github.com/prysmaticlabs/prysm@v1.4.4/tools/specs-checker/data/ssz/merkle-proofs.md

```python
def get_power_of_two_ceil(x: int) -> int:
    """
    Get the power of 2 for given input, or the closest higher power of 2 if the input is not a power of 2.
    Commonly used for "how many nodes do I need for a bottom tree layer fitting x elements?"
    Example: 0->1, 1->1, 2->2, 3->4, 4->4, 5->8, 6->8, 7->8, 8->8, 9->16.
    """
    if x <= 1:
        return 1
    elif x == 2:
        return 2
    else:
        return 2 * get_power_of_two_ceil((x + 1) // 2)
```
```python
def get_power_of_two_floor(x: int) -> int:
    """
    Get the power of 2 for given input, or the closest lower power of 2 if the input is not a power of 2.
    The zero case is a placeholder and not used for math with generalized indices.
    Commonly used for "what power of two makes up the root bit of the generalized index?"
    Example: 0->1, 1->1, 2->2, 3->2, 4->4, 5->4, 6->4, 7->4, 8->8, 9->8
    """
    if x <= 1:
        return 1
    if x == 2:
        return x
    else:
        return 2 * get_power_of_two_floor(x // 2)
```
```python
def merkle_tree(leaves: Sequence[Bytes32]) -> Sequence[Bytes32]:
    """
    Return an array representing the tree nodes by generalized index: 
    [0, 1, 2, 3, 4, 5, 6, 7], where each layer is a power of 2. The 0 index is ignored. The 1 index is the root.
    The result will be twice the size as the padded bottom layer for the input leaves.
    """
    bottom_length = get_power_of_two_ceil(len(leaves))
    o = [Bytes32()] * bottom_length + list(leaves) + [Bytes32()] * (bottom_length - len(leaves))
    for i in range(bottom_length - 1, 0, -1):
        o[i] = hash(o[i * 2] + o[i * 2 + 1])
    return o
```
```python
def item_length(typ: SSZType) -> int:
    """
    Return the number of bytes in a basic type, or 32 (a full hash) for compound types.
    """
    if issubclass(typ, BasicValue):
        return typ.byte_len
    else:
        return 32
```
```python
def get_elem_type(typ: Union[BaseBytes, BaseList, Container],
                  index_or_variable_name: Union[int, SSZVariableName]) -> SSZType:
    """
    Return the type of the element of an object of the given type with the given index
    or member variable name (eg. `7` for `x[7]`, `"foo"` for `x.foo`)
    """
    return typ.get_fields()[index_or_variable_name] if issubclass(typ, Container) else typ.elem_type
```
```python
def chunk_count(typ: SSZType) -> int:
    """
    Return the number of hashes needed to represent the top-level elements in the given type
    (eg. `x.foo` or `x[7]` but not `x[7].bar` or `x.foo.baz`). In all cases except lists/vectors
    of basic types, this is simply the number of top-level elements, as each element gets one
    hash. For lists/vectors of basic types, it is often fewer because multiple basic elements
    can be packed into one 32-byte chunk.
    """
    # typ.length describes the limit for list types, or the length for vector types.
    if issubclass(typ, BasicValue):
        return 1
    elif issubclass(typ, Bits):
        return (typ.length + 255) // 256
    elif issubclass(typ, Elements):
        return (typ.length * item_length(typ.elem_type) + 31) // 32
    elif issubclass(typ, Container):
        return len(typ.get_fields())
    else:
        raise Exception(f"Type not supported: {typ}")
```
```python
def get_item_position(typ: SSZType, index_or_variable_name: Union[int, SSZVariableName]) -> Tuple[int, int, int]:
    """
    Return three variables:
        (i) the index of the chunk in which the given element of the item is represented;
        (ii) the starting byte position within the chunk;
        (iii) the ending byte position within the chunk.
    For example: for a 6-item list of uint64 values, index=2 will return (0, 16, 24), index=5 will return (1, 8, 16)
    """
    if issubclass(typ, Elements):
        index = int(index_or_variable_name)
        start = index * item_length(typ.elem_type)
        return start // 32, start % 32, start % 32 + item_length(typ.elem_type)
    elif issubclass(typ, Container):
        variable_name = index_or_variable_name
        return typ.get_field_names().index(variable_name), 0, item_length(get_elem_type(typ, variable_name))
    else:
        raise Exception("Only lists/vectors/containers supported")
```
```python
def get_generalized_index(typ: SSZType, path: Sequence[Union[int, SSZVariableName]]) -> GeneralizedIndex:
    """
    Converts a path (eg. `[7, "foo", 3]` for `x[7].foo[3]`, `[12, "bar", "__len__"]` for
    `len(x[12].bar)`) into the generalized index representing its position in the Merkle tree.
    """
    root = GeneralizedIndex(1)
    for p in path:
        assert not issubclass(typ, BasicValue)  # If we descend to a basic type, the path cannot continue further
        if p == '__len__':
            typ = uint64
            assert issubclass(typ, (List, ByteList))
            root = GeneralizedIndex(root * 2 + 1)
        else:
            pos, _, _ = get_item_position(typ, p)
            base_index = (GeneralizedIndex(2) if issubclass(typ, (List, ByteList)) else GeneralizedIndex(1))
            root = GeneralizedIndex(root * base_index * get_power_of_two_ceil(chunk_count(typ)) + pos)
            typ = get_elem_type(typ, p)
    return root
```
```python
def concat_generalized_indices(*indices: GeneralizedIndex) -> GeneralizedIndex:
    """
    Given generalized indices i1 for A -> B, i2 for B -> C .... i_n for Y -> Z, returns
    the generalized index for A -> Z.
    """
    o = GeneralizedIndex(1)
    for i in indices:
        o = GeneralizedIndex(o * get_power_of_two_floor(i) + (i - get_power_of_two_floor(i)))
    return o
```
```python
def get_generalized_index_length(index: GeneralizedIndex) -> int:
    """
    Return the length of a path represented by a generalized index.
    """
    return int(log2(index))
```
```python
def get_generalized_index_bit(index: GeneralizedIndex, position: int) -> bool:
    """
    Return the given bit of a generalized index.
    """
    return (index & (1 << position)) > 0
```
```python
def generalized_index_sibling(index: GeneralizedIndex) -> GeneralizedIndex:
    return GeneralizedIndex(index ^ 1)
```
```python
def generalized_index_child(index: GeneralizedIndex, right_side: bool) -> GeneralizedIndex:
    return GeneralizedIndex(index * 2 + right_side)
```
```python
def generalized_index_parent(index: GeneralizedIndex) -> GeneralizedIndex:
    return GeneralizedIndex(index // 2)
```
```python
def get_branch_indices(tree_index: GeneralizedIndex) -> Sequence[GeneralizedIndex]:
    """
    Get the generalized indices of the sister chunks along the path from the chunk with the
    given tree index to the root.
    """
    o = [generalized_index_sibling(tree_index)]
    while o[-1] > 1:
        o.append(generalized_index_sibling(generalized_index_parent(o[-1])))
    return o[:-1]
```
```python
def get_path_indices(tree_index: GeneralizedIndex) -> Sequence[GeneralizedIndex]:
    """
    Get the generalized indices of the chunks along the path from the chunk with the
    given tree index to the root.
    """
    o = [tree_index]
    while o[-1] > 1:
        o.append(generalized_index_parent(o[-1]))
    return o[:-1]
```
```python
def get_helper_indices(indices: Sequence[GeneralizedIndex]) -> Sequence[GeneralizedIndex]:
    """
    Get the generalized indices of all "extra" chunks in the tree needed to prove the chunks with the given
    generalized indices. Note that the decreasing order is chosen deliberately to ensure equivalence to the
    order of hashes in a regular single-item Merkle proof in the single-item case.
    """
    all_helper_indices: Set[GeneralizedIndex] = set()
    all_path_indices: Set[GeneralizedIndex] = set()
    for index in indices:
        all_helper_indices = all_helper_indices.union(set(get_branch_indices(index)))
        all_path_indices = all_path_indices.union(set(get_path_indices(index)))

    return sorted(all_helper_indices.difference(all_path_indices), reverse=True)
```
```python
def calculate_merkle_root(leaf: Bytes32, proof: Sequence[Bytes32], index: GeneralizedIndex) -> Root:
    assert len(proof) == get_generalized_index_length(index)
    for i, h in enumerate(proof):
        if get_generalized_index_bit(index, i):
            leaf = hash(h + leaf)
        else:
            leaf = hash(leaf + h)
    return leaf
```
```python
def verify_merkle_proof(leaf: Bytes32, proof: Sequence[Bytes32], index: GeneralizedIndex, root: Root) -> bool:
    return calculate_merkle_root(leaf, proof, index) == root
```
```python
def calculate_multi_merkle_root(leaves: Sequence[Bytes32],
                                proof: Sequence[Bytes32],
                                indices: Sequence[GeneralizedIndex]) -> Root:
    assert len(leaves) == len(indices)
    helper_indices = get_helper_indices(indices)
    assert len(proof) == len(helper_indices)
    objects = {
        **{index: node for index, node in zip(indices, leaves)},
        **{index: node for index, node in zip(helper_indices, proof)}
    }
    keys = sorted(objects.keys(), reverse=True)
    pos = 0
    while pos < len(keys):
        k = keys[pos]
        if k in objects and k ^ 1 in objects and k // 2 not in objects:
            objects[GeneralizedIndex(k // 2)] = hash(
                objects[GeneralizedIndex((k | 1) ^ 1)] +
                objects[GeneralizedIndex(k | 1)]
            )
            keys.append(GeneralizedIndex(k // 2))
        pos += 1
    return objects[GeneralizedIndex(1)]
```
```python
def verify_merkle_multiproof(leaves: Sequence[Bytes32],
                             proof: Sequence[Bytes32],
                             indices: Sequence[GeneralizedIndex],
                             root: Root) -> bool:
    return calculate_multi_merkle_root(leaves, proof, indices) == root
```