dedekind::morphologies::Congruence< N, R > Struct Template Reference

The residue-class predicate \(x \equiv R \pmod N\): the characteristic map of the fiber of the reduction \(\mathbb{Z} \twoheadrightarrow \mathbb{Z}/N\mathbb{Z} =\) Modular<N> over the residue R. More...

Public Types
using	Domain = decltype(N)
using	Codomain = bool

Public Member Functions
template<typename X>
constexpr bool	operator() (const X &x) const

Detailed Description

template<auto N, decltype(N) R = decltype(N){}>
struct dedekind::morphologies::Congruence< N, R >

The residue-class predicate \(x \equiv R \pmod N\): the characteristic map of the fiber of the reduction \(\mathbb{Z} \twoheadrightarrow \mathbb{Z}/N\mathbb{Z} =\) Modular<N> over the residue R.

isEven is Congruence<2,0>. Because the predicate factors through the finite quotient Modular<N> (the reduction is the Modular<N> constructor), a quantifier over an infinite carrier reduces to exhausting the N residues: \(\exists\) and \(\forall\) both become decidable in finite time (see the sets:quantifier finite-carrier path and the numbers-level 𝔸<Cardinality> bridge). The modulus N is carried in the type, so the compiler sees the quotient; an opaque lambda could not.

It is a characteristic morphism (IsPredicate): Domain is the integer the reduction consumes, Codomain is bool. FIXME(#797): % may later subsume into a factor-through-quotient combinator (§3 morphisms / §4 quotients).

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

• src/main/modules/dedekind/morphologies/cyclic.cppm