cp_model.py

# Copyright 2010-2024 Google LLC
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

"""Methods for building and solving CP-SAT models.

The following two sections describe the main
methods for building and solving CP-SAT models.

* [`CpModel`](#cp_model.CpModel): Methods for creating
models, including variables and constraints.
* [`CPSolver`](#cp_model.CpSolver): Methods for solving
a model and evaluating solutions.

The following methods implement callbacks that the
solver calls each time it finds a new solution.

* [`CpSolverSolutionCallback`](#cp_model.CpSolverSolutionCallback):
  A general method for implementing callbacks.
* [`ObjectiveSolutionPrinter`](#cp_model.ObjectiveSolutionPrinter):
  Print objective values and elapsed time for intermediate solutions.
* [`VarArraySolutionPrinter`](#cp_model.VarArraySolutionPrinter):
  Print intermediate solutions (variable values, time).
* [`VarArrayAndObjectiveSolutionPrinter`]
      (#cp_model.VarArrayAndObjectiveSolutionPrinter):
  Print both intermediate solutions and objective values.

Additional methods for solving CP-SAT models:

* [`Constraint`](#cp_model.Constraint): A few utility methods for modifying
  constraints created by `CpModel`.
* [`LinearExpr`](#lineacp_model.LinearExpr): Methods for creating constraints
  and the objective from large arrays of coefficients.

Other methods and functions listed are primarily used for developing OR-Tools,
rather than for solving specific optimization problems.
"""

import collections
import itertools
import threading
import time
from typing import (
    Any,
    Callable,
    Dict,
    Iterable,
    List,
    NoReturn,
    Optional,
    Sequence,
    Tuple,
    Union,
    cast,
    overload,
)
import warnings

import numpy as np
import pandas as pd

from ortools.sat import cp_model_pb2
from ortools.sat import sat_parameters_pb2
from ortools.sat.python import cp_model_helper as cmh
from ortools.sat.python import swig_helper
from ortools.util.python import sorted_interval_list

Domain = sorted_interval_list.Domain

# The classes below allow linear expressions to be expressed naturally with the
# usual arithmetic operators + - * / and with constant numbers, which makes the
# python API very intuitive. See../ samples/*.py for examples.

INT_MIN = -(2**63)  # hardcoded to be platform independent.
INT_MAX = 2**63 - 1
INT32_MIN = -(2**31)
INT32_MAX = 2**31 - 1

# CpSolver status (exported to avoid importing cp_model_cp2).
UNKNOWN = cp_model_pb2.UNKNOWN
MODEL_INVALID = cp_model_pb2.MODEL_INVALID
FEASIBLE = cp_model_pb2.FEASIBLE
INFEASIBLE = cp_model_pb2.INFEASIBLE
OPTIMAL = cp_model_pb2.OPTIMAL

# Variable selection strategy
CHOOSE_FIRST = cp_model_pb2.DecisionStrategyProto.CHOOSE_FIRST
CHOOSE_LOWEST_MIN = cp_model_pb2.DecisionStrategyProto.CHOOSE_LOWEST_MIN
CHOOSE_HIGHEST_MAX = cp_model_pb2.DecisionStrategyProto.CHOOSE_HIGHEST_MAX
CHOOSE_MIN_DOMAIN_SIZE = cp_model_pb2.DecisionStrategyProto.CHOOSE_MIN_DOMAIN_SIZE
CHOOSE_MAX_DOMAIN_SIZE = cp_model_pb2.DecisionStrategyProto.CHOOSE_MAX_DOMAIN_SIZE

# Domain reduction strategy
SELECT_MIN_VALUE = cp_model_pb2.DecisionStrategyProto.SELECT_MIN_VALUE
SELECT_MAX_VALUE = cp_model_pb2.DecisionStrategyProto.SELECT_MAX_VALUE
SELECT_LOWER_HALF = cp_model_pb2.DecisionStrategyProto.SELECT_LOWER_HALF
SELECT_UPPER_HALF = cp_model_pb2.DecisionStrategyProto.SELECT_UPPER_HALF

# Search branching
AUTOMATIC_SEARCH = sat_parameters_pb2.SatParameters.AUTOMATIC_SEARCH
FIXED_SEARCH = sat_parameters_pb2.SatParameters.FIXED_SEARCH
PORTFOLIO_SEARCH = sat_parameters_pb2.SatParameters.PORTFOLIO_SEARCH
LP_SEARCH = sat_parameters_pb2.SatParameters.LP_SEARCH
PSEUDO_COST_SEARCH = sat_parameters_pb2.SatParameters.PSEUDO_COST_SEARCH
PORTFOLIO_WITH_QUICK_RESTART_SEARCH = (
    sat_parameters_pb2.SatParameters.PORTFOLIO_WITH_QUICK_RESTART_SEARCH
)
HINT_SEARCH = sat_parameters_pb2.SatParameters.HINT_SEARCH
PARTIAL_FIXED_SEARCH = sat_parameters_pb2.SatParameters.PARTIAL_FIXED_SEARCH
RANDOMIZED_SEARCH = sat_parameters_pb2.SatParameters.RANDOMIZED_SEARCH

# Type aliases
IntegralT = Union[int, np.int8, np.uint8, np.int32, np.uint32, np.int64, np.uint64]
IntegralTypes = (
    int,
    np.int8,
    np.uint8,
    np.int32,
    np.uint32,
    np.int64,
    np.uint64,
)
NumberT = Union[
    int,
    float,
    np.int8,
    np.uint8,
    np.int32,
    np.uint32,
    np.int64,
    np.uint64,
    np.double,
]
NumberTypes = (
    int,
    float,
    np.int8,
    np.uint8,
    np.int32,
    np.uint32,
    np.int64,
    np.uint64,
    np.double,
)

LiteralT = Union["IntVar", "_NotBooleanVariable", IntegralT, bool]
BoolVarT = Union["IntVar", "_NotBooleanVariable"]
VariableT = Union["IntVar", IntegralT]

# We need to add 'IntVar' for pytype.
LinearExprT = Union["LinearExpr", "IntVar", IntegralT]
ObjLinearExprT = Union["LinearExpr", NumberT]
BoundedLinearExprT = Union["BoundedLinearExpression", bool]

ArcT = Tuple[IntegralT, IntegralT, LiteralT]
_IndexOrSeries = Union[pd.Index, pd.Series]


def display_bounds(bounds: Sequence[int]) -> str:
    """Displays a flattened list of intervals."""
    out = ""
    for i in range(0, len(bounds), 2):
        if i != 0:
            out += ", "
        if bounds[i] == bounds[i + 1]:
            out += str(bounds[i])
        else:
            out += str(bounds[i]) + ".." + str(bounds[i + 1])
    return out


def short_name(model: cp_model_pb2.CpModelProto, i: int) -> str:
    """Returns a short name of an integer variable, or its negation."""
    if i < 0:
        return "not(%s)" % short_name(model, -i - 1)
    v = model.variables[i]
    if v.name:
        return v.name
    elif len(v.domain) == 2 and v.domain[0] == v.domain[1]:
        return str(v.domain[0])
    else:
        return "[%s]" % display_bounds(v.domain)


def short_expr_name(
    model: cp_model_pb2.CpModelProto, e: cp_model_pb2.LinearExpressionProto
) -> str:
    """Pretty-print LinearExpressionProto instances."""
    if not e.vars:
        return str(e.offset)
    if len(e.vars) == 1:
        var_name = short_name(model, e.vars[0])
        coeff = e.coeffs[0]
        result = ""
        if coeff == 1:
            result = var_name
        elif coeff == -1:
            result = f"-{var_name}"
        elif coeff != 0:
            result = f"{coeff} * {var_name}"
        if e.offset > 0:
            result = f"{result} + {e.offset}"
        elif e.offset < 0:
            result = f"{result} - {-e.offset}"
        return result
    # TODO(user): Support more than affine expressions.
    return str(e)


class LinearExpr:
    """Holds an integer linear expression.

    A linear expression is built from integer constants and variables.
    For example, `x + 2 * (y - z + 1)`.

    Linear expressions are used in CP-SAT models in constraints and in the
    objective:

    * You can define linear constraints as in:

    ```
    model.add(x + 2 * y <= 5)
    model.add(sum(array_of_vars) == 5)
    ```

    * In CP-SAT, the objective is a linear expression:

    ```
    model.minimize(x + 2 * y + z)
    ```

    * For large arrays, using the LinearExpr class is faster that using the python
    `sum()` function. You can create constraints and the objective from lists of
    linear expressions or coefficients as follows:

    ```
    model.minimize(cp_model.LinearExpr.sum(expressions))
    model.add(cp_model.LinearExpr.weighted_sum(expressions, coefficients) >= 0)
    ```
    """

    @classmethod
    def sum(cls, expressions: Sequence[LinearExprT]) -> LinearExprT:
        """Creates the expression sum(expressions)."""
        if len(expressions) == 1:
            return expressions[0]
        return _SumArray(expressions)

    @overload
    @classmethod
    def weighted_sum(
        cls,
        expressions: Sequence[LinearExprT],
        coefficients: Sequence[IntegralT],
    ) -> LinearExprT: ...

    @overload
    @classmethod
    def weighted_sum(
        cls,
        expressions: Sequence[ObjLinearExprT],
        coefficients: Sequence[NumberT],
    ) -> ObjLinearExprT: ...

    @classmethod
    def weighted_sum(cls, expressions, coefficients):
        """Creates the expression sum(expressions[i] * coefficients[i])."""
        if LinearExpr.is_empty_or_all_null(coefficients):
            return 0
        elif len(expressions) == 1:
            return expressions[0] * coefficients[0]
        else:
            return _WeightedSum(expressions, coefficients)

    @overload
    @classmethod
    def term(
        cls,
        expressions: LinearExprT,
        coefficients: IntegralT,
    ) -> LinearExprT: ...

    @overload
    @classmethod
    def term(
        cls,
        expressions: ObjLinearExprT,
        coefficients: NumberT,
    ) -> ObjLinearExprT: ...

    @classmethod
    def term(cls, expression, coefficient):
        """Creates `expression * coefficient`."""
        if cmh.is_zero(coefficient):
            return 0
        else:
            return expression * coefficient

    @classmethod
    def is_empty_or_all_null(cls, coefficients: Sequence[NumberT]) -> bool:
        for c in coefficients:
            if not cmh.is_zero(c):
                return False
        return True

    @classmethod
    def rebuild_from_linear_expression_proto(
        cls,
        model: cp_model_pb2.CpModelProto,
        proto: cp_model_pb2.LinearExpressionProto,
    ) -> LinearExprT:
        """Recreate a LinearExpr from a LinearExpressionProto."""
        offset = proto.offset
        num_elements = len(proto.vars)
        if num_elements == 0:
            return offset
        elif num_elements == 1:
            return (
                IntVar(model, proto.vars[0], None) * proto.coeffs[0] + offset
            )  # pytype: disable=bad-return-type
        else:
            variables = []
            coeffs = []
            all_ones = True
            for index, coeff in zip(proto.vars, proto.coeffs):
                variables.append(IntVar(model, index, None))
                coeffs.append(coeff)
                if not cmh.is_one(coeff):
                    all_ones = False
            if all_ones:
                return _SumArray(variables, offset)
            else:
                return _WeightedSum(variables, coeffs, offset)

    def get_integer_var_value_map(self) -> Tuple[Dict["IntVar", int], int]:
        """Scans the expression, and returns (var_coef_map, constant)."""
        coeffs: Dict["IntVar", int] = collections.defaultdict(int)
        constant = 0
        to_process: List[Tuple[LinearExprT, int]] = [(self, 1)]
        while to_process:  # Flatten to avoid recursion.
            expr: LinearExprT
            coeff: int
            expr, coeff = to_process.pop()
            if isinstance(expr, IntegralTypes):
                constant += coeff * int(expr)
            elif isinstance(expr, _ProductCst):
                to_process.append((expr.expression(), coeff * expr.coefficient()))
            elif isinstance(expr, _Sum):
                to_process.append((expr.left(), coeff))
                to_process.append((expr.right(), coeff))
            elif isinstance(expr, _SumArray):
                for e in expr.expressions():
                    to_process.append((e, coeff))
                constant += expr.constant() * coeff
            elif isinstance(expr, _WeightedSum):
                for e, c in zip(expr.expressions(), expr.coefficients()):
                    to_process.append((e, coeff * c))
                constant += expr.constant() * coeff
            elif isinstance(expr, IntVar):
                coeffs[expr] += coeff
            elif isinstance(expr, _NotBooleanVariable):
                constant += coeff
                coeffs[expr.negated()] -= coeff
            elif isinstance(expr, NumberTypes):
                raise TypeError(
                    f"Floating point constants are not supported in constraints: {expr}"
                )
            else:
                raise TypeError("Unrecognized linear expression: " + str(expr))

        return coeffs, constant

    def get_float_var_value_map(
        self,
    ) -> Tuple[Dict["IntVar", float], float, bool]:
        """Scans the expression. Returns (var_coef_map, constant, is_integer)."""
        coeffs: Dict["IntVar", Union[int, float]] = {}
        constant: Union[int, float] = 0
        to_process: List[Tuple[LinearExprT, Union[int, float]]] = [(self, 1)]
        while to_process:  # Flatten to avoid recursion.
            expr, coeff = to_process.pop()
            if isinstance(expr, IntegralTypes):  # Keep integrality.
                constant += coeff * int(expr)
            elif isinstance(expr, NumberTypes):
                constant += coeff * float(expr)
            elif isinstance(expr, _ProductCst):
                to_process.append((expr.expression(), coeff * expr.coefficient()))
            elif isinstance(expr, _Sum):
                to_process.append((expr.left(), coeff))
                to_process.append((expr.right(), coeff))
            elif isinstance(expr, _SumArray):
                for e in expr.expressions():
                    to_process.append((e, coeff))
                constant += expr.constant() * coeff
            elif isinstance(expr, _WeightedSum):
                for e, c in zip(expr.expressions(), expr.coefficients()):
                    to_process.append((e, coeff * c))
                constant += expr.constant() * coeff
            elif isinstance(expr, IntVar):
                if expr in coeffs:
                    coeffs[expr] += coeff
                else:
                    coeffs[expr] = coeff
            elif isinstance(expr, _NotBooleanVariable):
                constant += coeff
                if expr.negated() in coeffs:
                    coeffs[expr.negated()] -= coeff
                else:
                    coeffs[expr.negated()] = -coeff
            else:
                raise TypeError("Unrecognized linear expression: " + str(expr))
        is_integer = isinstance(constant, IntegralTypes)
        if is_integer:
            for coeff in coeffs.values():
                if not isinstance(coeff, IntegralTypes):
                    is_integer = False
                    break
        return coeffs, constant, is_integer

    def __hash__(self) -> int:
        return object.__hash__(self)

    def __abs__(self) -> NoReturn:
        raise NotImplementedError(
            "calling abs() on a linear expression is not supported, "
            "please use CpModel.add_abs_equality"
        )

    @overload
    def __add__(self, arg: "LinearExpr") -> "LinearExpr": ...

    @overload
    def __add__(self, arg: NumberT) -> "LinearExpr": ...

    def __add__(self, arg):
        if cmh.is_zero(arg):
            return self
        return _Sum(self, arg)

    @overload
    def __radd__(self, arg: "LinearExpr") -> "LinearExpr": ...

    @overload
    def __radd__(self, arg: NumberT) -> "LinearExpr": ...

    def __radd__(self, arg):
        return self.__add__(arg)

    @overload
    def __sub__(self, arg: "LinearExpr") -> "LinearExpr": ...

    @overload
    def __sub__(self, arg: NumberT) -> "LinearExpr": ...

    def __sub__(self, arg):
        if cmh.is_zero(arg):
            return self
        if isinstance(arg, NumberTypes):
            arg = cmh.assert_is_a_number(arg)
            return _Sum(self, -arg)
        else:
            return _Sum(self, -arg)

    @overload
    def __rsub__(self, arg: "LinearExpr") -> "LinearExpr": ...

    @overload
    def __rsub__(self, arg: NumberT) -> "LinearExpr": ...

    def __rsub__(self, arg):
        return _Sum(-self, arg)

    @overload
    def __mul__(self, arg: IntegralT) -> Union["LinearExpr", IntegralT]: ...

    @overload
    def __mul__(self, arg: NumberT) -> Union["LinearExpr", NumberT]: ...

    def __mul__(self, arg):
        arg = cmh.assert_is_a_number(arg)
        if cmh.is_one(arg):
            return self
        elif cmh.is_zero(arg):
            return 0
        return _ProductCst(self, arg)

    @overload
    def __rmul__(self, arg: IntegralT) -> Union["LinearExpr", IntegralT]: ...

    @overload
    def __rmul__(self, arg: NumberT) -> Union["LinearExpr", NumberT]: ...

    def __rmul__(self, arg):
        return self.__mul__(arg)

    def __div__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling / on a linear expression is not supported, "
            "please use CpModel.add_division_equality"
        )

    def __truediv__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling // on a linear expression is not supported, "
            "please use CpModel.add_division_equality"
        )

    def __mod__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling %% on a linear expression is not supported, "
            "please use CpModel.add_modulo_equality"
        )

    def __pow__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling ** on a linear expression is not supported, "
            "please use CpModel.add_multiplication_equality"
        )

    def __lshift__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling left shift on a linear expression is not supported"
        )

    def __rshift__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling right shift on a linear expression is not supported"
        )

    def __and__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling and on a linear expression is not supported, "
            "please use CpModel.add_bool_and"
        )

    def __or__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling or on a linear expression is not supported, "
            "please use CpModel.add_bool_or"
        )

    def __xor__(self, _) -> NoReturn:
        raise NotImplementedError(
            "calling xor on a linear expression is not supported, "
            "please use CpModel.add_bool_xor"
        )

    def __neg__(self) -> "LinearExpr":
        return _ProductCst(self, -1)

    def __bool__(self) -> NoReturn:
        raise NotImplementedError(
            "Evaluating a LinearExpr instance as a Boolean is not implemented."
        )

    def __eq__(self, arg: LinearExprT) -> BoundedLinearExprT:  # type: ignore[override]
        if arg is None:
            return False
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            return BoundedLinearExpression(self, [arg, arg])
        elif isinstance(arg, LinearExpr):
            return BoundedLinearExpression(self - arg, [0, 0])
        else:
            return False

    def __ge__(self, arg: LinearExprT) -> "BoundedLinearExpression":
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            return BoundedLinearExpression(self, [arg, INT_MAX])
        else:
            return BoundedLinearExpression(self - arg, [0, INT_MAX])

    def __le__(self, arg: LinearExprT) -> "BoundedLinearExpression":
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            return BoundedLinearExpression(self, [INT_MIN, arg])
        else:
            return BoundedLinearExpression(self - arg, [INT_MIN, 0])

    def __lt__(self, arg: LinearExprT) -> "BoundedLinearExpression":
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            if arg == INT_MIN:
                raise ArithmeticError("< INT_MIN is not supported")
            return BoundedLinearExpression(self, [INT_MIN, arg - 1])
        else:
            return BoundedLinearExpression(self - arg, [INT_MIN, -1])

    def __gt__(self, arg: LinearExprT) -> "BoundedLinearExpression":
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            if arg == INT_MAX:
                raise ArithmeticError("> INT_MAX is not supported")
            return BoundedLinearExpression(self, [arg + 1, INT_MAX])
        else:
            return BoundedLinearExpression(self - arg, [1, INT_MAX])

    def __ne__(self, arg: LinearExprT) -> BoundedLinearExprT:  # type: ignore[override]
        if arg is None:
            return True
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            if arg == INT_MAX:
                return BoundedLinearExpression(self, [INT_MIN, INT_MAX - 1])
            elif arg == INT_MIN:
                return BoundedLinearExpression(self, [INT_MIN + 1, INT_MAX])
            else:
                return BoundedLinearExpression(
                    self, [INT_MIN, arg - 1, arg + 1, INT_MAX]
                )
        elif isinstance(arg, LinearExpr):
            return BoundedLinearExpression(self - arg, [INT_MIN, -1, 1, INT_MAX])
        else:
            return True

    # Compatibility with pre PEP8
    # pylint: disable=invalid-name
    @classmethod
    def Sum(cls, expressions: Sequence[LinearExprT]) -> LinearExprT:
        """Creates the expression sum(expressions)."""
        return cls.sum(expressions)

    @overload
    @classmethod
    def WeightedSum(
        cls,
        expressions: Sequence[LinearExprT],
        coefficients: Sequence[IntegralT],
    ) -> LinearExprT: ...

    @overload
    @classmethod
    def WeightedSum(
        cls,
        expressions: Sequence[ObjLinearExprT],
        coefficients: Sequence[NumberT],
    ) -> ObjLinearExprT: ...

    @classmethod
    def WeightedSum(cls, expressions, coefficients):
        """Creates the expression sum(expressions[i] * coefficients[i])."""
        return cls.weighted_sum(expressions, coefficients)

    @overload
    @classmethod
    def Term(
        cls,
        expressions: LinearExprT,
        coefficients: IntegralT,
    ) -> LinearExprT: ...

    @overload
    @classmethod
    def Term(
        cls,
        expressions: ObjLinearExprT,
        coefficients: NumberT,
    ) -> ObjLinearExprT: ...

    @classmethod
    def Term(cls, expression, coefficient):
        """Creates `expression * coefficient`."""
        return cls.term(expression, coefficient)

    # pylint: enable=invalid-name


class _Sum(LinearExpr):
    """Represents the sum of two LinearExprs."""

    def __init__(self, left, right) -> None:
        for x in [left, right]:
            if not isinstance(x, (NumberTypes, LinearExpr)):
                raise TypeError("not an linear expression: " + str(x))
        self.__left = left
        self.__right = right

    def left(self):
        return self.__left

    def right(self):
        return self.__right

    def __str__(self):
        return f"({self.__left} + {self.__right})"

    def __repr__(self):
        return f"sum({self.__left!r}, {self.__right!r})"


class _ProductCst(LinearExpr):
    """Represents the product of a LinearExpr by a constant."""

    def __init__(self, expr, coeff) -> None:
        coeff = cmh.assert_is_a_number(coeff)
        if isinstance(expr, _ProductCst):
            self.__expr = expr.expression()
            self.__coef = expr.coefficient() * coeff
        else:
            self.__expr = expr
            self.__coef = coeff

    def __str__(self):
        if self.__coef == -1:
            return "-" + str(self.__expr)
        else:
            return "(" + str(self.__coef) + " * " + str(self.__expr) + ")"

    def __repr__(self):
        return f"ProductCst({self.__expr!r}, {self.__coef!r})"

    def coefficient(self):
        return self.__coef

    def expression(self):
        return self.__expr


class _SumArray(LinearExpr):
    """Represents the sum of a list of LinearExpr and a constant."""

    def __init__(self, expressions, constant=0) -> None:
        self.__expressions = []
        self.__constant = constant
        for x in expressions:
            if isinstance(x, NumberTypes):
                if cmh.is_zero(x):
                    continue
                x = cmh.assert_is_a_number(x)
                self.__constant += x
            elif isinstance(x, LinearExpr):
                self.__expressions.append(x)
            else:
                raise TypeError("not an linear expression: " + str(x))

    def __str__(self):
        constant_terms = (self.__constant,) if self.__constant != 0 else ()
        exprs_str = " + ".join(
            map(repr, itertools.chain(self.__expressions, constant_terms))
        )
        if not exprs_str:
            return "0"
        return f"({exprs_str})"

    def __repr__(self):
        exprs_str = ", ".join(map(repr, self.__expressions))
        return f"SumArray({exprs_str}, {self.__constant})"

    def expressions(self):
        return self.__expressions

    def constant(self):
        return self.__constant


class _WeightedSum(LinearExpr):
    """Represents sum(ai * xi) + b."""

    def __init__(self, expressions, coefficients, constant=0) -> None:
        self.__expressions = []
        self.__coefficients = []
        self.__constant = constant
        if len(expressions) != len(coefficients):
            raise TypeError(
                "In the LinearExpr.weighted_sum method, the expression array and the "
                " coefficient array must have the same length."
            )
        for e, c in zip(expressions, coefficients):
            c = cmh.assert_is_a_number(c)
            if cmh.is_zero(c):
                continue
            if isinstance(e, NumberTypes):
                e = cmh.assert_is_a_number(e)
                self.__constant += e * c
            elif isinstance(e, LinearExpr):
                self.__expressions.append(e)
                self.__coefficients.append(c)
            else:
                raise TypeError("not an linear expression: " + str(e))

    def __str__(self):
        output = None
        for expr, coeff in zip(self.__expressions, self.__coefficients):
            if not output and cmh.is_one(coeff):
                output = str(expr)
            elif not output and cmh.is_minus_one(coeff):
                output = "-" + str(expr)
            elif not output:
                output = f"{coeff} * {expr}"
            elif cmh.is_one(coeff):
                output += f" + {expr}"
            elif cmh.is_minus_one(coeff):
                output += f" - {expr}"
            elif coeff > 1:
                output += f" + {coeff} * {expr}"
            elif coeff < -1:
                output += f" - {-coeff} * {expr}"
        if output is None:
            output = str(self.__constant)
        elif self.__constant > 0:
            output += f" + {self.__constant}"
        elif self.__constant < 0:
            output += f" - {-self.__constant}"
        return output

    def __repr__(self):
        return (
            f"weighted_sum({self.__expressions!r}, {self.__coefficients!r},"
            f" {self.__constant})"
        )

    def expressions(self):
        return self.__expressions

    def coefficients(self):
        return self.__coefficients

    def constant(self):
        return self.__constant


class IntVar(LinearExpr):
    """An integer variable.

    An IntVar is an object that can take on any integer value within defined
    ranges. Variables appear in constraint like:

        x + y >= 5
        AllDifferent([x, y, z])

    Solving a model is equivalent to finding, for each variable, a single value
    from the set of initial values (called the initial domain), such that the
    model is feasible, or optimal if you provided an objective function.
    """

    def __init__(
        self,
        model: cp_model_pb2.CpModelProto,
        domain: Union[int, sorted_interval_list.Domain],
        name: Optional[str],
    ) -> None:
        """See CpModel.new_int_var below."""
        self.__index: int
        self.__var: cp_model_pb2.IntegerVariableProto
        self.__negation: Optional[_NotBooleanVariable] = None
        # Python do not support multiple __init__ methods.
        # This method is only called from the CpModel class.
        # We hack the parameter to support the two cases:
        # case 1:
        #     model is a CpModelProto, domain is a Domain, and name is a string.
        # case 2:
        #     model is a CpModelProto, domain is an index (int), and name is None.
        if isinstance(domain, IntegralTypes) and name is None:
            self.__index = int(domain)
            self.__var = model.variables[domain]
        else:
            self.__index = len(model.variables)
            self.__var = model.variables.add()
            self.__var.domain.extend(
                cast(sorted_interval_list.Domain, domain).flattened_intervals()
            )
            if name is not None:
                self.__var.name = name

    @property
    def index(self) -> int:
        """Returns the index of the variable in the model."""
        return self.__index

    @property
    def proto(self) -> cp_model_pb2.IntegerVariableProto:
        """Returns the variable protobuf."""
        return self.__var

    def is_equal_to(self, other: Any) -> bool:
        """Returns true if self == other in the python sense."""
        if not isinstance(other, IntVar):
            return False
        return self.index == other.index

    def __str__(self) -> str:
        if not self.__var.name:
            if (
                len(self.__var.domain) == 2
                and self.__var.domain[0] == self.__var.domain[1]
            ):
                # Special case for constants.
                return str(self.__var.domain[0])
            else:
                return "unnamed_var_%i" % self.__index
        return self.__var.name

    def __repr__(self) -> str:
        return "%s(%s)" % (self.__var.name, display_bounds(self.__var.domain))

    @property
    def name(self) -> str:
        if not self.__var or not self.__var.name:
            return ""
        return self.__var.name

    def negated(self) -> "_NotBooleanVariable":
        """Returns the negation of a Boolean variable.

        This method implements the logical negation of a Boolean variable.
        It is only valid if the variable has a Boolean domain (0 or 1).

        Note that this method is nilpotent: `x.negated().negated() == x`.
        """

        for bound in self.__var.domain:
            if bound < 0 or bound > 1:
                raise TypeError(
                    f"cannot call negated on a non boolean variable: {self}"
                )
        if self.__negation is None:
            self.__negation = _NotBooleanVariable(self)
        return self.__negation

    def __invert__(self) -> "_NotBooleanVariable":
        """Returns the logical negation of a Boolean variable."""
        return self.negated()

    # Pre PEP8 compatibility.
    # pylint: disable=invalid-name
    Not = negated

    def Name(self) -> str:
        return self.name

    def Proto(self) -> cp_model_pb2.IntegerVariableProto:
        return self.proto

    def Index(self) -> int:
        return self.index

    # pylint: enable=invalid-name


class _NotBooleanVariable(LinearExpr):
    """Negation of a boolean variable."""

    def __init__(self, boolvar: IntVar) -> None:
        self.__boolvar: IntVar = boolvar

    @property
    def index(self) -> int:
        return -self.__boolvar.index - 1

    def negated(self) -> IntVar:
        return self.__boolvar

    def __invert__(self) -> IntVar:
        """Returns the logical negation of a Boolean literal."""
        return self.negated()

    def __str__(self) -> str:
        return self.name

    @property
    def name(self) -> str:
        return "not(%s)" % str(self.__boolvar)

    def __bool__(self) -> NoReturn:
        raise NotImplementedError(
            "Evaluating a literal as a Boolean value is not implemented."
        )

    # Pre PEP8 compatibility.
    # pylint: disable=invalid-name
    def Not(self) -> "IntVar":
        return self.negated()

    def Index(self) -> int:
        return self.index

    # pylint: enable=invalid-name


class BoundedLinearExpression:
    """Represents a linear constraint: `lb <= linear expression <= ub`.

    The only use of this class is to be added to the CpModel through
    `CpModel.add(expression)`, as in:

        model.add(x + 2 * y -1 >= z)
    """

    def __init__(self, expr: LinearExprT, bounds: Sequence[int]) -> None:
        self.__expr: LinearExprT = expr
        self.__bounds: Sequence[int] = bounds

    def __str__(self):
        if len(self.__bounds) == 2:
            lb, ub = self.__bounds
            if lb > INT_MIN and ub < INT_MAX:
                if lb == ub:
                    return str(self.__expr) + " == " + str(lb)
                else:
                    return str(lb) + " <= " + str(self.__expr) + " <= " + str(ub)
            elif lb > INT_MIN:
                return str(self.__expr) + " >= " + str(lb)
            elif ub < INT_MAX:
                return str(self.__expr) + " <= " + str(ub)
            else:
                return "True (unbounded expr " + str(self.__expr) + ")"
        elif (
            len(self.__bounds) == 4
            and self.__bounds[0] == INT_MIN
            and self.__bounds[1] + 2 == self.__bounds[2]
            and self.__bounds[3] == INT_MAX
        ):
            return str(self.__expr) + " != " + str(self.__bounds[1] + 1)
        else:
            return str(self.__expr) + " in [" + display_bounds(self.__bounds) + "]"

    def expression(self) -> LinearExprT:
        return self.__expr

    def bounds(self) -> Sequence[int]:
        return self.__bounds

    def __bool__(self) -> bool:
        expr = self.__expr
        if isinstance(expr, LinearExpr):
            coeffs_map, constant = expr.get_integer_var_value_map()
            all_coeffs = set(coeffs_map.values())
            same_var = set([0])
            eq_bounds = [0, 0]
            different_vars = set([-1, 1])
            ne_bounds = [INT_MIN, -1, 1, INT_MAX]
            if (
                len(coeffs_map) == 1
                and all_coeffs == same_var
                and constant == 0
                and (self.__bounds == eq_bounds or self.__bounds == ne_bounds)
            ):
                return self.__bounds == eq_bounds
            if (
                len(coeffs_map) == 2
                and all_coeffs == different_vars
                and constant == 0
                and (self.__bounds == eq_bounds or self.__bounds == ne_bounds)
            ):
                return self.__bounds == ne_bounds

        raise NotImplementedError(
            f'Evaluating a BoundedLinearExpression "{self}" as a Boolean value'
            + " is not supported."
        )


class Constraint:
    """Base class for constraints.

    Constraints are built by the CpModel through the add<XXX> methods.
    Once created by the CpModel class, they are automatically added to the model.
    The purpose of this class is to allow specification of enforcement literals
    for this constraint.

        b = model.new_bool_var('b')
        x = model.new_int_var(0, 10, 'x')
        y = model.new_int_var(0, 10, 'y')

        model.add(x + 2 * y == 5).only_enforce_if(b.negated())
    """

    def __init__(
        self,
        cp_model: "CpModel",
    ) -> None:
        self.__index: int = len(cp_model.proto.constraints)
        self.__cp_model: "CpModel" = cp_model
        self.__constraint: cp_model_pb2.ConstraintProto = (
            cp_model.proto.constraints.add()
        )

    @overload
    def only_enforce_if(self, boolvar: Iterable[LiteralT]) -> "Constraint": ...

    @overload
    def only_enforce_if(self, *boolvar: LiteralT) -> "Constraint": ...

    def only_enforce_if(self, *boolvar) -> "Constraint":
        """Adds an enforcement literal to the constraint.

        This method adds one or more literals (that is, a boolean variable or its
        negation) as enforcement literals. The conjunction of all these literals
        determines whether the constraint is active or not. It acts as an
        implication, so if the conjunction is true, it implies that the constraint
        must be enforced. If it is false, then the constraint is ignored.

        BoolOr, BoolAnd, and linear constraints all support enforcement literals.

        Args:
          *boolvar: One or more Boolean literals.

        Returns:
          self.
        """
        for lit in expand_generator_or_tuple(boolvar):
            if (cmh.is_boolean(lit) and lit) or (
                isinstance(lit, IntegralTypes) and lit == 1
            ):
                # Always true. Do nothing.
                pass
            elif (cmh.is_boolean(lit) and not lit) or (
                isinstance(lit, IntegralTypes) and lit == 0
            ):
                self.__constraint.enforcement_literal.append(
                    self.__cp_model.new_constant(0).index
                )
            else:
                self.__constraint.enforcement_literal.append(
                    cast(Union[IntVar, _NotBooleanVariable], lit).index
                )
        return self

    def with_name(self, name: str) -> "Constraint":
        """Sets the name of the constraint."""
        if name:
            self.__constraint.name = name
        else:
            self.__constraint.ClearField("name")
        return self

    @property
    def name(self) -> str:
        """Returns the name of the constraint."""
        if not self.__constraint or not self.__constraint.name:
            return ""
        return self.__constraint.name

    @property
    def index(self) -> int:
        """Returns the index of the constraint in the model."""
        return self.__index

    @property
    def proto(self) -> cp_model_pb2.ConstraintProto:
        """Returns the constraint protobuf."""
        return self.__constraint

    # Pre PEP8 compatibility.
    # pylint: disable=invalid-name
    OnlyEnforceIf = only_enforce_if
    WithName = with_name

    def Name(self) -> str:
        return self.name

    def Index(self) -> int:
        return self.index

    def Proto(self) -> cp_model_pb2.ConstraintProto:
        return self.proto

    # pylint: enable=invalid-name


class IntervalVar:
    """Represents an Interval variable.

    An interval variable is both a constraint and a variable. It is defined by
    three integer variables: start, size, and end.

    It is a constraint because, internally, it enforces that start + size == end.

    It is also a variable as it can appear in specific scheduling constraints:
    NoOverlap, NoOverlap2D, Cumulative.

    Optionally, an enforcement literal can be added to this constraint, in which
    case these scheduling constraints will ignore interval variables with
    enforcement literals assigned to false. Conversely, these constraints will
    also set these enforcement literals to false if they cannot fit these
    intervals into the schedule.

    Raises:
      ValueError: if start, size, end are not defined, or have the wrong type.
    """

    def __init__(
        self,
        model: cp_model_pb2.CpModelProto,
        start: Union[cp_model_pb2.LinearExpressionProto, int],
        size: Optional[cp_model_pb2.LinearExpressionProto],
        end: Optional[cp_model_pb2.LinearExpressionProto],
        is_present_index: Optional[int],
        name: Optional[str],
    ) -> None:
        self.__model: cp_model_pb2.CpModelProto = model
        self.__index: int
        self.__ct: cp_model_pb2.ConstraintProto
        # As with the IntVar::__init__ method, we hack the __init__ method to
        # support two use cases:
        #   case 1: called when creating a new interval variable.
        #      {start|size|end} are linear expressions, is_present_index is either
        #      None or the index of a Boolean literal. name is a string
        #   case 2: called when querying an existing interval variable.
        #      start_index is an int, all parameters after are None.
        if isinstance(start, int):
            if size is not None:
                raise ValueError("size should be None")
            if end is not None:
                raise ValueError("end should be None")
            if is_present_index is not None:
                raise ValueError("is_present_index should be None")
            self.__index = cast(int, start)
            self.__ct = model.constraints[self.__index]
        else:
            self.__index = len(model.constraints)
            self.__ct = self.__model.constraints.add()
            if start is None:
                raise TypeError("start is not defined")
            self.__ct.interval.start.CopyFrom(start)
            if size is None:
                raise TypeError("size is not defined")
            self.__ct.interval.size.CopyFrom(size)
            if end is None:
                raise TypeError("end is not defined")
            self.__ct.interval.end.CopyFrom(end)
            if is_present_index is not None:
                self.__ct.enforcement_literal.append(is_present_index)
            if name:
                self.__ct.name = name

    @property
    def index(self) -> int:
        """Returns the index of the interval constraint in the model."""
        return self.__index

    @property
    def proto(self) -> cp_model_pb2.IntervalConstraintProto:
        """Returns the interval protobuf."""
        return self.__ct.interval

    def __str__(self):
        return self.__ct.name

    def __repr__(self):
        interval = self.__ct.interval
        if self.__ct.enforcement_literal:
            return "%s(start = %s, size = %s, end = %s, is_present = %s)" % (
                self.__ct.name,
                short_expr_name(self.__model, interval.start),
                short_expr_name(self.__model, interval.size),
                short_expr_name(self.__model, interval.end),
                short_name(self.__model, self.__ct.enforcement_literal[0]),
            )
        else:
            return "%s(start = %s, size = %s, end = %s)" % (
                self.__ct.name,
                short_expr_name(self.__model, interval.start),
                short_expr_name(self.__model, interval.size),
                short_expr_name(self.__model, interval.end),
            )

    @property
    def name(self) -> str:
        if not self.__ct or not self.__ct.name:
            return ""
        return self.__ct.name

    def start_expr(self) -> LinearExprT:
        return LinearExpr.rebuild_from_linear_expression_proto(
            self.__model, self.__ct.interval.start
        )

    def size_expr(self) -> LinearExprT:
        return LinearExpr.rebuild_from_linear_expression_proto(
            self.__model, self.__ct.interval.size
        )

    def end_expr(self) -> LinearExprT:
        return LinearExpr.rebuild_from_linear_expression_proto(
            self.__model, self.__ct.interval.end
        )

    # Pre PEP8 compatibility.
    # pylint: disable=invalid-name
    def Name(self) -> str:
        return self.name

    def Index(self) -> int:
        return self.index

    def Proto(self) -> cp_model_pb2.IntervalConstraintProto:
        return self.proto

    StartExpr = start_expr
    SizeExpr = size_expr
    EndExpr = end_expr

    # pylint: enable=invalid-name


def object_is_a_true_literal(literal: LiteralT) -> bool:
    """Checks if literal is either True, or a Boolean literals fixed to True."""
    if isinstance(literal, IntVar):
        proto = literal.proto
        return len(proto.domain) == 2 and proto.domain[0] == 1 and proto.domain[1] == 1
    if isinstance(literal, _NotBooleanVariable):
        proto = literal.negated().proto
        return len(proto.domain) == 2 and proto.domain[0] == 0 and proto.domain[1] == 0
    if isinstance(literal, IntegralTypes):
        return int(literal) == 1
    return False


def object_is_a_false_literal(literal: LiteralT) -> bool:
    """Checks if literal is either False, or a Boolean literals fixed to False."""
    if isinstance(literal, IntVar):
        proto = literal.proto
        return len(proto.domain) == 2 and proto.domain[0] == 0 and proto.domain[1] == 0
    if isinstance(literal, _NotBooleanVariable):
        proto = literal.negated().proto
        return len(proto.domain) == 2 and proto.domain[0] == 1 and proto.domain[1] == 1
    if isinstance(literal, IntegralTypes):
        return int(literal) == 0
    return False


class CpModel:
    """Methods for building a CP model.

    Methods beginning with:

    * ```New``` create integer, boolean, or interval variables.
    * ```add``` create new constraints and add them to the model.
    """

    def __init__(self) -> None:
        self.__model: cp_model_pb2.CpModelProto = cp_model_pb2.CpModelProto()
        self.__constant_map: Dict[IntegralT, int] = {}

    # Naming.
    @property
    def name(self) -> str:
        """Returns the name of the model."""
        if not self.__model or not self.__model.name:
            return ""
        return self.__model.name

    @name.setter
    def name(self, name: str):
        """Sets the name of the model."""
        self.__model.name = name

    # Integer variable.

    def new_int_var(self, lb: IntegralT, ub: IntegralT, name: str) -> IntVar:
        """Create an integer variable with domain [lb, ub].

        The CP-SAT solver is limited to integer variables. If you have fractional
        values, scale them up so that they become integers; if you have strings,
        encode them as integers.

        Args:
          lb: Lower bound for the variable.
          ub: Upper bound for the variable.
          name: The name of the variable.

        Returns:
          a variable whose domain is [lb, ub].
        """

        return IntVar(self.__model, sorted_interval_list.Domain(lb, ub), name)

    def new_int_var_from_domain(
        self, domain: sorted_interval_list.Domain, name: str
    ) -> IntVar:
        """Create an integer variable from a domain.

        A domain is a set of integers specified by a collection of intervals.
        For example, `model.new_int_var_from_domain(cp_model.
             Domain.from_intervals([[1, 2], [4, 6]]), 'x')`

        Args:
          domain: An instance of the Domain class.
          name: The name of the variable.

        Returns:
            a variable whose domain is the given domain.
        """
        return IntVar(self.__model, domain, name)

    def new_bool_var(self, name: str) -> IntVar:
        """Creates a 0-1 variable with the given name."""
        return IntVar(self.__model, sorted_interval_list.Domain(0, 1), name)

    def new_constant(self, value: IntegralT) -> IntVar:
        """Declares a constant integer."""
        return IntVar(self.__model, self.get_or_make_index_from_constant(value), None)

    def new_int_var_series(
        self,
        name: str,
        index: pd.Index,
        lower_bounds: Union[IntegralT, pd.Series],
        upper_bounds: Union[IntegralT, pd.Series],
    ) -> pd.Series:
        """Creates a series of (scalar-valued) variables with the given name.

        Args:
          name (str): Required. The name of the variable set.
          index (pd.Index): Required. The index to use for the variable set.
          lower_bounds (Union[int, pd.Series]): A lower bound for variables in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          upper_bounds (Union[int, pd.Series]): An upper bound for variables in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.

        Returns:
          pd.Series: The variable set indexed by its corresponding dimensions.

        Raises:
          TypeError: if the `index` is invalid (e.g. a `DataFrame`).
          ValueError: if the `name` is not a valid identifier or already exists.
          ValueError: if the `lowerbound` is greater than the `upperbound`.
          ValueError: if the index of `lower_bound`, or `upper_bound` does not match
          the input index.
        """
        if not isinstance(index, pd.Index):
            raise TypeError("Non-index object is used as index")
        if not name.isidentifier():
            raise ValueError("name={} is not a valid identifier".format(name))
        if (
            isinstance(lower_bounds, IntegralTypes)
            and isinstance(upper_bounds, IntegralTypes)
            and lower_bounds > upper_bounds
        ):
            raise ValueError(
                f"lower_bound={lower_bounds} is greater than"
                f" upper_bound={upper_bounds} for variable set={name}"
            )

        lower_bounds = _convert_to_integral_series_and_validate_index(
            lower_bounds, index
        )
        upper_bounds = _convert_to_integral_series_and_validate_index(
            upper_bounds, index
        )
        return pd.Series(
            index=index,
            data=[
                # pylint: disable=g-complex-comprehension
                IntVar(
                    model=self.__model,
                    name=f"{name}[{i}]",
                    domain=sorted_interval_list.Domain(
                        lower_bounds[i], upper_bounds[i]
                    ),
                )
                for i in index
            ],
        )

    def new_bool_var_series(
        self,
        name: str,
        index: pd.Index,
    ) -> pd.Series:
        """Creates a series of (scalar-valued) variables with the given name.

        Args:
          name (str): Required. The name of the variable set.
          index (pd.Index): Required. The index to use for the variable set.

        Returns:
          pd.Series: The variable set indexed by its corresponding dimensions.

        Raises:
          TypeError: if the `index` is invalid (e.g. a `DataFrame`).
          ValueError: if the `name` is not a valid identifier or already exists.
        """
        return self.new_int_var_series(
            name=name, index=index, lower_bounds=0, upper_bounds=1
        )

    # Linear constraints.

    def add_linear_constraint(
        self, linear_expr: LinearExprT, lb: IntegralT, ub: IntegralT
    ) -> Constraint:
        """Adds the constraint: `lb <= linear_expr <= ub`."""
        return self.add_linear_expression_in_domain(
            linear_expr, sorted_interval_list.Domain(lb, ub)
        )

    def add_linear_expression_in_domain(
        self, linear_expr: LinearExprT, domain: sorted_interval_list.Domain
    ) -> Constraint:
        """Adds the constraint: `linear_expr` in `domain`."""
        if isinstance(linear_expr, LinearExpr):
            ct = Constraint(self)
            model_ct = self.__model.constraints[ct.index]
            coeffs_map, constant = linear_expr.get_integer_var_value_map()
            for t in coeffs_map.items():
                if not isinstance(t[0], IntVar):
                    raise TypeError("Wrong argument" + str(t))
                c = cmh.assert_is_int64(t[1])
                model_ct.linear.vars.append(t[0].index)
                model_ct.linear.coeffs.append(c)
            model_ct.linear.domain.extend(
                [
                    cmh.capped_subtraction(x, constant)
                    for x in domain.flattened_intervals()
                ]
            )
            return ct
        if isinstance(linear_expr, IntegralTypes):
            if not domain.contains(int(linear_expr)):
                return self.add_bool_or([])  # Evaluate to false.
            else:
                return self.add_bool_and([])  # Evaluate to true.
        raise TypeError(
            "not supported: CpModel.add_linear_expression_in_domain("
            + str(linear_expr)
            + " "
            + str(domain)
            + ")"
        )

    @overload
    def add(self, ct: BoundedLinearExpression) -> Constraint: ...

    @overload
    def add(self, ct: Union[bool, np.bool_]) -> Constraint: ...

    def add(self, ct):
        """Adds a `BoundedLinearExpression` to the model.

        Args:
          ct: A [`BoundedLinearExpression`](#boundedlinearexpression).

        Returns:
          An instance of the `Constraint` class.
        """
        if isinstance(ct, BoundedLinearExpression):
            return self.add_linear_expression_in_domain(
                ct.expression(),
                sorted_interval_list.Domain.from_flat_intervals(ct.bounds()),
            )
        if ct and cmh.is_boolean(ct):
            return self.add_bool_or([True])
        if not ct and cmh.is_boolean(ct):
            return self.add_bool_or([])  # Evaluate to false.
        raise TypeError("not supported: CpModel.add(" + str(ct) + ")")

    # General Integer Constraints.

    @overload
    def add_all_different(self, expressions: Iterable[LinearExprT]) -> Constraint: ...

    @overload
    def add_all_different(self, *expressions: LinearExprT) -> Constraint: ...

    def add_all_different(self, *expressions):
        """Adds AllDifferent(expressions).

        This constraint forces all expressions to have different values.

        Args:
          *expressions: simple expressions of the form a * var + constant.

        Returns:
          An instance of the `Constraint` class.
        """
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        expanded = expand_generator_or_tuple(expressions)
        model_ct.all_diff.exprs.extend(
            self.parse_linear_expression(x) for x in expanded
        )
        return ct

    def add_element(
        self, index: VariableT, variables: Sequence[VariableT], target: VariableT
    ) -> Constraint:
        """Adds the element constraint: `variables[index] == target`.

        Args:
          index: The index of the variable that's being constrained.
          variables: A list of variables.
          target: The value that the variable must be equal to.

        Returns:
          An instance of the `Constraint` class.
        """

        if not variables:
            raise ValueError("add_element expects a non-empty variables array")

        if isinstance(index, IntegralTypes):
            variable: VariableT = list(variables)[int(index)]
            return self.add(variable == target)

        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.element.index = self.get_or_make_index(index)
        model_ct.element.vars.extend([self.get_or_make_index(x) for x in variables])
        model_ct.element.target = self.get_or_make_index(target)
        return ct

    def add_circuit(self, arcs: Sequence[ArcT]) -> Constraint:
        """Adds Circuit(arcs).

        Adds a circuit constraint from a sparse list of arcs that encode the graph.

        A circuit is a unique Hamiltonian path in a subgraph of the total
        graph. In case a node 'i' is not in the path, then there must be a
        loop arc 'i -> i' associated with a true literal. Otherwise
        this constraint will fail.

        Args:
          arcs: a list of arcs. An arc is a tuple (source_node, destination_node,
            literal). The arc is selected in the circuit if the literal is true.
            Both source_node and destination_node must be integers between 0 and the
            number of nodes - 1.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          ValueError: If the list of arcs is empty.
        """
        if not arcs:
            raise ValueError("add_circuit expects a non-empty array of arcs")
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        for arc in arcs:
            tail = cmh.assert_is_int32(arc[0])
            head = cmh.assert_is_int32(arc[1])
            lit = self.get_or_make_boolean_index(arc[2])
            model_ct.circuit.tails.append(tail)
            model_ct.circuit.heads.append(head)
            model_ct.circuit.literals.append(lit)
        return ct

    def add_multiple_circuit(self, arcs: Sequence[ArcT]) -> Constraint:
        """Adds a multiple circuit constraint, aka the 'VRP' constraint.

        The direct graph where arc #i (from tails[i] to head[i]) is present iff
        literals[i] is true must satisfy this set of properties:
        - #incoming arcs == 1 except for node 0.
        - #outgoing arcs == 1 except for node 0.
        - for node zero, #incoming arcs == #outgoing arcs.
        - There are no duplicate arcs.
        - Self-arcs are allowed except for node 0.
        - There is no cycle in this graph, except through node 0.

        Args:
          arcs: a list of arcs. An arc is a tuple (source_node, destination_node,
            literal). The arc is selected in the circuit if the literal is true.
            Both source_node and destination_node must be integers between 0 and the
            number of nodes - 1.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          ValueError: If the list of arcs is empty.
        """
        if not arcs:
            raise ValueError("add_multiple_circuit expects a non-empty array of arcs")
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        for arc in arcs:
            tail = cmh.assert_is_int32(arc[0])
            head = cmh.assert_is_int32(arc[1])
            lit = self.get_or_make_boolean_index(arc[2])
            model_ct.routes.tails.append(tail)
            model_ct.routes.heads.append(head)
            model_ct.routes.literals.append(lit)
        return ct

    def add_allowed_assignments(
        self,
        variables: Sequence[VariableT],
        tuples_list: Iterable[Sequence[IntegralT]],
    ) -> Constraint:
        """Adds AllowedAssignments(variables, tuples_list).

        An AllowedAssignments constraint is a constraint on an array of variables,
        which requires that when all variables are assigned values, the resulting
        array equals one of the  tuples in `tuple_list`.

        Args:
          variables: A list of variables.
          tuples_list: A list of admissible tuples. Each tuple must have the same
            length as the variables, and the ith value of a tuple corresponds to the
            ith variable.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          TypeError: If a tuple does not have the same size as the list of
              variables.
          ValueError: If the array of variables is empty.
        """

        if not variables:
            raise ValueError(
                "add_allowed_assignments expects a non-empty variables array"
            )

        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.table.vars.extend([self.get_or_make_index(x) for x in variables])
        arity = len(variables)
        for t in tuples_list:
            if len(t) != arity:
                raise TypeError("Tuple " + str(t) + " has the wrong arity")
            ar = []
            for v in t:
                ar.append(cmh.assert_is_int64(v))
            model_ct.table.values.extend(ar)
        return ct

    def add_forbidden_assignments(
        self,
        variables: Sequence[VariableT],
        tuples_list: Iterable[Sequence[IntegralT]],
    ) -> Constraint:
        """Adds add_forbidden_assignments(variables, [tuples_list]).

        A ForbiddenAssignments constraint is a constraint on an array of variables
        where the list of impossible combinations is provided in the tuples list.

        Args:
          variables: A list of variables.
          tuples_list: A list of forbidden tuples. Each tuple must have the same
            length as the variables, and the *i*th value of a tuple corresponds to
            the *i*th variable.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          TypeError: If a tuple does not have the same size as the list of
                     variables.
          ValueError: If the array of variables is empty.
        """

        if not variables:
            raise ValueError(
                "add_forbidden_assignments expects a non-empty variables array"
            )

        index = len(self.__model.constraints)
        ct = self.add_allowed_assignments(variables, tuples_list)
        self.__model.constraints[index].table.negated = True
        return ct

    def add_automaton(
        self,
        transition_variables: Sequence[VariableT],
        starting_state: IntegralT,
        final_states: Sequence[IntegralT],
        transition_triples: Sequence[Tuple[IntegralT, IntegralT, IntegralT]],
    ) -> Constraint:
        """Adds an automaton constraint.

        An automaton constraint takes a list of variables (of size *n*), an initial
        state, a set of final states, and a set of transitions. A transition is a
        triplet (*tail*, *transition*, *head*), where *tail* and *head* are states,
        and *transition* is the label of an arc from *head* to *tail*,
        corresponding to the value of one variable in the list of variables.

        This automaton will be unrolled into a flow with *n* + 1 phases. Each phase
        contains the possible states of the automaton. The first state contains the
        initial state. The last phase contains the final states.

        Between two consecutive phases *i* and *i* + 1, the automaton creates a set
        of arcs. For each transition (*tail*, *transition*, *head*), it will add
        an arc from the state *tail* of phase *i* and the state *head* of phase
        *i* + 1. This arc is labeled by the value *transition* of the variables
        `variables[i]`. That is, this arc can only be selected if `variables[i]`
        is assigned the value *transition*.

        A feasible solution of this constraint is an assignment of variables such
        that, starting from the initial state in phase 0, there is a path labeled by
        the values of the variables that ends in one of the final states in the
        final phase.

        Args:
          transition_variables: A non-empty list of variables whose values
            correspond to the labels of the arcs traversed by the automaton.
          starting_state: The initial state of the automaton.
          final_states: A non-empty list of admissible final states.
          transition_triples: A list of transitions for the automaton, in the
            following format (current_state, variable_value, next_state).

        Returns:
          An instance of the `Constraint` class.

        Raises:
          ValueError: if `transition_variables`, `final_states`, or
            `transition_triples` are empty.
        """

        if not transition_variables:
            raise ValueError(
                "add_automaton expects a non-empty transition_variables array"
            )
        if not final_states:
            raise ValueError("add_automaton expects some final states")

        if not transition_triples:
            raise ValueError("add_automaton expects some transition triples")

        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.automaton.vars.extend(
            [self.get_or_make_index(x) for x in transition_variables]
        )
        starting_state = cmh.assert_is_int64(starting_state)
        model_ct.automaton.starting_state = starting_state
        for v in final_states:
            v = cmh.assert_is_int64(v)
            model_ct.automaton.final_states.append(v)
        for t in transition_triples:
            if len(t) != 3:
                raise TypeError("Tuple " + str(t) + " has the wrong arity (!= 3)")
            tail = cmh.assert_is_int64(t[0])
            label = cmh.assert_is_int64(t[1])
            head = cmh.assert_is_int64(t[2])
            model_ct.automaton.transition_tail.append(tail)
            model_ct.automaton.transition_label.append(label)
            model_ct.automaton.transition_head.append(head)
        return ct

    def add_inverse(
        self,
        variables: Sequence[VariableT],
        inverse_variables: Sequence[VariableT],
    ) -> Constraint:
        """Adds Inverse(variables, inverse_variables).

        An inverse constraint enforces that if `variables[i]` is assigned a value
        `j`, then `inverse_variables[j]` is assigned a value `i`. And vice versa.

        Args:
          variables: An array of integer variables.
          inverse_variables: An array of integer variables.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          TypeError: if variables and inverse_variables have different lengths, or
              if they are empty.
        """

        if not variables or not inverse_variables:
            raise TypeError("The Inverse constraint does not accept empty arrays")
        if len(variables) != len(inverse_variables):
            raise TypeError(
                "In the inverse constraint, the two array variables and"
                " inverse_variables must have the same length."
            )
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.inverse.f_direct.extend([self.get_or_make_index(x) for x in variables])
        model_ct.inverse.f_inverse.extend(
            [self.get_or_make_index(x) for x in inverse_variables]
        )
        return ct

    def add_reservoir_constraint(
        self,
        times: Iterable[LinearExprT],
        level_changes: Iterable[LinearExprT],
        min_level: int,
        max_level: int,
    ) -> Constraint:
        """Adds Reservoir(times, level_changes, min_level, max_level).

        Maintains a reservoir level within bounds. The water level starts at 0, and
        at any time, it must be between min_level and max_level.

        If the affine expression `times[i]` is assigned a value t, then the current
        level changes by `level_changes[i]`, which is constant, at time t.

         Note that min level must be <= 0, and the max level must be >= 0. Please
         use fixed level_changes to simulate initial state.

         Therefore, at any time:
             sum(level_changes[i] if times[i] <= t) in [min_level, max_level]

        Args:
          times: A list of 1-var affine expressions (a * x + b) which specify the
            time of the filling or emptying the reservoir.
          level_changes: A list of integer values that specifies the amount of the
            emptying or filling. Currently, variable demands are not supported.
          min_level: At any time, the level of the reservoir must be greater or
            equal than the min level.
          max_level: At any time, the level of the reservoir must be less or equal
            than the max level.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          ValueError: if max_level < min_level.

          ValueError: if max_level < 0.

          ValueError: if min_level > 0
        """

        if max_level < min_level:
            raise ValueError("Reservoir constraint must have a max_level >= min_level")

        if max_level < 0:
            raise ValueError("Reservoir constraint must have a max_level >= 0")

        if min_level > 0:
            raise ValueError("Reservoir constraint must have a min_level <= 0")

        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.reservoir.time_exprs.extend(
            [self.parse_linear_expression(x) for x in times]
        )
        model_ct.reservoir.level_changes.extend(
            [self.parse_linear_expression(x) for x in level_changes]
        )
        model_ct.reservoir.min_level = min_level
        model_ct.reservoir.max_level = max_level
        return ct

    def add_reservoir_constraint_with_active(
        self,
        times: Iterable[LinearExprT],
        level_changes: Iterable[LinearExprT],
        actives: Iterable[LiteralT],
        min_level: int,
        max_level: int,
    ) -> Constraint:
        """Adds Reservoir(times, level_changes, actives, min_level, max_level).

        Maintains a reservoir level within bounds. The water level starts at 0, and
        at any time, it must be between min_level and max_level.

        If the variable `times[i]` is assigned a value t, and `actives[i]` is
        `True`, then the current level changes by `level_changes[i]`, which is
        constant,
        at time t.

         Note that min level must be <= 0, and the max level must be >= 0. Please
         use fixed level_changes to simulate initial state.

         Therefore, at any time:
             sum(level_changes[i] * actives[i] if times[i] <= t) in [min_level,
             max_level]


        The array of boolean variables 'actives', if defined, indicates which
        actions are actually performed.

        Args:
          times: A list of 1-var affine expressions (a * x + b) which specify the
            time of the filling or emptying the reservoir.
          level_changes: A list of integer values that specifies the amount of the
            emptying or filling. Currently, variable demands are not supported.
          actives: a list of boolean variables. They indicates if the
            emptying/refilling events actually take place.
          min_level: At any time, the level of the reservoir must be greater or
            equal than the min level.
          max_level: At any time, the level of the reservoir must be less or equal
            than the max level.

        Returns:
          An instance of the `Constraint` class.

        Raises:
          ValueError: if max_level < min_level.

          ValueError: if max_level < 0.

          ValueError: if min_level > 0
        """

        if max_level < min_level:
            raise ValueError("Reservoir constraint must have a max_level >= min_level")

        if max_level < 0:
            raise ValueError("Reservoir constraint must have a max_level >= 0")

        if min_level > 0:
            raise ValueError("Reservoir constraint must have a min_level <= 0")

        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.reservoir.time_exprs.extend(
            [self.parse_linear_expression(x) for x in times]
        )
        model_ct.reservoir.level_changes.extend(
            [self.parse_linear_expression(x) for x in level_changes]
        )
        model_ct.reservoir.active_literals.extend(
            [self.get_or_make_boolean_index(x) for x in actives]
        )
        model_ct.reservoir.min_level = min_level
        model_ct.reservoir.max_level = max_level
        return ct

    def add_map_domain(
        self, var: IntVar, bool_var_array: Iterable[IntVar], offset: IntegralT = 0
    ):
        """Adds `var == i + offset <=> bool_var_array[i] == true for all i`."""

        for i, bool_var in enumerate(bool_var_array):
            b_index = bool_var.index
            var_index = var.index
            model_ct = self.__model.constraints.add()
            model_ct.linear.vars.append(var_index)
            model_ct.linear.coeffs.append(1)
            offset_as_int = int(offset)
            model_ct.linear.domain.extend([offset_as_int + i, offset_as_int + i])
            model_ct.enforcement_literal.append(b_index)

            model_ct = self.__model.constraints.add()
            model_ct.linear.vars.append(var_index)
            model_ct.linear.coeffs.append(1)
            model_ct.enforcement_literal.append(-b_index - 1)
            if offset + i - 1 >= INT_MIN:
                model_ct.linear.domain.extend([INT_MIN, offset_as_int + i - 1])
            if offset + i + 1 <= INT_MAX:
                model_ct.linear.domain.extend([offset_as_int + i + 1, INT_MAX])

    def add_implication(self, a: LiteralT, b: LiteralT) -> Constraint:
        """Adds `a => b` (`a` implies `b`)."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.bool_or.literals.append(self.get_or_make_boolean_index(b))
        model_ct.enforcement_literal.append(self.get_or_make_boolean_index(a))
        return ct

    @overload
    def add_bool_or(self, literals: Iterable[LiteralT]) -> Constraint: ...

    @overload
    def add_bool_or(self, *literals: LiteralT) -> Constraint: ...

    def add_bool_or(self, *literals):
        """Adds `Or(literals) == true`: sum(literals) >= 1."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.bool_or.literals.extend(
            [
                self.get_or_make_boolean_index(x)
                for x in expand_generator_or_tuple(literals)
            ]
        )
        return ct

    @overload
    def add_at_least_one(self, literals: Iterable[LiteralT]) -> Constraint: ...

    @overload
    def add_at_least_one(self, *literals: LiteralT) -> Constraint: ...

    def add_at_least_one(self, *literals):
        """Same as `add_bool_or`: `sum(literals) >= 1`."""
        return self.add_bool_or(*literals)

    @overload
    def add_at_most_one(self, literals: Iterable[LiteralT]) -> Constraint: ...

    @overload
    def add_at_most_one(self, *literals: LiteralT) -> Constraint: ...

    def add_at_most_one(self, *literals):
        """Adds `AtMostOne(literals)`: `sum(literals) <= 1`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.at_most_one.literals.extend(
            [
                self.get_or_make_boolean_index(x)
                for x in expand_generator_or_tuple(literals)
            ]
        )
        return ct

    @overload
    def add_exactly_one(self, literals: Iterable[LiteralT]) -> Constraint: ...

    @overload
    def add_exactly_one(self, *literals: LiteralT) -> Constraint: ...

    def add_exactly_one(self, *literals):
        """Adds `ExactlyOne(literals)`: `sum(literals) == 1`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.exactly_one.literals.extend(
            [
                self.get_or_make_boolean_index(x)
                for x in expand_generator_or_tuple(literals)
            ]
        )
        return ct

    @overload
    def add_bool_and(self, literals: Iterable[LiteralT]) -> Constraint: ...

    @overload
    def add_bool_and(self, *literals: LiteralT) -> Constraint: ...

    def add_bool_and(self, *literals):
        """Adds `And(literals) == true`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.bool_and.literals.extend(
            [
                self.get_or_make_boolean_index(x)
                for x in expand_generator_or_tuple(literals)
            ]
        )
        return ct

    @overload
    def add_bool_xor(self, literals: Iterable[LiteralT]) -> Constraint: ...

    @overload
    def add_bool_xor(self, *literals: LiteralT) -> Constraint: ...

    def add_bool_xor(self, *literals):
        """Adds `XOr(literals) == true`.

        In contrast to add_bool_or and add_bool_and, it does not support
            .only_enforce_if().

        Args:
          *literals: the list of literals in the constraint.

        Returns:
          An `Constraint` object.
        """
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.bool_xor.literals.extend(
            [
                self.get_or_make_boolean_index(x)
                for x in expand_generator_or_tuple(literals)
            ]
        )
        return ct

    def add_min_equality(
        self, target: LinearExprT, exprs: Iterable[LinearExprT]
    ) -> Constraint:
        """Adds `target == Min(exprs)`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.lin_max.exprs.extend(
            [self.parse_linear_expression(x, True) for x in exprs]
        )
        model_ct.lin_max.target.CopyFrom(self.parse_linear_expression(target, True))
        return ct

    def add_max_equality(
        self, target: LinearExprT, exprs: Iterable[LinearExprT]
    ) -> Constraint:
        """Adds `target == Max(exprs)`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.lin_max.exprs.extend([self.parse_linear_expression(x) for x in exprs])
        model_ct.lin_max.target.CopyFrom(self.parse_linear_expression(target))
        return ct

    def add_division_equality(
        self, target: LinearExprT, num: LinearExprT, denom: LinearExprT
    ) -> Constraint:
        """Adds `target == num // denom` (integer division rounded towards 0)."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.int_div.exprs.append(self.parse_linear_expression(num))
        model_ct.int_div.exprs.append(self.parse_linear_expression(denom))
        model_ct.int_div.target.CopyFrom(self.parse_linear_expression(target))
        return ct

    def add_abs_equality(self, target: LinearExprT, expr: LinearExprT) -> Constraint:
        """Adds `target == Abs(expr)`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.lin_max.exprs.append(self.parse_linear_expression(expr))
        model_ct.lin_max.exprs.append(self.parse_linear_expression(expr, True))
        model_ct.lin_max.target.CopyFrom(self.parse_linear_expression(target))
        return ct

    def add_modulo_equality(
        self, target: LinearExprT, expr: LinearExprT, mod: LinearExprT
    ) -> Constraint:
        """Adds `target = expr % mod`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.int_mod.exprs.append(self.parse_linear_expression(expr))
        model_ct.int_mod.exprs.append(self.parse_linear_expression(mod))
        model_ct.int_mod.target.CopyFrom(self.parse_linear_expression(target))
        return ct

    def add_multiplication_equality(
        self,
        target: LinearExprT,
        *expressions: Union[Iterable[LinearExprT], LinearExprT],
    ) -> Constraint:
        """Adds `target == expressions[0] * .. * expressions[n]`."""
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.int_prod.exprs.extend(
            [
                self.parse_linear_expression(expr)
                for expr in expand_generator_or_tuple(expressions)
            ]
        )
        model_ct.int_prod.target.CopyFrom(self.parse_linear_expression(target))
        return ct

    # Scheduling support

    def new_interval_var(
        self, start: LinearExprT, size: LinearExprT, end: LinearExprT, name: str
    ) -> IntervalVar:
        """Creates an interval variable from start, size, and end.

        An interval variable is a constraint, that is itself used in other
        constraints like NoOverlap.

        Internally, it ensures that `start + size == end`.

        Args:
          start: The start of the interval. It must be of the form a * var + b.
          size: The size of the interval. It must be of the form a * var + b.
          end: The end of the interval. It must be of the form a * var + b.
          name: The name of the interval variable.

        Returns:
          An `IntervalVar` object.
        """

        start_expr = self.parse_linear_expression(start)
        size_expr = self.parse_linear_expression(size)
        end_expr = self.parse_linear_expression(end)
        if len(start_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: start must be 1-var affine or constant."
            )
        if len(size_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: size must be 1-var affine or constant."
            )
        if len(end_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: end must be 1-var affine or constant."
            )
        return IntervalVar(self.__model, start_expr, size_expr, end_expr, None, name)

    def new_interval_var_series(
        self,
        name: str,
        index: pd.Index,
        starts: Union[LinearExprT, pd.Series],
        sizes: Union[LinearExprT, pd.Series],
        ends: Union[LinearExprT, pd.Series],
    ) -> pd.Series:
        """Creates a series of interval variables with the given name.

        Args:
          name (str): Required. The name of the variable set.
          index (pd.Index): Required. The index to use for the variable set.
          starts (Union[LinearExprT, pd.Series]): The start of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          sizes (Union[LinearExprT, pd.Series]): The size of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          ends (Union[LinearExprT, pd.Series]): The ends of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.

        Returns:
          pd.Series: The interval variable set indexed by its corresponding
          dimensions.

        Raises:
          TypeError: if the `index` is invalid (e.g. a `DataFrame`).
          ValueError: if the `name` is not a valid identifier or already exists.
          ValueError: if the all the indexes do not match.
        """
        if not isinstance(index, pd.Index):
            raise TypeError("Non-index object is used as index")
        if not name.isidentifier():
            raise ValueError("name={} is not a valid identifier".format(name))

        starts = _convert_to_linear_expr_series_and_validate_index(starts, index)
        sizes = _convert_to_linear_expr_series_and_validate_index(sizes, index)
        ends = _convert_to_linear_expr_series_and_validate_index(ends, index)
        interval_array = []
        for i in index:
            interval_array.append(
                self.new_interval_var(
                    start=starts[i],
                    size=sizes[i],
                    end=ends[i],
                    name=f"{name}[{i}]",
                )
            )
        return pd.Series(index=index, data=interval_array)

    def new_fixed_size_interval_var(
        self, start: LinearExprT, size: IntegralT, name: str
    ) -> IntervalVar:
        """Creates an interval variable from start, and a fixed size.

        An interval variable is a constraint, that is itself used in other
        constraints like NoOverlap.

        Args:
          start: The start of the interval. It must be of the form a * var + b.
          size: The size of the interval. It must be an integer value.
          name: The name of the interval variable.

        Returns:
          An `IntervalVar` object.
        """
        size = cmh.assert_is_int64(size)
        start_expr = self.parse_linear_expression(start)
        size_expr = self.parse_linear_expression(size)
        end_expr = self.parse_linear_expression(start + size)
        if len(start_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: start must be affine or constant."
            )
        return IntervalVar(self.__model, start_expr, size_expr, end_expr, None, name)

    def new_fixed_size_interval_var_series(
        self,
        name: str,
        index: pd.Index,
        starts: Union[LinearExprT, pd.Series],
        sizes: Union[IntegralT, pd.Series],
    ) -> pd.Series:
        """Creates a series of interval variables with the given name.

        Args:
          name (str): Required. The name of the variable set.
          index (pd.Index): Required. The index to use for the variable set.
          starts (Union[LinearExprT, pd.Series]): The start of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          sizes (Union[IntegralT, pd.Series]): The fixed size of each interval in
            the set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.

        Returns:
          pd.Series: The interval variable set indexed by its corresponding
          dimensions.

        Raises:
          TypeError: if the `index` is invalid (e.g. a `DataFrame`).
          ValueError: if the `name` is not a valid identifier or already exists.
          ValueError: if the all the indexes do not match.
        """
        if not isinstance(index, pd.Index):
            raise TypeError("Non-index object is used as index")
        if not name.isidentifier():
            raise ValueError("name={} is not a valid identifier".format(name))

        starts = _convert_to_linear_expr_series_and_validate_index(starts, index)
        sizes = _convert_to_integral_series_and_validate_index(sizes, index)
        interval_array = []
        for i in index:
            interval_array.append(
                self.new_fixed_size_interval_var(
                    start=starts[i],
                    size=sizes[i],
                    name=f"{name}[{i}]",
                )
            )
        return pd.Series(index=index, data=interval_array)

    def new_optional_interval_var(
        self,
        start: LinearExprT,
        size: LinearExprT,
        end: LinearExprT,
        is_present: LiteralT,
        name: str,
    ) -> IntervalVar:
        """Creates an optional interval var from start, size, end, and is_present.

        An optional interval variable is a constraint, that is itself used in other
        constraints like NoOverlap. This constraint is protected by a presence
        literal that indicates if it is active or not.

        Internally, it ensures that `is_present` implies `start + size ==
        end`.

        Args:
          start: The start of the interval. It must be of the form a * var + b.
          size: The size of the interval. It must be of the form a * var + b.
          end: The end of the interval. It must be of the form a * var + b.
          is_present: A literal that indicates if the interval is active or not. A
            inactive interval is simply ignored by all constraints.
          name: The name of the interval variable.

        Returns:
          An `IntervalVar` object.
        """

        # Creates the IntervalConstraintProto object.
        is_present_index = self.get_or_make_boolean_index(is_present)
        start_expr = self.parse_linear_expression(start)
        size_expr = self.parse_linear_expression(size)
        end_expr = self.parse_linear_expression(end)
        if len(start_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: start must be affine or constant."
            )
        if len(size_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: size must be affine or constant."
            )
        if len(end_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: end must be affine or constant."
            )
        return IntervalVar(
            self.__model, start_expr, size_expr, end_expr, is_present_index, name
        )

    def new_optional_interval_var_series(
        self,
        name: str,
        index: pd.Index,
        starts: Union[LinearExprT, pd.Series],
        sizes: Union[LinearExprT, pd.Series],
        ends: Union[LinearExprT, pd.Series],
        are_present: Union[LiteralT, pd.Series],
    ) -> pd.Series:
        """Creates a series of interval variables with the given name.

        Args:
          name (str): Required. The name of the variable set.
          index (pd.Index): Required. The index to use for the variable set.
          starts (Union[LinearExprT, pd.Series]): The start of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          sizes (Union[LinearExprT, pd.Series]): The size of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          ends (Union[LinearExprT, pd.Series]): The ends of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          are_present (Union[LiteralT, pd.Series]): The performed literal of each
            interval in the set. If a `pd.Series` is passed in, it will be based on
            the corresponding values of the pd.Series.

        Returns:
          pd.Series: The interval variable set indexed by its corresponding
          dimensions.

        Raises:
          TypeError: if the `index` is invalid (e.g. a `DataFrame`).
          ValueError: if the `name` is not a valid identifier or already exists.
          ValueError: if the all the indexes do not match.
        """
        if not isinstance(index, pd.Index):
            raise TypeError("Non-index object is used as index")
        if not name.isidentifier():
            raise ValueError("name={} is not a valid identifier".format(name))

        starts = _convert_to_linear_expr_series_and_validate_index(starts, index)
        sizes = _convert_to_linear_expr_series_and_validate_index(sizes, index)
        ends = _convert_to_linear_expr_series_and_validate_index(ends, index)
        are_present = _convert_to_literal_series_and_validate_index(are_present, index)

        interval_array = []
        for i in index:
            interval_array.append(
                self.new_optional_interval_var(
                    start=starts[i],
                    size=sizes[i],
                    end=ends[i],
                    is_present=are_present[i],
                    name=f"{name}[{i}]",
                )
            )
        return pd.Series(index=index, data=interval_array)

    def new_optional_fixed_size_interval_var(
        self,
        start: LinearExprT,
        size: IntegralT,
        is_present: LiteralT,
        name: str,
    ) -> IntervalVar:
        """Creates an interval variable from start, and a fixed size.

        An interval variable is a constraint, that is itself used in other
        constraints like NoOverlap.

        Args:
          start: The start of the interval. It must be of the form a * var + b.
          size: The size of the interval. It must be an integer value.
          is_present: A literal that indicates if the interval is active or not. A
            inactive interval is simply ignored by all constraints.
          name: The name of the interval variable.

        Returns:
          An `IntervalVar` object.
        """
        size = cmh.assert_is_int64(size)
        start_expr = self.parse_linear_expression(start)
        size_expr = self.parse_linear_expression(size)
        end_expr = self.parse_linear_expression(start + size)
        if len(start_expr.vars) > 1:
            raise TypeError(
                "cp_model.new_interval_var: start must be affine or constant."
            )
        is_present_index = self.get_or_make_boolean_index(is_present)
        return IntervalVar(
            self.__model,
            start_expr,
            size_expr,
            end_expr,
            is_present_index,
            name,
        )

    def new_optional_fixed_size_interval_var_series(
        self,
        name: str,
        index: pd.Index,
        starts: Union[LinearExprT, pd.Series],
        sizes: Union[IntegralT, pd.Series],
        are_present: Union[LiteralT, pd.Series],
    ) -> pd.Series:
        """Creates a series of interval variables with the given name.

        Args:
          name (str): Required. The name of the variable set.
          index (pd.Index): Required. The index to use for the variable set.
          starts (Union[LinearExprT, pd.Series]): The start of each interval in the
            set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          sizes (Union[IntegralT, pd.Series]): The fixed size of each interval in
            the set. If a `pd.Series` is passed in, it will be based on the
            corresponding values of the pd.Series.
          are_present (Union[LiteralT, pd.Series]): The performed literal of each
            interval in the set. If a `pd.Series` is passed in, it will be based on
            the corresponding values of the pd.Series.

        Returns:
          pd.Series: The interval variable set indexed by its corresponding
          dimensions.

        Raises:
          TypeError: if the `index` is invalid (e.g. a `DataFrame`).
          ValueError: if the `name` is not a valid identifier or already exists.
          ValueError: if the all the indexes do not match.
        """
        if not isinstance(index, pd.Index):
            raise TypeError("Non-index object is used as index")
        if not name.isidentifier():
            raise ValueError("name={} is not a valid identifier".format(name))

        starts = _convert_to_linear_expr_series_and_validate_index(starts, index)
        sizes = _convert_to_integral_series_and_validate_index(sizes, index)
        are_present = _convert_to_literal_series_and_validate_index(are_present, index)
        interval_array = []
        for i in index:
            interval_array.append(
                self.new_optional_fixed_size_interval_var(
                    start=starts[i],
                    size=sizes[i],
                    is_present=are_present[i],
                    name=f"{name}[{i}]",
                )
            )
        return pd.Series(index=index, data=interval_array)

    def add_no_overlap(self, interval_vars: Iterable[IntervalVar]) -> Constraint:
        """Adds NoOverlap(interval_vars).

        A NoOverlap constraint ensures that all present intervals do not overlap
        in time.

        Args:
          interval_vars: The list of interval variables to constrain.

        Returns:
          An instance of the `Constraint` class.
        """
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.no_overlap.intervals.extend(
            [self.get_interval_index(x) for x in interval_vars]
        )
        return ct

    def add_no_overlap_2d(
        self,
        x_intervals: Iterable[IntervalVar],
        y_intervals: Iterable[IntervalVar],
    ) -> Constraint:
        """Adds NoOverlap2D(x_intervals, y_intervals).

        A NoOverlap2D constraint ensures that all present rectangles do not overlap
        on a plane. Each rectangle is aligned with the X and Y axis, and is defined
        by two intervals which represent its projection onto the X and Y axis.

        Furthermore, one box is optional if at least one of the x or y interval is
        optional.

        Args:
          x_intervals: The X coordinates of the rectangles.
          y_intervals: The Y coordinates of the rectangles.

        Returns:
          An instance of the `Constraint` class.
        """
        ct = Constraint(self)
        model_ct = self.__model.constraints[ct.index]
        model_ct.no_overlap_2d.x_intervals.extend(
            [self.get_interval_index(x) for x in x_intervals]
        )
        model_ct.no_overlap_2d.y_intervals.extend(
            [self.get_interval_index(x) for x in y_intervals]
        )
        return ct

    def add_cumulative(
        self,
        intervals: Iterable[IntervalVar],
        demands: Iterable[LinearExprT],
        capacity: LinearExprT,
    ) -> Constraint:
        """Adds Cumulative(intervals, demands, capacity).

        This constraint enforces that:

            for all t:
              sum(demands[i]
                if (start(intervals[i]) <= t < end(intervals[i])) and
                (intervals[i] is present)) <= capacity

        Args:
          intervals: The list of intervals.
          demands: The list of demands for each interval. Each demand must be >= 0.
            Each demand can be a 1-var affine expression (a * x + b).
          capacity: The maximum capacity of the cumulative constraint. It can be a
            1-var affine expression (a * x + b).

        Returns:
          An instance of the `Constraint` class.
        """
        cumulative = Constraint(self)
        model_ct = self.__model.constraints[cumulative.index]
        model_ct.cumulative.intervals.extend(
            [self.get_interval_index(x) for x in intervals]
        )
        for d in demands:
            model_ct.cumulative.demands.append(self.parse_linear_expression(d))
        model_ct.cumulative.capacity.CopyFrom(self.parse_linear_expression(capacity))
        return cumulative

    # Support for model cloning.
    def clone(self) -> "CpModel":
        """Reset the model, and creates a new one from a CpModelProto instance."""
        clone = CpModel()
        clone.proto.CopyFrom(self.proto)
        clone.rebuild_constant_map()
        return clone

    def rebuild_constant_map(self):
        """Internal method used during model cloning."""
        for i, var in enumerate(self.__model.variables):
            if len(var.domain) == 2 and var.domain[0] == var.domain[1]:
                self.__constant_map[var.domain[0]] = i

    def get_bool_var_from_proto_index(self, index: int) -> IntVar:
        """Returns an already created Boolean variable from its index."""
        if index < 0 or index >= len(self.__model.variables):
            raise ValueError(
                f"get_bool_var_from_proto_index: out of bound index {index}"
            )
        var = self.__model.variables[index]
        if len(var.domain) != 2 or var.domain[0] < 0 or var.domain[1] > 1:
            raise ValueError(
                f"get_bool_var_from_proto_index: index {index} does not reference"
                + " a Boolean variable"
            )

        return IntVar(self.__model, index, None)

    def get_int_var_from_proto_index(self, index: int) -> IntVar:
        """Returns an already created integer variable from its index."""
        if index < 0 or index >= len(self.__model.variables):
            raise ValueError(
                f"get_int_var_from_proto_index: out of bound index {index}"
            )
        return IntVar(self.__model, index, None)

    def get_interval_var_from_proto_index(self, index: int) -> IntervalVar:
        """Returns an already created interval variable from its index."""
        if index < 0 or index >= len(self.__model.constraints):
            raise ValueError(
                f"get_interval_var_from_proto_index: out of bound index {index}"
            )
        ct = self.__model.constraints[index]
        if not ct.HasField("interval"):
            raise ValueError(
                f"get_interval_var_from_proto_index: index {index} does not"
                " reference an" + " interval variable"
            )

        return IntervalVar(self.__model, index, None, None, None, None)

    # Helpers.

    def __str__(self):
        return str(self.__model)

    @property
    def proto(self) -> cp_model_pb2.CpModelProto:
        """Returns the underlying CpModelProto."""
        return self.__model

    def negated(self, index: int) -> int:
        return -index - 1

    def get_or_make_index(self, arg: VariableT) -> int:
        """Returns the index of a variable, its negation, or a number."""
        if isinstance(arg, IntVar):
            return arg.index
        if (
            isinstance(arg, _ProductCst)
            and isinstance(arg.expression(), IntVar)
            and arg.coefficient() == -1
        ):
            return -arg.expression().index - 1
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_int64(arg)
            return self.get_or_make_index_from_constant(arg)
        raise TypeError("NotSupported: model.get_or_make_index(" + str(arg) + ")")

    def get_or_make_boolean_index(self, arg: LiteralT) -> int:
        """Returns an index from a boolean expression."""
        if isinstance(arg, IntVar):
            self.assert_is_boolean_variable(arg)
            return arg.index
        if isinstance(arg, _NotBooleanVariable):
            self.assert_is_boolean_variable(arg.negated())
            return arg.index
        if isinstance(arg, IntegralTypes):
            arg = cmh.assert_is_zero_or_one(arg)
            return self.get_or_make_index_from_constant(arg)
        if cmh.is_boolean(arg):
            return self.get_or_make_index_from_constant(int(arg))
        raise TypeError(f"not supported: model.get_or_make_boolean_index({arg})")

    def get_interval_index(self, arg: IntervalVar) -> int:
        if not isinstance(arg, IntervalVar):
            raise TypeError("NotSupported: model.get_interval_index(%s)" % arg)
        return arg.index

    def get_or_make_index_from_constant(self, value: IntegralT) -> int:
        if value in self.__constant_map:
            return self.__constant_map[value]
        index = len(self.__model.variables)
        self.__model.variables.add(domain=[value, value])
        self.__constant_map[value] = index
        return index

    def var_index_to_var_proto(
        self, var_index: int
    ) -> cp_model_pb2.IntegerVariableProto:
        if var_index >= 0:
            return self.__model.variables[var_index]
        else:
            return self.__model.variables[-var_index - 1]

    def parse_linear_expression(
        self, linear_expr: LinearExprT, negate: bool = False
    ) -> cp_model_pb2.LinearExpressionProto:
        """Returns a LinearExpressionProto built from a LinearExpr instance."""
        result: cp_model_pb2.LinearExpressionProto = (
            cp_model_pb2.LinearExpressionProto()
        )
        mult = -1 if negate else 1
        if isinstance(linear_expr, IntegralTypes):
            result.offset = int(linear_expr) * mult
            return result

        if isinstance(linear_expr, IntVar):
            result.vars.append(self.get_or_make_index(linear_expr))
            result.coeffs.append(mult)
            return result

        coeffs_map, constant = cast(LinearExpr, linear_expr).get_integer_var_value_map()
        result.offset = constant * mult
        for t in coeffs_map.items():
            if not isinstance(t[0], IntVar):
                raise TypeError("Wrong argument" + str(t))
            c = cmh.assert_is_int64(t[1])
            result.vars.append(t[0].index)
            result.coeffs.append(c * mult)
        return result

    def _set_objective(self, obj: ObjLinearExprT, minimize: bool):
        """Sets the objective of the model."""
        self.clear_objective()
        if isinstance(obj, IntVar):
            self.__model.objective.vars.append(obj.index)
            self.__model.objective.offset = 0
            if minimize:
                self.__model.objective.coeffs.append(1)
                self.__model.objective.scaling_factor = 1
            else:
                self.__model.objective.coeffs.append(-1)
                self.__model.objective.scaling_factor = -1
        elif isinstance(obj, LinearExpr):
            coeffs_map, constant, is_integer = obj.get_float_var_value_map()
            if is_integer:
                if minimize:
                    self.__model.objective.scaling_factor = 1
                    self.__model.objective.offset = constant
                else:
                    self.__model.objective.scaling_factor = -1
                    self.__model.objective.offset = -constant
                for v, c in coeffs_map.items():
                    c_as_int = int(c)
                    self.__model.objective.vars.append(v.index)
                    if minimize:
                        self.__model.objective.coeffs.append(c_as_int)
                    else:
                        self.__model.objective.coeffs.append(-c_as_int)
            else:
                self.__model.floating_point_objective.maximize = not minimize
                self.__model.floating_point_objective.offset = constant
                for v, c in coeffs_map.items():
                    self.__model.floating_point_objective.coeffs.append(c)
                    self.__model.floating_point_objective.vars.append(v.index)
        elif isinstance(obj, IntegralTypes):
            self.__model.objective.offset = int(obj)
            self.__model.objective.scaling_factor = 1
        else:
            raise TypeError("TypeError: " + str(obj) + " is not a valid objective")

    def minimize(self, obj: ObjLinearExprT):
        """Sets the objective of the model to minimize(obj)."""
        self._set_objective(obj, minimize=True)

    def maximize(self, obj: ObjLinearExprT):
        """Sets the objective of the model to maximize(obj)."""
        self._set_objective(obj, minimize=False)

    def has_objective(self) -> bool:
        return self.__model.HasField("objective") or self.__model.HasField(
            "floating_point_objective"
        )

    def clear_objective(self):
        self.__model.ClearField("objective")
        self.__model.ClearField("floating_point_objective")

    def add_decision_strategy(
        self,
        variables: Sequence[IntVar],
        var_strategy: cp_model_pb2.DecisionStrategyProto.VariableSelectionStrategy,
        domain_strategy: cp_model_pb2.DecisionStrategyProto.DomainReductionStrategy,
    ) -> None:
        """Adds a search strategy to the model.

        Args:
          variables: a list of variables this strategy will assign.
          var_strategy: heuristic to choose the next variable to assign.
          domain_strategy: heuristic to reduce the domain of the selected variable.
            Currently, this is advanced code: the union of all strategies added to
            the model must be complete, i.e. instantiates all variables. Otherwise,
            solve() will fail.
        """

        strategy: cp_model_pb2.DecisionStrategyProto = (
            self.__model.search_strategy.add()
        )
        for v in variables:
            expr = strategy.exprs.add()
            if v.index >= 0:
                expr.vars.append(v.index)
                expr.coeffs.append(1)
            else:
                expr.vars.append(self.negated(v.index))
                expr.coeffs.append(-1)
                expr.offset = 1

        strategy.variable_selection_strategy = var_strategy
        strategy.domain_reduction_strategy = domain_strategy

    def model_stats(self) -> str:
        """Returns a string containing some model statistics."""
        return swig_helper.CpSatHelper.model_stats(self.__model)

    def validate(self) -> str:
        """Returns a string indicating that the model is invalid."""
        return swig_helper.CpSatHelper.validate_model(self.__model)

    def export_to_file(self, file: str) -> bool:
        """Write the model as a protocol buffer to 'file'.

        Args:
          file: file to write the model to. If the filename ends with 'txt', the
            model will be written as a text file, otherwise, the binary format will
            be used.

        Returns:
          True if the model was correctly written.
        """
        return swig_helper.CpSatHelper.write_model_to_file(self.__model, file)

    def add_hint(self, var: IntVar, value: int) -> None:
        """Adds 'var == value' as a hint to the solver."""
        self.__model.solution_hint.vars.append(self.get_or_make_index(var))
        self.__model.solution_hint.values.append(value)

    def clear_hints(self):
        """Removes any solution hint from the model."""
        self.__model.ClearField("solution_hint")

    def add_assumption(self, lit: LiteralT) -> None:
        """Adds the literal to the model as assumptions."""
        self.__model.assumptions.append(self.get_or_make_boolean_index(lit))

    def add_assumptions(self, literals: Iterable[LiteralT]) -> None:
        """Adds the literals to the model as assumptions."""
        for lit in literals:
            self.add_assumption(lit)

    def clear_assumptions(self) -> None:
        """Removes all assumptions from the model."""
        self.__model.ClearField("assumptions")

    # Helpers.
    def assert_is_boolean_variable(self, x: LiteralT) -> None:
        if isinstance(x, IntVar):
            var = self.__model.variables[x.index]
            if len(var.domain) != 2 or var.domain[0] < 0 or var.domain[1] > 1:
                raise TypeError("TypeError: " + str(x) + " is not a boolean variable")
        elif not isinstance(x, _NotBooleanVariable):
            raise TypeError("TypeError: " + str(x) + " is not a boolean variable")

    # Compatibility with pre PEP8
    # pylint: disable=invalid-name

    def Name(self) -> str:
        return self.name

    def SetName(self, name: str) -> None:
        self.name = name

    def Proto(self) -> cp_model_pb2.CpModelProto:
        return self.proto

    NewIntVar = new_int_var
    NewIntVarFromDomain = new_int_var_from_domain
    NewBoolVar = new_bool_var
    NewConstant = new_constant
    NewIntVarSeries = new_int_var_series
    NewBoolVarSeries = new_bool_var_series
    AddLinearConstraint = add_linear_constraint
    AddLinearExpressionInDomain = add_linear_expression_in_domain
    Add = add
    AddAllDifferent = add_all_different
    AddElement = add_element
    AddCircuit = add_circuit
    AddMultipleCircuit = add_multiple_circuit
    AddAllowedAssignments = add_allowed_assignments
    AddForbiddenAssignments = add_forbidden_assignments
    AddAutomaton = add_automaton
    AddInverse = add_inverse
    AddReservoirConstraint = add_reservoir_constraint
    AddReservoirConstraintWithActive = add_reservoir_constraint_with_active
    AddImplication = add_implication
    AddBoolOr = add_bool_or
    AddAtLeastOne = add_at_least_one
    AddAtMostOne = add_at_most_one
    AddExactlyOne = add_exactly_one
    AddBoolAnd = add_bool_and
    AddBoolXOr = add_bool_xor
    AddMinEquality = add_min_equality
    AddMaxEquality = add_max_equality
    AddDivisionEquality = add_division_equality
    AddAbsEquality = add_abs_equality
    AddModuloEquality = add_modulo_equality
    AddMultiplicationEquality = add_multiplication_equality
    NewIntervalVar = new_interval_var
    NewIntervalVarSeries = new_interval_var_series
    NewFixedSizeIntervalVar = new_fixed_size_interval_var
    NewOptionalIntervalVar = new_optional_interval_var
    NewOptionalIntervalVarSeries = new_optional_interval_var_series
    NewOptionalFixedSizeIntervalVar = new_optional_fixed_size_interval_var
    NewOptionalFixedSizeIntervalVarSeries = new_optional_fixed_size_interval_var_series
    AddNoOverlap = add_no_overlap
    AddNoOverlap2D = add_no_overlap_2d
    AddCumulative = add_cumulative
    Clone = clone
    GetBoolVarFromProtoIndex = get_bool_var_from_proto_index
    GetIntVarFromProtoIndex = get_int_var_from_proto_index
    GetIntervalVarFromProtoIndex = get_interval_var_from_proto_index
    Minimize = minimize
    Maximize = maximize
    HasObjective = has_objective
    ClearObjective = clear_objective
    AddDecisionStrategy = add_decision_strategy
    ModelStats = model_stats
    Validate = validate
    ExportToFile = export_to_file
    AddHint = add_hint
    ClearHints = clear_hints
    AddAssumption = add_assumption
    AddAssumptions = add_assumptions
    ClearAssumptions = clear_assumptions

    # pylint: enable=invalid-name


@overload
def expand_generator_or_tuple(
    args: Union[Tuple[LiteralT, ...], Iterable[LiteralT]]
) -> Union[Iterable[LiteralT], LiteralT]: ...


@overload
def expand_generator_or_tuple(
    args: Union[Tuple[LinearExprT, ...], Iterable[LinearExprT]]
) -> Union[Iterable[LinearExprT], LinearExprT]: ...


def expand_generator_or_tuple(args):
    if hasattr(args, "__len__"):  # Tuple
        if len(args) != 1:
            return args
        if isinstance(args[0], (NumberTypes, LinearExpr)):
            return args
    # Generator
    return args[0]


def evaluate_linear_expr(
    expression: LinearExprT, solution: cp_model_pb2.CpSolverResponse
) -> int:
    """Evaluate a linear expression against a solution."""
    if isinstance(expression, IntegralTypes):
        return int(expression)
    if not isinstance(expression, LinearExpr):
        raise TypeError("Cannot interpret %s as a linear expression." % expression)

    value = 0
    to_process = [(expression, 1)]
    while to_process:
        expr, coeff = to_process.pop()
        if isinstance(expr, IntegralTypes):
            value += int(expr) * coeff
        elif isinstance(expr, _ProductCst):
            to_process.append((expr.expression(), coeff * expr.coefficient()))
        elif isinstance(expr, _Sum):
            to_process.append((expr.left(), coeff))
            to_process.append((expr.right(), coeff))
        elif isinstance(expr, _SumArray):
            for e in expr.expressions():
                to_process.append((e, coeff))
            value += expr.constant() * coeff
        elif isinstance(expr, _WeightedSum):
            for e, c in zip(expr.expressions(), expr.coefficients()):
                to_process.append((e, coeff * c))
            value += expr.constant() * coeff
        elif isinstance(expr, IntVar):
            value += coeff * solution.solution[expr.index]
        elif isinstance(expr, _NotBooleanVariable):
            value += coeff * (1 - solution.solution[expr.negated().index])
        else:
            raise TypeError(f"Cannot interpret {expr} as a linear expression.")

    return value


def evaluate_boolean_expression(
    literal: LiteralT, solution: cp_model_pb2.CpSolverResponse
) -> bool:
    """Evaluate a boolean expression against a solution."""
    if isinstance(literal, IntegralTypes):
        return bool(literal)
    elif isinstance(literal, IntVar) or isinstance(literal, _NotBooleanVariable):
        index: int = cast(Union[IntVar, _NotBooleanVariable], literal).index
        if index >= 0:
            return bool(solution.solution[index])
        else:
            return not solution.solution[-index - 1]
    else:
        raise TypeError(f"Cannot interpret {literal} as a boolean expression.")


class CpSolver:
    """Main solver class.

    The purpose of this class is to search for a solution to the model provided
    to the solve() method.

    Once solve() is called, this class allows inspecting the solution found
    with the value() and boolean_value() methods, as well as general statistics
    about the solve procedure.
    """

    def __init__(self) -> None:
        self.__solution: Optional[cp_model_pb2.CpSolverResponse] = None
        self.parameters: sat_parameters_pb2.SatParameters = (
            sat_parameters_pb2.SatParameters()
        )
        self.log_callback: Optional[Callable[[str], None]] = None
        self.best_bound_callback: Optional[Callable[[float], None]] = None
        self.__solve_wrapper: Optional[swig_helper.SolveWrapper] = None
        self.__lock: threading.Lock = threading.Lock()

    def solve(
        self,
        model: CpModel,
        solution_callback: Optional["CpSolverSolutionCallback"] = None,
    ) -> cp_model_pb2.CpSolverStatus:
        """Solves a problem and passes each solution to the callback if not null."""
        with self.__lock:
            self.__solve_wrapper = swig_helper.SolveWrapper()

        self.__solve_wrapper.set_parameters(self.parameters)
        if solution_callback is not None:
            self.__solve_wrapper.add_solution_callback(solution_callback)

        if self.log_callback is not None:
            self.__solve_wrapper.add_log_callback(self.log_callback)

        if self.best_bound_callback is not None:
            self.__solve_wrapper.add_best_bound_callback(self.best_bound_callback)

        solution: cp_model_pb2.CpSolverResponse = self.__solve_wrapper.solve(
            model.proto
        )
        self.__solution = solution

        if solution_callback is not None:
            self.__solve_wrapper.clear_solution_callback(solution_callback)

        with self.__lock:
            self.__solve_wrapper = None

        return solution.status

    def stop_search(self) -> None:
        """Stops the current search asynchronously."""
        with self.__lock:
            if self.__solve_wrapper:
                self.__solve_wrapper.stop_search()

    def value(self, expression: LinearExprT) -> int:
        """Returns the value of a linear expression after solve."""
        return evaluate_linear_expr(expression, self._solution)

    def values(self, variables: _IndexOrSeries) -> pd.Series:
        """Returns the values of the input variables.

        If `variables` is a `pd.Index`, then the output will be indexed by the
        variables. If `variables` is a `pd.Series` indexed by the underlying
        dimensions, then the output will be indexed by the same underlying
        dimensions.

        Args:
          variables (Union[pd.Index, pd.Series]): The set of variables from which to
            get the values.

        Returns:
          pd.Series: The values of all variables in the set.
        """
        solution = self._solution
        return _attribute_series(
            func=lambda v: solution.solution[v.index],
            values=variables,
        )

    def boolean_value(self, literal: LiteralT) -> bool:
        """Returns the boolean value of a literal after solve."""
        return evaluate_boolean_expression(literal, self._solution)

    def boolean_values(self, variables: _IndexOrSeries) -> pd.Series:
        """Returns the values of the input variables.

        If `variables` is a `pd.Index`, then the output will be indexed by the
        variables. If `variables` is a `pd.Series` indexed by the underlying
        dimensions, then the output will be indexed by the same underlying
        dimensions.

        Args:
          variables (Union[pd.Index, pd.Series]): The set of variables from which to
            get the values.

        Returns:
          pd.Series: The values of all variables in the set.
        """
        solution = self._solution
        return _attribute_series(
            func=lambda literal: evaluate_boolean_expression(literal, solution),
            values=variables,
        )

    @property
    def objective_value(self) -> float:
        """Returns the value of the objective after solve."""
        return self._solution.objective_value

    @property
    def best_objective_bound(self) -> float:
        """Returns the best lower (upper) bound found when min(max)imizing."""
        return self._solution.best_objective_bound

    @property
    def num_booleans(self) -> int:
        """Returns the number of boolean variables managed by the SAT solver."""
        return self._solution.num_booleans

    @property
    def num_conflicts(self) -> int:
        """Returns the number of conflicts since the creation of the solver."""
        return self._solution.num_conflicts

    @property
    def num_branches(self) -> int:
        """Returns the number of search branches explored by the solver."""
        return self._solution.num_branches

    @property
    def wall_time(self) -> float:
        """Returns the wall time in seconds since the creation of the solver."""
        return self._solution.wall_time

    @property
    def user_time(self) -> float:
        """Returns the user time in seconds since the creation of the solver."""
        return self._solution.user_time

    @property
    def response_proto(self) -> cp_model_pb2.CpSolverResponse:
        """Returns the response object."""
        return self._solution

    def response_stats(self) -> str:
        """Returns some statistics on the solution found as a string."""
        return swig_helper.CpSatHelper.solver_response_stats(self._solution)

    def sufficient_assumptions_for_infeasibility(self) -> Sequence[int]:
        """Returns the indices of the infeasible assumptions."""
        return self._solution.sufficient_assumptions_for_infeasibility

    def status_name(self, status: Optional[Any] = None) -> str:
        """Returns the name of the status returned by solve()."""
        if status is None:
            status = self._solution.status
        return cp_model_pb2.CpSolverStatus.Name(status)

    def solution_info(self) -> str:
        """Returns some information on the solve process.

        Returns some information on how the solution was found, or the reason
        why the model or the parameters are invalid.

        Raises:
          RuntimeError: if solve() has not been called.
        """
        return self._solution.solution_info

    @property
    def _solution(self) -> cp_model_pb2.CpSolverResponse:
        """Checks solve() has been called, and returns the solution."""
        if self.__solution is None:
            raise RuntimeError("solve() has not been called.")
        return self.__solution

    # Compatibility with pre PEP8
    # pylint: disable=invalid-name

    def BestObjectiveBound(self) -> float:
        return self.best_objective_bound

    def BooleanValue(self, literal: LiteralT) -> bool:
        return self.boolean_value(literal)

    def BooleanValues(self, variables: _IndexOrSeries) -> pd.Series:
        return self.boolean_values(variables)

    def NumBooleans(self) -> int:
        return self.num_booleans

    def NumConflicts(self) -> int:
        return self.num_conflicts

    def NumBranches(self) -> int:
        return self.num_branches

    def ObjectiveValue(self) -> float:
        return self.objective_value

    def ResponseProto(self) -> cp_model_pb2.CpSolverResponse:
        return self.response_proto

    def ResponseStats(self) -> str:
        return self.response_stats()

    def Solve(
        self,
        model: CpModel,
        solution_callback: Optional["CpSolverSolutionCallback"] = None,
    ) -> cp_model_pb2.CpSolverStatus:
        return self.solve(model, solution_callback)

    def SolutionInfo(self) -> str:
        return self.solution_info()

    def StatusName(self, status: Optional[Any] = None) -> str:
        return self.status_name(status)

    def StopSearch(self) -> None:
        self.stop_search()

    def SufficientAssumptionsForInfeasibility(self) -> Sequence[int]:
        return self.sufficient_assumptions_for_infeasibility()

    def UserTime(self) -> float:
        return self.user_time

    def Value(self, expression: LinearExprT) -> int:
        return self.value(expression)

    def Values(self, variables: _IndexOrSeries) -> pd.Series:
        return self.values(variables)

    def WallTime(self) -> float:
        return self.wall_time

    def SolveWithSolutionCallback(
        self, model: CpModel, callback: "CpSolverSolutionCallback"
    ) -> cp_model_pb2.CpSolverStatus:
        """DEPRECATED Use solve() with the callback argument."""
        warnings.warn(
            "solve_with_solution_callback is deprecated; use solve() with"
            + "the callback argument.",
            DeprecationWarning,
        )
        return self.solve(model, callback)

    def SearchForAllSolutions(
        self, model: CpModel, callback: "CpSolverSolutionCallback"
    ) -> cp_model_pb2.CpSolverStatus:
        """DEPRECATED Use solve() with the right parameter.

        Search for all solutions of a satisfiability problem.

        This method searches for all feasible solutions of a given model.
        Then it feeds the solution to the callback.

        Note that the model cannot contain an objective.

        Args:
          model: The model to solve.
          callback: The callback that will be called at each solution.

        Returns:
          The status of the solve:

          * *FEASIBLE* if some solutions have been found
          * *INFEASIBLE* if the solver has proved there are no solution
          * *OPTIMAL* if all solutions have been found
        """
        warnings.warn(
            "search_for_all_solutions is deprecated; use solve() with"
            + "enumerate_all_solutions = True.",
            DeprecationWarning,
        )
        if model.has_objective():
            raise TypeError(
                "Search for all solutions is only defined on satisfiability problems"
            )
        # Store old parameter.
        enumerate_all = self.parameters.enumerate_all_solutions
        self.parameters.enumerate_all_solutions = True

        status: cp_model_pb2.CpSolverStatus = self.solve(model, callback)

        # Restore parameter.
        self.parameters.enumerate_all_solutions = enumerate_all
        return status


# pylint: enable=invalid-name


class CpSolverSolutionCallback(swig_helper.SolutionCallback):
    """Solution callback.

    This class implements a callback that will be called at each new solution
    found during search.

    The method on_solution_callback() will be called by the solver, and must be
    implemented. The current solution can be queried using the boolean_value()
    and value() methods.

    These methods returns the same information as their counterpart in the
    `CpSolver` class.
    """

    def __init__(self) -> None:
        swig_helper.SolutionCallback.__init__(self)

    def OnSolutionCallback(self) -> None:
        """Proxy for the same method in snake case."""
        self.on_solution_callback()

    def boolean_value(self, lit: LiteralT) -> bool:
        """Returns the boolean value of a boolean literal.

        Args:
            lit: A boolean variable or its negation.

        Returns:
            The Boolean value of the literal in the solution.

        Raises:
            RuntimeError: if `lit` is not a boolean variable or its negation.
        """
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        if isinstance(lit, IntegralTypes):
            return bool(lit)
        if isinstance(lit, IntVar) or isinstance(lit, _NotBooleanVariable):
            return self.SolutionBooleanValue(
                cast(Union[IntVar, _NotBooleanVariable], lit).index
            )
        if cmh.is_boolean(lit):
            return bool(lit)
        raise TypeError(f"Cannot interpret {lit} as a boolean expression.")

    def value(self, expression: LinearExprT) -> int:
        """Evaluates an linear expression in the current solution.

        Args:
            expression: a linear expression of the model.

        Returns:
            An integer value equal to the evaluation of the linear expression
            against the current solution.

        Raises:
            RuntimeError: if 'expression' is not a LinearExpr.
        """
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")

        value = 0
        to_process = [(expression, 1)]
        while to_process:
            expr, coeff = to_process.pop()
            if isinstance(expr, IntegralTypes):
                value += int(expr) * coeff
            elif isinstance(expr, _ProductCst):
                to_process.append((expr.expression(), coeff * expr.coefficient()))
            elif isinstance(expr, _Sum):
                to_process.append((expr.left(), coeff))
                to_process.append((expr.right(), coeff))
            elif isinstance(expr, _SumArray):
                for e in expr.expressions():
                    to_process.append((e, coeff))
                    value += expr.constant() * coeff
            elif isinstance(expr, _WeightedSum):
                for e, c in zip(expr.expressions(), expr.coefficients()):
                    to_process.append((e, coeff * c))
                value += expr.constant() * coeff
            elif isinstance(expr, IntVar):
                value += coeff * self.SolutionIntegerValue(expr.index)
            elif isinstance(expr, _NotBooleanVariable):
                value += coeff * (1 - self.SolutionIntegerValue(expr.negated().index))
            else:
                raise TypeError(
                    f"cannot interpret {expression} as a linear expression."
                )

        return value

    def has_response(self) -> bool:
        return self.HasResponse()

    def stop_search(self) -> None:
        """Stops the current search asynchronously."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        self.StopSearch()

    @property
    def objective_value(self) -> float:
        """Returns the value of the objective after solve."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.ObjectiveValue()

    @property
    def best_objective_bound(self) -> float:
        """Returns the best lower (upper) bound found when min(max)imizing."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.BestObjectiveBound()

    @property
    def num_booleans(self) -> int:
        """Returns the number of boolean variables managed by the SAT solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.NumBooleans()

    @property
    def num_conflicts(self) -> int:
        """Returns the number of conflicts since the creation of the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.NumConflicts()

    @property
    def num_branches(self) -> int:
        """Returns the number of search branches explored by the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.NumBranches()

    @property
    def num_integer_propagations(self) -> int:
        """Returns the number of integer propagations done by the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.NumIntegerPropagations()

    @property
    def num_boolean_propagations(self) -> int:
        """Returns the number of Boolean propagations done by the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.NumBooleanPropagations()

    @property
    def deterministic_time(self) -> float:
        """Returns the determistic time in seconds since the creation of the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.DeterministicTime()

    @property
    def wall_time(self) -> float:
        """Returns the wall time in seconds since the creation of the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.WallTime()

    @property
    def user_time(self) -> float:
        """Returns the user time in seconds since the creation of the solver."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.UserTime()

    @property
    def response_proto(self) -> cp_model_pb2.CpSolverResponse:
        """Returns the response object."""
        if not self.has_response():
            raise RuntimeError("solve() has not been called.")
        return self.Response()

    # Compatibility with pre PEP8
    # pylint: disable=invalid-name
    Value = value
    BooleanValue = boolean_value
    # pylint: enable=invalid-name


class ObjectiveSolutionPrinter(CpSolverSolutionCallback):
    """Display the objective value and time of intermediate solutions."""

    def __init__(self) -> None:
        CpSolverSolutionCallback.__init__(self)
        self.__solution_count = 0
        self.__start_time = time.time()

    def on_solution_callback(self) -> None:
        """Called on each new solution."""
        current_time = time.time()
        obj = self.objective_value
        print(
            "Solution %i, time = %0.2f s, objective = %i"
            % (self.__solution_count, current_time - self.__start_time, obj)
        )
        self.__solution_count += 1

    def solution_count(self) -> int:
        """Returns the number of solutions found."""
        return self.__solution_count


class VarArrayAndObjectiveSolutionPrinter(CpSolverSolutionCallback):
    """Print intermediate solutions (objective, variable values, time)."""

    def __init__(self, variables: Sequence[IntVar]) -> None:
        CpSolverSolutionCallback.__init__(self)
        self.__variables: Sequence[IntVar] = variables
        self.__solution_count: int = 0
        self.__start_time: float = time.time()

    def on_solution_callback(self) -> None:
        """Called on each new solution."""
        current_time = time.time()
        obj = self.objective_value
        print(
            "Solution %i, time = %0.2f s, objective = %i"
            % (self.__solution_count, current_time - self.__start_time, obj)
        )
        for v in self.__variables:
            print("  %s = %i" % (v, self.value(v)), end=" ")
        print()
        self.__solution_count += 1

    @property
    def solution_count(self) -> int:
        """Returns the number of solutions found."""
        return self.__solution_count


class VarArraySolutionPrinter(CpSolverSolutionCallback):
    """Print intermediate solutions (variable values, time)."""

    def __init__(self, variables: Sequence[IntVar]) -> None:
        CpSolverSolutionCallback.__init__(self)
        self.__variables: Sequence[IntVar] = variables
        self.__solution_count: int = 0
        self.__start_time: float = time.time()

    def on_solution_callback(self) -> None:
        """Called on each new solution."""
        current_time = time.time()
        print(
            "Solution %i, time = %0.2f s"
            % (self.__solution_count, current_time - self.__start_time)
        )
        for v in self.__variables:
            print("  %s = %i" % (v, self.value(v)), end=" ")
        print()
        self.__solution_count += 1

    @property
    def solution_count(self) -> int:
        """Returns the number of solutions found."""
        return self.__solution_count


def _get_index(obj: _IndexOrSeries) -> pd.Index:
    """Returns the indices of `obj` as a `pd.Index`."""
    if isinstance(obj, pd.Series):
        return obj.index
    return obj


def _attribute_series(
    *,
    func: Callable[[IntVar], IntegralT],
    values: _IndexOrSeries,
) -> pd.Series:
    """Returns the attributes of `values`.

    Args:
      func: The function to call for getting the attribute data.
      values: The values that the function will be applied (element-wise) to.

    Returns:
      pd.Series: The attribute values.
    """
    return pd.Series(
        data=[func(v) for v in values],
        index=_get_index(values),
    )


def _convert_to_integral_series_and_validate_index(
    value_or_series: Union[IntegralT, pd.Series], index: pd.Index
) -> pd.Series:
    """Returns a pd.Series of the given index with the corresponding values.

    Args:
      value_or_series: the values to be converted (if applicable).
      index: the index of the resulting pd.Series.

    Returns:
      pd.Series: The set of values with the given index.

    Raises:
      TypeError: If the type of `value_or_series` is not recognized.
      ValueError: If the index does not match.
    """
    if isinstance(value_or_series, IntegralTypes):
        result = pd.Series(data=value_or_series, index=index)
    elif isinstance(value_or_series, pd.Series):
        if value_or_series.index.equals(index):
            result = value_or_series
        else:
            raise ValueError("index does not match")
    else:
        raise TypeError("invalid type={}".format(type(value_or_series)))
    return result


def _convert_to_linear_expr_series_and_validate_index(
    value_or_series: Union[LinearExprT, pd.Series], index: pd.Index
) -> pd.Series:
    """Returns a pd.Series of the given index with the corresponding values.

    Args:
      value_or_series: the values to be converted (if applicable).
      index: the index of the resulting pd.Series.

    Returns:
      pd.Series: The set of values with the given index.

    Raises:
      TypeError: If the type of `value_or_series` is not recognized.
      ValueError: If the index does not match.
    """
    if isinstance(value_or_series, IntegralTypes):
        result = pd.Series(data=value_or_series, index=index)
    elif isinstance(value_or_series, pd.Series):
        if value_or_series.index.equals(index):
            result = value_or_series
        else:
            raise ValueError("index does not match")
    else:
        raise TypeError("invalid type={}".format(type(value_or_series)))
    return result


def _convert_to_literal_series_and_validate_index(
    value_or_series: Union[LiteralT, pd.Series], index: pd.Index
) -> pd.Series:
    """Returns a pd.Series of the given index with the corresponding values.

    Args:
      value_or_series: the values to be converted (if applicable).
      index: the index of the resulting pd.Series.

    Returns:
      pd.Series: The set of values with the given index.

    Raises:
      TypeError: If the type of `value_or_series` is not recognized.
      ValueError: If the index does not match.
    """
    if isinstance(value_or_series, IntegralTypes):
        result = pd.Series(data=value_or_series, index=index)
    elif isinstance(value_or_series, pd.Series):
        if value_or_series.index.equals(index):
            result = value_or_series
        else:
            raise ValueError("index does not match")
    else:
        raise TypeError("invalid type={}".format(type(value_or_series)))
    return result