gnat 0.1.8

Type-level numbers without extra bounds
Documentation 
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

use super::*;

// SubIfGe(L, R) := if L >= R { L - R } else { L }
type _SubIfGe<L, R> = If<
    _Lt<L, R>, // Ge is implemented on top of Lt, so use Lt directly
    L,
    _SubUnchecked<L, R>,
>;

// NaiveRem(L, R) := 2 * (H(L) % R) + P(L), where R > 0
#[apply(nat_expr)]
pub type _NaiveRem<L: NatExpr, R: NatExpr> = If<
    L,
    PushBit<
        _RemUnchecked<_H<L>, R>, //
        _P<L>,
    >,
    // NaiveRem(0, R) = 2 * (0 % R) + 0 = 0
    crate::lit!(0),
>;

// RemUnchecked(L, R) := L % R, where R > 0
//
// H := H(L), P := P(L), R > 0
//
// L % R = (2 * H + P) % R
//       = (H + H + P) % R
//       = ((H % R) + (H % R) + P) % R
//       = (2 * (H % R) + P) % R
//       = NaiveRem(L, R) % R
//
//      H % R <= R - 1
// thus NaiveRem(L, R) = 2 * (H % R) + P <= 2 * (H % R) + 1 <= 2 * R - 1
// thus RemUnchecked(L, R) = L % R = NaiveRem(L, R) % R = SubIfGe(NaiveRem(L, R), R)
pub(crate) type _RemUnchecked<L, R> = _SubIfGe<_NaiveRem<L, R>, R>;

// DivUnchecked(L, R) := L / R, where R > 0
//
// H := H(L), P := P(L)
//
// Note that H = (H / R) * R + H % R, and
// For any X, Y: (X * R + Y) / R = X + Y / R
//
// L / R = (2 * H + P) / R
//       = (2 * ((H / R) * R + H % R) + P) / R
//       = (2 * (H / R) * R + 2 * (H % R) + P) / R
//       = (2 * (H / R) * R + 2 * (H % R) + P) / R
//       = 2 * (H / R) + (2 * (H % R) + P) / R
//       = 2 * (H / R) + NaiveRem(L, R) / R
//
// Since we still have NaiveRem(L, R) <= 2 * R - 1, it follows that
// NaiveRem(L, R) / R = if NaiveRem(L, R) >= R { 1 } else { 0 }
#[apply(nat_expr)]
pub type _DivUnchecked<L: NatExpr, R: NatExpr> = If<
    L,
    PushBit<
        _DivUnchecked<_H<L>, R>, //
        IsZero<_Lt<_NaiveRem<L, R>, R>>,
    >,
    // 0 / R = 0
    crate::lit!(0),
>;

#[apply(nat_expr)]
pub type _Rem<L: NatExpr, R: NatExpr> = If<
    R,
    _RemUnchecked<L, R>,
    // Return the dividend for division by zero
    L,
>;

/// Type-level [`%`](std::ops::Rem).
///
/// The remainder `Rem(L, R)` is always given by `L - (L / R) * R` (but calculated more efficiently).
///
/// # Examples
#[doc = op_examples!(
    Rem,
    (5, 2) == 1,
    (11, 4) == 3,
)]
/// Since division by zero always yields zero, the remainder is equal to the dividend.
#[doc = op_examples!(
    Rem,
    (0, 0) == 0,
    (7, 0) == 7,
)]
#[doc(alias = "%")]
#[doc(alias = "modulo")]
#[apply(opaque)]
#[apply(test_op!
    test_rem,
    L % R,
    ..,
    1..
)]
pub type Rem<L, R> = _Rem;

#[apply(nat_expr)]
pub type _Div<L: NatExpr, R: NatExpr> = If<
    R,
    _DivUnchecked<L, R>,
    // Division by zero is defined as zero
    crate::lit!(0),
>;

/// Type-level [`/`](std::ops::Div)
///
///
/// # Examples
#[doc = op_examples!(
    Div,
    (5, 2) == 2,
    (11, 4) == 2,
)]
/// Division by zero is defined to yield zero.
#[doc = op_examples!(
    Div,
    (0, 0) == 0,
    (7, 0) == 0,
)]
#[doc(alias = "/")]
#[apply(opaque)]
#[apply(test_op!
    test_div,
    L / R,
    ..,
    1..,
)]
pub type Div<L, R> = _Div;