# Path: blackjax/mcmc/metrics.py
# def gaussian_euclidean(
#     inverse_mass_matrix: Array,
# ) -> Tuple[Callable, EuclideanKineticEnergy, Callable]:
#     def momentum_generator(rng_key: PRNGKey, position: PyTree) -> PyTree:
#     def kinetic_energy(momentum: PyTree) -> float:
#     def is_turning(
#         momentum_left: PyTree, momentum_right: PyTree, momentum_sum: PyTree
#     ) -> bool:
# 
# Path: blackjax/types.py



from typing import Callable, NamedTuple
from blackjax.mcmc.metrics import EuclideanKineticEnergy
from blackjax.types import PyTree
import jax
"""Symplectic, time-reversible, integrators for Hamiltonian trajectories."""



__all__ = ["mclachlan", "velocity_verlet", "yoshida"]


class IntegratorState(NamedTuple):
    """State of the trajectory integration.

    We keep the gradient of the potential energy to speedup computations.
    """

    position: PyTree
    momentum: PyTree
    potential_energy: float
    potential_energy_grad: PyTree


EuclideanIntegrator = Callable[[IntegratorState, float], IntegratorState]


def new_integrator_state(potential_fn, position, momentum):
    potential_energy, potential_energy_grad = jax.value_and_grad(potential_fn)(position)
    return IntegratorState(position, momentum, potential_energy, potential_energy_grad)


def velocity_verlet(

# Path: blackjax/mcmc/metrics.py
# def gaussian_euclidean(
#     inverse_mass_matrix: Array,
# ) -> Tuple[Callable, EuclideanKineticEnergy, Callable]:
#     def momentum_generator(rng_key: PRNGKey, position: PyTree) -> PyTree:
#     def kinetic_energy(momentum: PyTree) -> float:
#     def is_turning(
#         momentum_left: PyTree, momentum_right: PyTree, momentum_sum: PyTree
#     ) -> bool:
# 
# Path: blackjax/types.py



# Path: blackjax/mcmc/integrators.py
from typing import Callable, NamedTuple
from blackjax.mcmc.metrics import EuclideanKineticEnergy
from blackjax.types import PyTree
import jax

"""Symplectic, time-reversible, integrators for Hamiltonian trajectories."""



__all__ = ["mclachlan", "velocity_verlet", "yoshida"]


class IntegratorState(NamedTuple):
    """State of the trajectory integration.

    We keep the gradient of the potential energy to speedup computations.
    """

    position: PyTree
    momentum: PyTree
    potential_energy: float
    potential_energy_grad: PyTree


EuclideanIntegrator = Callable[[IntegratorState, float], IntegratorState]


def new_integrator_state(potential_fn, position, momentum):
    potential_energy, potential_energy_grad = jax.value_and_grad(potential_fn)(position)
    return IntegratorState(position, momentum, potential_energy, potential_energy_grad)


def velocity_verlet(

    potential_fn: Callable, kinetic_energy_fn: EuclideanKineticEnergy