Non-empty DUAL-trees.

A non-empty DUAL-tree paired with a cached u value. These should never be constructed directly; instead, use pullU.

Rose (n-ary) trees with both upwards- (i.e. cached) and downwards-traveling (i.e. accumulating) monoidal annotations. Abstractly, a DUALTree is a rose (n-ary) tree with data (of type l) at leaves, data (of type a) at internal nodes, and two types of monoidal annotations, one (of type u) travelling "up" the tree and one (of type d) traveling "down". See the documentation at the top of this file for full details.

DUALTree comes with some instances:

• Functor, for modifying leaf data. Note that fmap of course cannot alter any u annotations.
• Semigroup. DUALTreeNEs form a semigroup where (<>) corresponds to adjoining two trees under a common parent root, with sconcat specialized to put all the trees under a single parent. Note that this does not satisfy associativity up to structural equality, but only up to observational equivalence under flatten. Technically using foldDUAL directly enables one to observe the difference, but it is understood that foldDUAL should be used only in ways such that reassociation of subtrees "does not matter".
• Monoid. The identity is the empty tree.

The empty DUAL-tree. This is a synonym for mempty, but with a more general type.

Construct a leaf node from a u annotation along with a leaf datum.

Construct a leaf node from a u annotation.

Add an internal data value at the root of a tree. Note that this only works on non-empty trees; on empty trees this function is the identity.

Apply a d annotation at the root of a tree, transforming all u annotations by the action of d.

Add a u annotation to the root, combining it (on the left) with the existing cached u annotation. This function is provided just for convenience; applyUpre u t = leafU u <> t.

Add a u annotation to the root, combining it (on the right) with the existing cached u annotation. This function is provided just for convenience; applyUpost u t = t <> leafU u.

Map a function (which must be a monoid homomorphism, and commute with the action of d) over all the u annotations in a non-empty DUAL-tree.

Map a function (which must be a monoid homomorphism, and commute with the action of d) over all the u annotations in a non-empty DUAL-tree paired with its cached u value.

Map a function over all the u annotations in a DUAL-tree. The function must be a monoid homomorphism, and must commute with the action of d on u. That is, to use mapU f safely it must be the case that

• f mempty == mempty

• f (u1 <> u2) == f u1 <> f u2

• f (act d u) == act d (f u)

Decompose a DUAL-tree into either Nothing (if empty) or a top-level cached u annotation paired with a non-empty DUAL-tree.

Get the u annotation at the root, or Nothing if the tree is empty.

Fold for non-empty DUAL-trees.

Fold for DUAL-trees. It is given access to the internal and leaf data, internal d values, and the accumulated d values at each leaf. It is also allowed to replace "u-only" leaves with a constant value. In particular, however, it is not given access to any of the u annotations, the idea being that those are used only for constructing trees. If you do need access to u values, you can duplicate the values you need in the internal data nodes.

Be careful not to mix up the d values at internal nodes with the d values at leaves. Each d value at a leaf satisfies the property that it is the mconcat of all internal d values along the path from the root to the leaf.

The result is Nothing if and only if the tree is empty.

A specialized fold provided for convenience: flatten a tree into a list of leaves along with their d annotations, ignoring internal data values.