gnat 0.1.10

Type-level numbers without extra bounds
Documentation 
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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;