacements, kept_paths} ->
        case find_synonymous_relationship_path(resource, kept_paths, path) do
          nil ->
            {replacements, [path | kept_paths]}

          synonymous_path ->
            Map.put(replacements, path, synonymous_path)
        end
      end)

    do_consolidate_relationships(expression, replacements, resource)
  end

  defp do_consolidate_relationships(
         %BooleanExpression{op: op, left: left, right: right},
         replacements,
         resource
       ) do
    BooleanExpression.new(
      op,
      do_consolidate_relationships(left, replacements, resource),
      do_consolidate_relationships(right, replacements, resource)
    )
  end

  defp do_consolidate_relationships(%Not{expression: expression}, replacements, resource) do
    Not.new(do_consolidate_relationships(expression, replacements, resource))
  end

  defp do_consolidate_relationships(
         %Ash.Query.Exists{at_path: at_path, path: path, expr: expr} = exists,
         replacements,
         resource
       ) do
    exists =
      case Map.fetch(replacements, at_path) do
        {:ok, replacement} when not is_nil(replacement) ->
          %{exists | at_path: replacement}

        :error ->
          exists
      end

    related = Ash.Resource.Info.related(resource, at_path)

    {replacements, _all_relationship_paths} =
      expr
      |> Filter.relationship_paths(true)
      |> Enum.uniq()
      |> Enum.reduce({%{}, []}, fn path, {replacements, kept_paths} ->
        case find_synonymous_relationship_path(related, kept_paths, path) do
          nil ->
            {replacements, [path | kept_paths]}

          synonymous_path ->
            Map.put(replacements, path, synonymous_path)
        end
      end)

    exists =
      case Map.fetch(replacements, path) do
        {:ok, replacement} when not is_nil(replacement) ->
          %{exists | path: replacement}

        :error ->
          exists
      end

    full_related = Ash.Resource.Info.related(related, path)

    %{exists | expr: consolidate_relationships(expr, full_related)}
  end

  defp do_consolidate_relationships(
         %Ash.Query.Ref{relationship_path: path} = ref,
         replacements,
         _resource
       )
       when path != [] do
    case Map.fetch(replacements, path) do
      {:ok, replacement} when not is_nil(replacement) -> %{ref | relationship_path: replacement}
      :error -> ref
    end
  end

  defp do_consolidate_relationships(
         %{__function__?: true, arguments: args} = func,
         replacements,
         resource
       ) do
    %{func | arguments: Enum.map(args, &do_consolidate_relationships(&1, replacements, resource))}
  end

  defp do_consolidate_relationships(
         %{__operator__?: true, left: left, right: right} = op,
         replacements,
         resource
       ) do
    %{
      op
      | left: do_consolidate_relationships(left, replacements, resource),
        right: do_consolidate_relationships(right, replacements, resource)
    }
  end

  defp do_consolidate_relationships(other, _, _), do: other

  defp find_synonymous_relationship_path(resource, paths, path) do
    Enum.find_value(paths, fn candidate_path ->
      if synonymous_relationship_paths?(resource, candidate_path, path) do
        candidate_path
      else
        false
      end
    end)
  end

  @doc "Returns `true` if the relationship paths are synonymous from a data perspective"
  def synonymous_relationship_paths?(
        left_resource,
        candidate,
        search,
        right_resource \\ nil
      )

  def synonymous_relationship_paths?(_, [], [], _), do: true
  def synonymous_relationship_paths?(_, [], _, _), do: false
  def synonymous_relationship_paths?(_, _, [], _), do: false

  def synonymous_relationship_paths?(
        left_resource,
        [candidate_first | candidate_rest],
        [first | rest],
        right_resource
      ) do
    right_resource = right_resource || left_resource
    relationship = Ash.Resource.Info.relationship(left_resource, first)
    candidate_relationship = Ash.Resource.Info.relationship(right_resource, candidate_first)

    cond do
      !relationship || !candidate_relationship ->
        false

      relationship.type == :many_to_many && candidate_relationship.type == :has_many ->
        synonymous_relationship_paths?(left_resource, [relationship.join_relationship], [
          candidate_first
        ]) && !Enum.empty?(candidate_rest) &&
          synonymous_relationship_paths?(
            left_resource,
            candidate_rest,
            rest,
            right_resource
          )

      relationship.type == :has_many && candidate_relationship.type == :many_to_many ->
        synonymous_relationship_paths?(left_resource, [relationship.name], [
          candidate_relationship.join_relationship
        ]) && !Enum.empty?(rest) &&
          synonymous_relationship_paths?(
            left_resource,
            candidate_rest,
            rest,
            right_resource
          )

      true ->
        comparison_keys = [
          :source_attribute,
          :destination_attribute,
          :source_attribute_on_join_resource,
          :destination_attribute_on_join_resource,
          :destination_attribute,
          :destination,
          :manual,
          :sort,
          :filter
        ]

        Map.take(relationship, comparison_keys) ==
          Map.take(candidate_relationship, comparison_keys) and
          synonymous_relationship_paths?(relationship.destination, candidate_rest, rest)
    end
  end

  defp build_expr_with_predicate_information(expression) do
    expression = fully_simplify(expression)

    all_predicates =
      expression
      |> Filter.list_predicates()
      |> Enum.uniq()

    comparison_expressions =
      all_predicates
      |> Enum.filter(fn %module{} ->
        :erlang.function_exported(module, :compare, 2)
      end)
      |> Enum.reduce([], fn predicate, new_expressions ->
        all_predicates
        |> Enum.reject(&Kernel.==(&1, predicate))
        |> Enum.filter(&shares_ref?(&1, predicate))
        |> Enum.reduce(new_expressions, fn other_predicate, new_expressions ->
          # With predicate as a and other_predicate as b
          case Ash.Filter.Predicate.compare(predicate, other_predicate) do
            :right_includes_left ->
              # b || !a

              [b(other_predicate or not predicate) | new_expressions]

            :left_includes_right ->
              # a || ! b
              [b(predicate or not other_predicate) | new_expressions]

            :mutually_inclusive ->
              # (a && b) || (! a && ! b)
              [
                b((predicate and other_predicate) or (not predicate and not other_predicate))
                | new_expressions
              ]

            :mutually_exclusive ->
              [b(not (other_predicate and predicate)) | new_expressions]

            :mutually_exclusive_and_collectively_exhaustive ->
              [
                b(
                  not (other_predicate and predicate) and
                    not (not other_predicate and not predicate)
                )
                | new_expressions
              ]

            _other ->
              # If we can't tell, we assume that both could be true
              new_expressions
          end
        end)
      end)
      |> Enum.uniq()

    expression = filter_to_expr(expression)

    expression_with_comparisons =
      Enum.reduce(comparison_expressions, expression, fn comparison_expression, expression ->
        b(comparison_expression and expression)
      end)

    all_predicates
    |> Enum.map(& &1.__struct__)
    |> Enum.uniq()
    |> Enum.flat_map(fn struct ->
      if :erlang.function_exported(struct, :bulk_compare, 1) do
        struct.bulk_compare(all_predicates)
      else
        []
      end
    end)
    |> Enum.reduce(expression_with_comparisons, fn comparison_expression, expression ->
      b(comparison_expression and expression)
    end)
  end

  defp fully_simplify(expression) do
    expression
    |> do_fully_simplify()
    |> lift_equals_out_of_in()
    |> do_fully_simplify()
  end

  defp do_fully_simplify(expression) do
    expression
    |> simplify()
    |> case do
      ^expression ->
        expression

      simplified ->
        fully_simplify(simplified)
    end
  end

  def lift_equals_out_of_in(expression) do
    case find_non_equal_overlap(expression) do
      nil ->
        expression

      non_equal_overlap ->
        expression
        |> split_in_expressions(non_equal_overlap)
        |> lift_equals_out_of_in()
    end
  end

  def find_non_equal_overlap(expression) do
    Ash.Filter.find(expression, fn sub_expr ->
      Ash.Filter.find(expression, fn sub_expr2 ->
        # if has_call_or_expression?(sub_expr) || has_call_or_expression?(sub_expr2) do
        #   false
        # else
        overlap?(sub_expr, sub_expr2)
        # end
      end)
    end)
  end

  defp new_in(base, right) do
    case MapSet.size(right) do
      1 ->
        %Ash.Query.Operator.Eq{left: base.left, right: Enum.at(right, 0)}

      _ ->
        %Ash.Query.Operator.In{left: base.left, right: right}
    end
  end

  def split_in_expressions(
        %Ash.Query.Operator.In{right: right} = sub_expr,
        %Ash.Query.Operator.Eq{right: value} = non_equal_overlap
      ) do
    if overlap?(non_equal_overlap, sub_expr) do
      Ash.Query.BooleanExpression.new(
        :or,
        new_in(sub_expr, MapSet.delete(right, value)),
        non_equal_overlap
      )
    else
      sub_expr
    end
  end

  def split_in_expressions(
        %Ash.Query.Operator.In{} = sub_expr,
        %Ash.Query.Operator.In{right: right} = non_equal_overlap
      ) do
    if overlap?(sub_expr, non_equal_overlap) do
      diff = MapSet.difference(sub_expr.right, right)

      if MapSet.size(diff) == 0 do
        Enum.reduce(sub_expr.right, nil, fn var, acc ->
          BooleanExpression.new(:or, %Ash.Query.Operator.Eq{left: sub_expr.left, right: var}, acc)
        end)
      else
        new_right = new_in(sub_expr, MapSet.intersection(sub_expr.right, right))

        Ash.Query.BooleanExpression.new(
          :or,
          new_in(sub_expr, diff),
          new_right
        )
      end
    else
      sub_expr
    end
  end

  def split_in_expressions(nil, _), do: nil

  def split_in_expressions(%Ash.Filter{expression: expression} = filter, non_equal_overlap),
    do: %{filter | expression: split_in_expressions(expression, non_equal_overlap)}

  def split_in_expressions(%Not{expression: expression} = not_expr, non_equal_overlap),
    do: %{not_expr | expression: split_in_expressions(expression, non_equal_overlap)}

  def split_in_expressions(
        %BooleanExpression{left: left, right: right} = expr,
        non_equal_overlap
      ),
      do: %{
        expr
        | left: split_in_expressions(left, non_equal_overlap),
          right: split_in_expressions(right, non_equal_overlap)
      }

  def split_in_expressions(other, _), do: other

  defp overlap?(
         %Ash.Query.Operator.In{left: left, right: %MapSet{} = left_right},
         %Ash.Query.Operator.In{left: left, right: %MapSet{} = right_right}
       ) do
    if MapSet.equal?(left_right, right_right) do
      false
    else
      overlap? =
        left_right
        |> MapSet.intersection(right_right)
        |> MapSet.size()
        |> Kernel.>(0)

      if overlap? do
        true
      else
        false
      end
    end
  end

  defp overlap?(_, %Ash.Query.Operator.Eq{right: %Ref{}}),
    do: false

  defp overlap?(%Ash.Query.Operator.Eq{right: %Ref{}}, _),
    do: false

  defp overlap?(
         %Ash.Query.Operator.Eq{left: left, right: left_right},
         %Ash.Query.Operator.In{left: left, right: %MapSet{} = right_right}
       ) do
    MapSet.member?(right_right, left_right)
  end

  defp overlap?(_left, _right) do
    false
  end

  @doc "Returns a statement expressing that the predicates are mutually exclusive."
  def mutually_exclusive(predicates, acc \\ [])
  def mutually_exclusive([], acc), do: acc

  def mutually_exclusive([predicate | rest], acc) do
    new_acc =
      Enum.reduce(rest, acc, fn other_predicate, acc ->
        [b(not (predicate and other_predicate)) | acc]
      end)

    mutually_exclusive(rest, new_acc)
  end

  @doc "Returns a statement expressing that the predicates are mutually exclusive and collectively exhaustive."
  def mutually_exclusive_and_collectively_exhaustive([]), do: []

  def mutually_exclusive_and_collectively_exhaustive([_]), do: []

  def mutually_exclusive_and_collectively_exhaustive(predicates) do
    mutually_exclusive(predicates) ++
      Enum.flat_map(predicates, fn predicate ->
        other_predicates = Enum.reject(predicates, &(&1 == predicate))

        other_predicates_union =
          Enum.reduce(other_predicates, nil, fn other_predicate, expr ->
            if expr do
              b(expr or other_predicate)
            else
              other_predicate
            end
          end)

        b(
          not (predicate and other_predicates_union) and
            not (not predicate and not other_predicates_union)
        )
      end)
  end

  @doc "Returns `b(not (left and right))`"
  def left_excludes_right(left, right) do
    b(not (left and right))
  end

  @doc "Returns `b(not (right and left))`"
  def right_excludes_left(left, right) do
    b(not (right and left))
  end

  @doc "Returns a statement expressing that the predicates are mutually inclusive"
  def mutually_inclusive(predicates, acc \\ [])
  def mutually_inclusive([], acc), do: acc

  def mutually_inclusive([predicate | rest], acc) do
    new_acc =
      Enum.reduce(rest, acc, fn other_predicate, acc ->
        [b((predicate and other_predicate) or (not predicate and not other_predicate)) | acc]
      end)

    mutually_exclusive(rest, new_acc)
  end

  @doc "Returns `b(not (right and not left))`"
  def right_implies_left(left, right) do
    b(not (right and not left))
  end

  @doc "Returns `b(not (left and not right))`"
  def left_implies_right(left, right) do
    b(not (left and not right))
  end

  defp shares_ref?(left, right) do
    any_refs_in_common?(refs(left), refs(right))
  end

  defp any_refs_in_common?(left_refs, right_refs) do
    Enum.any?(left_refs, &(&1 in right_refs))
  end

  defp refs(%{__operator__?: true, left: left, right: right}) do
    Enum.filter([left, right], &match?(%Ref{}, &1)) |> Enum.map(&Map.put(&1, :input?, false))
  end

  defp refs(%{__function__?: true, arguments: arguments}) do
    Enum.filter(arguments, &match?(%Ref{}, &1)) |> Enum.map(&Map.put(&1, :input?, false))
  end

  defp refs(_), do: []

  defp simplify(%BooleanExpression{op: op, left: left, right: right}) do
    BooleanExpression.new(op, simplify(left), simplify(right))
  end

  defp simplify(%Not{expression: expression}) do
    Not.new(simplify(expression))
  end

  defp simplify(%Ash.Query.Exists{expr: expr} = exists) do
    %{exists | expr: simplify(expr)}
  end

  defp simplify(%mod{__predicate__?: true} = predicate) do
    if :erlang.function_exported(mod, :simplify, 1) do
      predicate
      |> mod.simplify()
      |> Kernel.||(predicate)
    else
      predicate
    end
  end

  defp simplify(other), do: other

  @doc """
  Transforms a statement to Conjunctive Normal Form(CNF), as lists of lists of integers.
  """
  def to_cnf(expression) do
    expression_with_constants = b(true and not false and expression)

    {bindings, expression} = extract_bindings(expression_with_constants)

    expression
    |> to_conjunctive_normal_form()
    |> lift_clauses()
    |> negations_to_negative_numbers()
    |> Enum.map(fn scenario ->
      Enum.sort_by(scenario, fn item ->
        {abs(item), item}
      end)
    end)
    |> group_predicates(bindings)
    |> rebind()
    |> unique_clauses()
  end

  defp unique_clauses({clauses, bindings}) do
    {Enum.uniq(clauses), bindings}
  end

  defp group_predicates(expression, bindings) do
    case expression do
      [_] ->
        {expression, bindings}

      scenarios ->
        Enum.reduce(scenarios, {[], bindings}, fn scenario, {new_scenarios, bindings} ->
          {scenario, bindings} = group_scenario_predicates(scenario, scenarios, bindings)
          {[scenario | new_scenarios], bindings}
        end)
    end
  end

  defp group_scenario_predicates(scenario, all_scenarios, bindings) do
    scenario
    |> Ash.SatSolver.Utils.ordered_sublists()
    |> Enum.filter(&can_be_used_as_group?(&1, all_scenarios, bindings))
    |> Enum.sort_by(&length/1)
    |> remove_overlapping()
    |> Enum.reduce({scenario, bindings}, fn group, {scenario, bindings} ->
      bindings = add_group_binding(bindings, group)

      {Ash.SatSolver.Utils.replace_ordered_sublist(scenario, group, bindings[:groups][group]),
       bindings}
    end)
  end

  defp remove_overlapping([]), do: []

  defp remove_overlapping([item | rest]) do
    if Enum.any?(item, fn n ->
         Enum.any?(rest, &(n in &1 or -n in &1))
       end) do
      remove_overlapping(rest)
    else
      [item | remove_overlapping(rest)]
    end
  end

  @doc "Remaps integers back to clauses"
  def unbind(expression, %{temp_bindings: temp_bindings, old_bindings: old_bindings}) do
    expression =
      Enum.flat_map(expression, fn statement ->
        Enum.flat_map(statement, fn var ->
          neg? = var < 0
          old_binding = temp_bindings[abs(var)]

          case old_bindings[:reverse_groups][old_binding] do
            nil ->
              if neg? do
                [-old_binding]
              else
                [old_binding]
              end

            group ->
              if neg? do
                Enum.map(group, &(-&1))
              else
                [{:expand, group}]
              end
          end
        end)
        |> expand_groups()
      end)

    {expression, old_bindings}
  end

  defp expand_groups(expression) do
    do_expand_groups(expression)
  end

  defp do_expand_groups([]), do: [[]]

  defp do_expand_groups([{:expand, group} | rest]) do
    Enum.flat_map(group, fn var ->
      Enum.map(do_expand_groups(rest), fn future ->
        [var | future]
      end)
    end)
  end

  defp do_expand_groups([var | rest]) do
    Enum.map(do_expand_groups(rest), fn future ->
      [var | future]
    end)
  end

  defp rebind({expression, bindings}) do
    {expression, temp_bindings} =
      Enum.reduce(expression, {[], %{current: 0}}, fn statement, {statements, acc} ->
        {statement, acc} =
          Enum.reduce(statement, {[], acc}, fn var, {statement, acc} ->
            case acc[:reverse][abs(var)] do
              nil ->
                binding = acc.current + 1

                value =
                  if var < 0 do
                    -binding
                  else
                    binding
                  end

                {[value | statement],
                 acc
                 |> Map.put(:current, binding)
                 |> Map.update(:reverse, %{abs(var) => binding}, &Map.put(&1, abs(var), binding))
                 |> Map.put(binding, abs(var))}

              value ->
                value =
                  if var < 0 do
                    -value
                  else
                    value
                  end

                {[value | statement], acc}
            end
          end)

        {[Enum.reverse(statement) | statements], acc}
      end)

    bindings_with_old_bindings = %{temp_bindings: temp_bindings, old_bindings: bindings}

    {expression, bindings_with_old_bindings}
  end

  defp can_be_used_as_group?(group, scenarios, bindings) do
    Map.has_key?(bindings[:groups] || %{}, group) ||
      Enum.all?(scenarios, fn scenario ->
        has_no_overlap?(scenario, group) || group_in_scenario?(scenario, group)
      end)
  end

  defp has_no_overlap?(scenario, group) do
    not Enum.any?(group, fn group_predicate ->
      Enum.any?(scenario, fn scenario_predicate ->
        abs(group_predicate) == abs(scenario_predicate)
      end)
    end)
  end

  defp group_in_scenario?(scenario, group) do
    Ash.SatSolver.Utils.is_ordered_sublist_of?(group, scenario)
  end

  defp add_group_binding(bindings, group) do
    if bindings[:groups][group] do
      bindings
    else
      binding = bindings[:current]

      bindings
      |> Map.put_new(:reverse_groups, %{})
      |> Map.update!(:reverse_groups, &Map.put(&1, binding, group))
      |> Map.put_new(:groups, %{})
      |> Map.update!(:groups, &Map.put(&1, group, binding))
      |> Map.put(:current, binding + 1)
    end
  end

  cond do
    Application.compile_env(:ash, :sat_testing) ->
      def solve_expression(cnf) do
        Module.concat([System.get_env("SAT_SOLVER") || "Picosat"]).solve(cnf)
      end

    Code.ensure_loaded?(Picosat) ->
      def solve_expression(cnf) do
        Picosat.solve(cnf)
      end

    Code.ensure_loaded?(SimpleSat) ->
      def solve_expression(cnf) do
        SimpleSat.solve(cnf)
      end

    true ->
      def solve_expression(_cnf) do
        if Code.ensure_loaded?(Picosat) || Code.ensure_loaded?(SimpleSat) do
          raise """
          No SAT solver available, although one was loaded. This typically means you need to run `mix deps.compile ash --force`

          If that doesn't work, please ensure that one of the following dependencies is present in your application to use sat solver features:

          * `:picosat_elixir` (recommended) - A NIF wrapper around the PicoSAT SAT solver. Fast, production ready, battle tested.
          * `:simple_sat` - A pure Elixir SAT solver. Slower than PicoSAT, but no NIF dependency.
          """
        else
          raise """
          No SAT solver available.

          Please add one of the following dependencies to your application to use sat solver features:

          * `:picosat_elixir` (recommended) - A NIF wrapper around the PicoSAT SAT solver. Fast, production ready, battle tested.
          * `:simple_sat` - A pure Elixir SAT solver. Slower than PicoSAT, but no NIF dependency.
          """
        end
      end
  end

  def contains?([], _), do: false

  def contains?([_ | t] = l1, l2) do
    List.starts_with?(l1, l2) or contains?(t, l2)
  end

  def solutions_to_predicate_values(solution, bindings) do
    Enum.reduce(solution, %{true: [], false: []}, fn var, state ->
      fact = Map.get(bindings, abs(var))

      if is_nil(fact) do
        raise Ash.Error.Framework.AssumptionFailed.exception(
                message: """
                A fact from the sat solver had no corresponding bound fact:

                Bindings:
                  #{inspect(bindings)}

                Missing:
                  #{inspect(var)}
                """
              )
      end

      Map.put(state, fact, var > 0)
    end)
  end

  defp extract_bindings(expr, bindings \\ %{current: 1})

  defp extract_bindings({operator, left, right}, bindings) do
    {bindings, left_extracted} = extract_bindings(left, bindings)
    {bindings, right_extracted} = extract_bindings(right, bindings)

    {bindings, {operator, left_extracted, right_extracted}}
  end

  defp extract_bindings({:not, value}, bindings) do
    {bindings, extracted} = extract_bindings(value, bindings)

    {bindings, b(not extracted)}
  end

  defp extract_bindings(value, %{current: current} = bindings) do
    current_binding =
      Enum.find(bindings, fn {key, binding_value} ->
        key != :current && binding_value == value
      end)

    case current_binding do
      nil ->
        new_bindings =
          bindings
          |> Map.put(:current, current + 1)
          |> Map.put(current, value)

        {new_bindings, current}

      {binding, _} ->
        {bindings, binding}
    end
  end

  # A helper function for formatting to the same output we'd give to picosat
  @doc false
  def to_picosat(clauses, variable_count) do
    clause_count = Enum.count(clauses)

    formatted_input =
      Enum.map_join(clauses, "\n", fn clause ->
        format_clause(clause) <> " 0"
      end)

    "p cnf #{variable_count} #{clause_count}\n" <> formatted_input
  end

  defp negations_to_negative_numbers(clauses) do
    Enum.map(
      clauses,
      fn
        {:not, var} when is_integer(var) ->
          [negate_var(var)]

        var when is_integer(var) ->
          [var]

        clause ->
          Enum.map(clause, fn
            {:not, var} -> negate_var(var)
            var -> var
          end)
      end
    )
  end

  defp negate_var(var, multiplier \\ -1)

  defp negate_var({:not, value}, multiplier) do
    negate_var(value, multiplier * -1)
  end

  defp negate_var(value, multiplier), do: value * multiplier

  defp format_clause(clause) do
    Enum.map_join(clause, " ", fn
      {:not, var} -> "-#{var}"
      var -> "#{var}"
    end)
  end

  defp lift_clauses({:and, left, right}) do
    lift_clauses(left) ++ lift_clauses(right)
  end

  defp lift_clauses({:or, left, right}) do
    [lift_or_clauses(left) ++ lift_or_clauses(right)]
  end

  defp lift_clauses(value), do: [[value]]

  defp lift_or_clauses({:or, left, right}) do
    lift_or_clauses(left) ++ lift_or_clauses(right)
  end

  defp lift_or_clauses(value), do: [value]

  defp to_conjunctive_normal_form(expression) do
    expression
    |> demorgans_law()
    |> distributive_law()
  end

  defp distributive_law(expression) do
    distributive_law_applied = apply_distributive_law(expression)

    if expression == distributive_law_applied do
      expression
    else
      distributive_law(distributive_law_applied)
    end
  end

  defp apply_distributive_law({:or, left, {:and, right1, right2}}) do
    left_distributed = apply_distributive_law(left)

    {:and, {:or, left_distributed, apply_distributive_law(right1)},
     {:or, left_distributed, apply_distributive_law(right2)}}
  end

  defp apply_distributive_law({:or, {:and, left1, left2}, right}) do
    right_distributed = apply_distributive_law(right)

    {:and, {:or, apply_distributive_law(left1), right_distributed},
     {:or, apply_distributive_law(left2), right_distributed}}
  end

  defp apply_distributive_law({:not, expression}) do
    {:not, apply_distributive_law(expression)}
  end

  defp apply_distributive_law({operator, left, right}) when operator in [:and, :or] do
    {operator, apply_distributive_law(left), apply_distributive_law(right)}
  end

  defp apply_distributive_law(var) when is_integer(var) do
    var
  end

  defp demorgans_law(expression) do
    demorgans_law_applied = apply_demorgans_law(expression)

    if expression == demorgans_law_applied do
      expression
    else
      demorgans_law(demorgans_law_applied)
    end
  end

  defp apply_demorgans_law({:not, {:and, left, right}}) do
    {:or, {:not, apply_demorgans_law(left)}, {:not, apply_demorgans_law(right)}}
  end

  defp apply_demorgans_law({:not, {:or, left, right}}) do
    {:and, {:not, left}, {:not, right}}
  end

  defp apply_demorgans_law({operator, left, right}) when operator in [:or, :and] do
    {operator, apply_demorgans_law(left), apply_demorgans_law(right)}
  end

  defp apply_demorgans_law({:not, expression}) do
    {:not, apply_demorgans_law(expression)}
  end

  defp apply_demorgans_law(var) when is_integer(var) do
    var
  end
end