dedekind::sets::SignedExtensionalCardinal< S > Struct Template Reference

Signed one-word integer: two's complement in the signed word S. More...

Classes
struct	CheckedResult
	A wrapped result and whether the exact result overflowed. More...

Public Types
using	signed_type = S
using	word_type = std::make_unsigned_t<S>
using	magnitude_type = ExtensionalCardinal<word_type>

Public Member Functions
template<std::integral I>
constexpr	SignedExtensionalCardinal (I v) noexcept
	Construction from any integral source, modulo 2^w.
constexpr bool	negative () const noexcept
	Is the value negative?
constexpr magnitude_type	magnitude () const noexcept
	|value| as an unsigned word (exact, also for min).
constexpr	operator std::size_t () const noexcept
	Explicit projection to std::size_t — the magnitude (sign is dropped). Sibling to ExtensionalCardinal::operator std::size_t() ; used at the same call sites in :sequences:path 's helpers.
constexpr SignedExtensionalCardinal	operator- () const noexcept
	The additive inverse in ℤ/2^w (-min wraps to min).
template<std::signed_integral Target>
constexpr	operator Target () const noexcept
	Explicit conversion to a signed integral type (modular, like the carrier). Used by the IR-fixture showcases to extract a compile-time-known rational's numerator as a concrete machine integer for the emitted ret i64 ... witness.

Static Public Member Functions
static constexpr CheckedResult	checked_add (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
static constexpr CheckedResult	checked_sub (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
static constexpr CheckedResult	checked_mul (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept

Public Attributes
signed_type	value {}

Friends
constexpr friend bool	operator== (const SignedExtensionalCardinal &, const SignedExtensionalCardinal &)=default
constexpr friend std::strong_ordering	operator<=> (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
template<std::integral T>
constexpr friend bool	operator== (const SignedExtensionalCardinal &lhs, T rhs) noexcept
	Heterogeneous comparison with built-in integrals, exact across signedness (compared in 128 bits). Without it, SEC == int is ambiguous between the SEC and SignedCardinality overloads, each one user-defined conversion away.
template<std::integral T>
constexpr friend bool	operator== (T lhs, const SignedExtensionalCardinal &rhs) noexcept
template<std::integral T>
constexpr friend std::strong_ordering	operator<=> (const SignedExtensionalCardinal &lhs, T rhs) noexcept
template<std::integral T>
constexpr friend std::strong_ordering	operator<=> (T lhs, const SignedExtensionalCardinal &rhs) noexcept
constexpr friend SignedExtensionalCardinal	operator+ (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
constexpr friend SignedExtensionalCardinal	operator- (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
constexpr friend SignedExtensionalCardinal	operator* (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
constexpr friend SignedExtensionalCardinal	operator/ (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
	Division truncating toward zero (C++ semantics). Division by zero yields zero by convention; min / -1 wraps to min.
constexpr friend SignedExtensionalCardinal	operator% (const SignedExtensionalCardinal &lhs, const SignedExtensionalCardinal &rhs) noexcept
	Remainder with the sign of the dividend (C++ semantics: a == (a / b) * b + a % b). Modulo zero yields lhs.

Detailed Description

template<typename S = std::ptrdiff_t>
requires std::signed_integral<S> && IsWrappingWord<std::make_unsigned_t<S>>
struct dedekind::sets::SignedExtensionalCardinal< S >

Signed one-word integer: two's complement in the signed word S.

One structural field, value, of type S. Arithmetic runs through the overflow builtins on that type, which yield the wrapped result, so the carrier is exactly the residue ring ℤ/2^w read in the signed range [-2^(w-1), 2^(w-1)-1]: a genuine commutative ring and cyclic group, which the registrations below claim. -min wraps to min, and min / -1 is min (the ring answer), so no operation is undefined.

It satisfies IsInteger (see registrations after the numbers:integer module pulls this file), making it the signed backing carrier for fixed-width Rational<Z>. The SignedCardinality variant downstream escalates to ±ℵ_0 where this carrier wraps.

Structural on purpose: the public field makes the type usable as a non-type template parameter wherever Rational<Z> is.

Template Parameters

S	The signed word (default std::ptrdiff_t, the signed twin of std::size_t). Gated on being a signed built-in integer whose unsigned twin is an IsWrappingWord: the raw signed type is not a ring (overflow is UB), but this carrier's builtin-based arithmetic is exactly the ring of its unsigned twin.

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/sets/cardinality.cppm