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
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196

use crate::ring::*;
use crate::divisibility::*;

///
/// Trait for rings that are principal ideal rings, i.e. every ideal is generated
/// by a single element. 
/// 
pub trait PrincipalIdealRing: DivisibilityRing {

    ///
    /// Computes a Bezout identity for the generator `g` of the ideal `(lhs, rhs)`
    /// as `g = s * lhs + t * rhs`.
    /// 
    /// More concretely, this returns (s, t, g) such that g is a generator 
    /// of the ideal `(lhs, rhs)` and `g = s * lhs + t * rhs`. This `g` is also known
    /// as the greatest common divisor of `lhs` and `rhs`, since `g | lhs, rhs` and
    /// for every `g'` with this property, have `g' | g`. Note that this `g` is only
    /// unique up to multiplication by units.
    /// 
    fn extended_ideal_gen(&self, lhs: &Self::Element, rhs: &Self::Element) -> (Self::Element, Self::Element, Self::Element);

    ///
    /// Computes a generator `g` of the ideal `(lhs, rhs) = (g)`, also known as greatest
    /// common divisor.
    /// 
    /// If you require also a Bezout identiy, i.e. `g = s * lhs + t * rhs`, consider
    /// using [`PrincipalIdealRing::extended_ideal_gen()`].
    /// 
    fn ideal_gen(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element {
        self.extended_ideal_gen(lhs, rhs).2
    }

    ///
    /// Computes a generator of the ideal `(lhs) ∩ (rhs)`, also known as least common
    /// multiple.
    /// 
    /// In other words, computes a ring element `g` such that `lhs, rhs | g` and for every
    /// `g'` with this property, have `g | g'`. Note that such an `g` is only unique up to
    /// multiplication by units.
    /// 
    fn lcm(&self, lhs: &Self::Element, rhs: &Self::Element) -> Self::Element {
        self.checked_left_div(&self.mul_ref(lhs, rhs), &self.ideal_gen(lhs, rhs)).unwrap()
    }
}

///
/// Trait for [`RingStore`]s that store [`PrincipalIdealRing`]s. Mainly used
/// to provide a convenient interface to the `PrincipalIdealRing`-functions.
/// 
pub trait PrincipalIdealRingStore: RingStore
    where Self::Type: PrincipalIdealRing
{
    delegate!{ PrincipalIdealRing, fn extended_ideal_gen(&self, lhs: &El<Self>, rhs: &El<Self>) -> (El<Self>, El<Self>, El<Self>) }
    delegate!{ PrincipalIdealRing, fn ideal_gen(&self, lhs: &El<Self>, rhs: &El<Self>) -> El<Self> }
    delegate!{ PrincipalIdealRing, fn lcm(&self, lhs: &El<Self>, rhs: &El<Self>) -> El<Self> }

    ///
    /// Alias for [`PrincipalIdealRingStore::ideal_gen()`].
    /// 
    fn gcd(&self, lhs: &El<Self>, rhs: &El<Self>) -> El<Self> {
        self.ideal_gen(lhs, rhs)
    }
}

impl<R> PrincipalIdealRingStore for R
    where R: RingStore,
        R::Type: PrincipalIdealRing
{}

///
/// Trait for rings that support euclidean division.
/// 
/// In other words, there is a degree function d(.) 
/// returning nonnegative integers such that for every `x, y` 
/// with `y != 0` there are `q, r` with `x = qy + r` and 
/// `d(r) < d(y)`. Note that `q, r` do not have to be unique, 
/// and implementations are free to use any choice.
/// 
/// # Example
/// ```
/// # use feanor_math::assert_el_eq;
/// # use feanor_math::ring::*;
/// # use feanor_math::pid::*;
/// # use feanor_math::primitive_int::*;
/// let ring = StaticRing::<i64>::RING;
/// let (q, r) = ring.euclidean_div_rem(14, &6);
/// assert_el_eq!(ring, 14, ring.add(ring.mul(q, 6), r));
/// assert!(ring.euclidean_deg(&r) < ring.euclidean_deg(&6));
/// ```
/// 
pub trait EuclideanRing: PrincipalIdealRing {

    ///
    /// Computes euclidean division with remainder.
    /// 
    /// In general, the euclidean division of `lhs` by `rhs` is a tuple `(q, r)` such that
    /// `lhs = q * rhs + r`, and `r` is "smaller" than "rhs". The notion of smallness is
    /// given by the smallness of the euclidean degree function [`EuclideanRing::euclidean_deg()`].
    /// 
    fn euclidean_div_rem(&self, lhs: Self::Element, rhs: &Self::Element) -> (Self::Element, Self::Element);

    ///
    /// Defines how "small" an element is. For details, see [`EuclideanRing`].
    /// 
    fn euclidean_deg(&self, val: &Self::Element) -> Option<usize>;

    ///
    /// Computes euclidean division without remainder.
    /// 
    /// In general, the euclidean division of `lhs` by `rhs` is a tuple `(q, r)` such that
    /// `lhs = q * rhs + r`, and `r` is "smaller" than "rhs". The notion of smallness is
    /// given by the smallness of the euclidean degree function [`EuclideanRing::euclidean_deg()`].
    /// 
    fn euclidean_div(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element {
        self.euclidean_div_rem(lhs, rhs).0
    }

    ///
    /// Computes only the remainder of euclidean division.
    /// 
    /// In general, the euclidean division of `lhs` by `rhs` is a tuple `(q, r)` such that
    /// `lhs = q * rhs + r`, and `r` is "smaller" than "rhs". The notion of smallness is
    /// given by the smallness of the euclidean degree function [`EuclideanRing::euclidean_deg()`].
    /// 
    fn euclidean_rem(&self, lhs: Self::Element, rhs: &Self::Element) -> Self::Element {
        self.euclidean_div_rem(lhs, rhs).1
    }
}

///
/// [`RingStore`] for [`EuclideanRing`]s
/// 
pub trait EuclideanRingStore: RingStore + DivisibilityRingStore
    where Self::Type: EuclideanRing
{
    delegate!{ EuclideanRing, fn euclidean_div_rem(&self, lhs: El<Self>, rhs: &El<Self>) -> (El<Self>, El<Self>) }
    delegate!{ EuclideanRing, fn euclidean_div(&self, lhs: El<Self>, rhs: &El<Self>) -> El<Self> }
    delegate!{ EuclideanRing, fn euclidean_rem(&self, lhs: El<Self>, rhs: &El<Self>) -> El<Self> }
    delegate!{ EuclideanRing, fn euclidean_deg(&self, val: &El<Self>) -> Option<usize> }
}

impl<R> EuclideanRingStore for R
    where R: RingStore, R::Type: EuclideanRing
{}

#[cfg(any(test, feature = "generic_tests"))]
pub mod generic_tests {
    use super::*;
    use crate::ring::El;

    pub fn test_euclidean_ring_axioms<R: EuclideanRingStore, I: Iterator<Item = El<R>>>(ring: R, edge_case_elements: I) 
        where R::Type: EuclideanRing
    {
        assert!(ring.is_commutative());
        assert!(ring.is_noetherian());
        let elements = edge_case_elements.collect::<Vec<_>>();
        for a in &elements {
            for b in &elements {
                if ring.is_zero(b) {
                    continue;
                }
                let (q, r) = ring.euclidean_div_rem(ring.clone_el(a), b);
                assert!(ring.euclidean_deg(b).is_none() || ring.euclidean_deg(&r).unwrap_or(usize::MAX) < ring.euclidean_deg(b).unwrap());
                assert_el_eq!(ring, a, ring.add(ring.mul(q, ring.clone_el(b)), r));
            }
        }
    }

    pub fn test_principal_ideal_ring_axioms<R: PrincipalIdealRingStore, I: Iterator<Item = El<R>>>(ring: R, edge_case_elements: I)
        where R::Type: PrincipalIdealRing
    {
        assert!(ring.is_commutative());
        assert!(ring.is_noetherian());
        let elements = edge_case_elements.collect::<Vec<_>>();
        for a in &elements {
            for b in &elements {
                for c in &elements {
                    let g1 = ring.mul_ref(a, b);
                    let g2 = ring.mul_ref(a, c);
                    let (s, t, g) = ring.extended_ideal_gen(&g1, &g2);
                    assert!(ring.checked_div(&g, a).is_some(), "Wrong ideal generator: ({}) contains the ideal I = ({}, {}), but extended_ideal_gen() found a generator I = ({}) that does not satisfy {} | {}", ring.format(a), ring.format(&g1), ring.format(&g2), ring.format(&g), ring.format(a), ring.format(&g));
                    assert_el_eq!(ring, g, ring.add(ring.mul_ref(&s, &g1), ring.mul_ref(&t, &g2)));
                }
            }
        }
        for a in &elements {
            for b in &elements {
                let g1 = ring.mul_ref(a, b);
                let g2 = ring.mul_ref_fst(a, ring.add_ref_fst(b, ring.one()));
                let (s, t, g) = ring.extended_ideal_gen(&g1, &g2);
                assert!(ring.checked_div(&g, a).is_some() && ring.checked_div(a, &g).is_some(), "Expected ideals ({}) and I = ({}, {}) to be equal, but extended_ideal_gen() returned generator {} of I", ring.format(a), ring.format(&g1), ring.format(&g2), ring.format(&g));
                assert_el_eq!(ring, g, ring.add(ring.mul_ref(&s, &g1), ring.mul_ref(&t, &g2)));
            }
        }
    }
}