1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165

use std::fmt::Display;

use crate::flow::comparator::*;
use crate::func::{context::*, *};

use super::keyed::*;

pub mod avl;
pub mod balancing;
pub mod bound;
pub mod bounds;

pub type Split<BT> = (
    <BT as BinaryTrees>::Tree,
    <BT as BinaryTrees>::Tree,
    <BT as Keyed>::Key,
);

pub type KeySplit<BT> = (<BT as BinaryTrees>::Tree, <BT as BinaryTrees>::Tree);

pub type BTWrap<'a, BT, A> = WrapC<'a, A, BT>;

pub trait BinaryTrees: Keyed {
    type Node: Send;
    type Reference: Send;
    type Tree: Send;

    fn split(&self, node: &Self::Node) -> Split<Self>;
    fn equal(&self, rl: &Self::Reference, rr: &Self::Reference) -> bool;
    fn refer(&self, tree: &Self::Tree) -> Option<Self::Reference>;
}

pub trait MonadTrees<'a>: MonadContext<'a> + BinaryTrees {
    fn resolve(&self, reference: &Self::Reference) -> BTWrap<'a, Self, Self::Node>;
}

pub trait BinaryTreesTreeOf<'a>: MonadTrees<'a> {
    fn tree_of(&self, node: Self::Node) -> BTWrap<'a, Self, Self::Tree>;

    fn tree_bind(self, fnode: BTWrap<'a, Self, Self::Node>) -> BTWrap<'a, Self, Self::Tree> {
        Self::bind(fnode, move |node| self.tree_of(node))
    }
}

pub trait BinaryTreesEmpty<'a>: MonadTrees<'a> {
    fn empty(&self) -> Self::Tree;

    fn split_key_empty(&self, tree: Self::Tree, key: Self::Key)
        -> BTWrap<'a, Self, KeySplit<Self>>;
}

pub trait BinaryTreesMutable<'a>:
    BinaryTreesEmpty<'a> + BinaryTreesTreeOf<'a> + WithComparator
{
    fn join_key(
        self,
        tl: Self::Tree,
        key: Self::Key,
        tr: Self::Tree,
    ) -> BTWrap<'a, Self, Self::Node>;

    fn join_key_tree(
        self,
        tl: Self::Tree,
        key: Self::Key,
        tr: Self::Tree,
    ) -> BTWrap<'a, Self, Self::Tree> {
        self.clone().tree_bind(self.join_key(tl, key, tr))
    }

    fn join(self, tl: Self::Tree, tr: Self::Tree) -> BTWrap<'a, Self, Self::Tree> {
        let Some(rl) = self.refer(&tl) else {
            return Self::pure(tr);
        };
        let Some(rr) = self.refer(&tr) else {
            return Self::pure(tl);
        };
        Self::T::bind2(self.resolve(&rl), self.resolve(&rr), move |nl, nr| {
            let (tll, tlr, kl) = self.split(&nl);
            let (trl, trr, kr) = self.split(&nr);
            let ft = self.clone().join(tlr, trl);
            let ft = {
                let ctx = self.clone();
                Self::bind(ft, move |t| ctx.join_key_tree(tll, kl, t))
            };
            let ft = {
                let ctx = self;
                Self::bind(ft, move |t| ctx.join_key_tree(t, kr, trr))
            };
            ft
        })
    }

    fn split_key_node(self, node: Self::Node, key: Self::Key) -> BTWrap<'a, Self, KeySplit<Self>> {
        let (tl, tr, k) = self.split(&node);
        match self.comparator().compare(&key, &k) {
            Comparison::L => Self::bind(self.clone().split_key(tl, key), move |(tll, tlr)| {
                Self::fmap(self.join_key_tree(tlr, k, tr), |t| (tll, t))
            }),
            Comparison::E => Self::pure((tl, tr)),
            Comparison::R => Self::bind(self.clone().split_key(tr, key), move |(trl, trr)| {
                Self::fmap(self.join_key_tree(tl, k, trl), |t| (t, trr))
            }),
        }
    }

    fn split_key(self, tree: Self::Tree, key: Self::Key) -> BTWrap<'a, Self, KeySplit<Self>> {
        match self.refer(&tree) {
            Some(reference) => Self::bind(self.resolve(&reference), |node| {
                self.split_key_node(node, key)
            }),
            None => self.split_key_empty(tree, key),
        }
    }

    fn add(self, tree: Self::Tree, key: Self::Key) -> BTWrap<'a, Self, Self::Node> {
        Self::bind(self.clone().split_key(tree, key.clone()), |(tl, tr)| {
            self.join_key(tl, key, tr)
        })
    }

    fn add_tree(self, tree: Self::Tree, key: Self::Key) -> BTWrap<'a, Self, Self::Tree> {
        self.clone().tree_bind(self.add(tree, key))
    }

    fn remove(self, tree: Self::Tree, key: Self::Key) -> BTWrap<'a, Self, Self::Tree> {
        Self::bind(self.clone().split_key(tree, key), |(tl, tr)| {
            self.join(tl, tr)
        })
    }
}

#[derive(Debug)]
pub enum HeightError {
    LeafHeight(u64),
    NodeHeight,
}

impl Display for HeightError {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        match self {
            Self::NodeHeight => write!(f, "invalid node height: 0"),
            Self::LeafHeight(height) => {
                write!(f, "invalid leaf height: {height}!=0")
            }
        }
    }
}

pub trait TreesHeight: BinaryTrees {
    fn height(&self, tree: &Self::Tree) -> u64;
}

pub trait TreesHeightError<'a>: MonadTrees<'a> + TreesHeight {
    fn height_error<T: 'a + Send>(&self, error: HeightError) -> BTWrap<'a, Self, T>;
}

pub trait BinaryTreesTryJoin<'a>: MonadTrees<'a> {
    fn try_join(
        &self,
        tl: Self::Tree,
        key: Self::Key,
        tr: Self::Tree,
    ) -> BTWrap<'a, Self, Self::Node>;
}