The orbit of a seed under the NNO step, \(n \mapsto S^{n}(a)\): the unique morphism the universal property gives from ℕ into the carrier ( \(f(0) = a,\; f(n+1) = S(f(n))\)), as a sequence — the unfold iterate(a, Successor<N>{}) named, so the sequence-shape traits can attach to it. Its shape classifies the carrier's posture at the bound: an injective infinite stream on an unbounded chain (the NNO proper), monotone and bounded on a saturating one — eventually constant at ⊤ when the chain is finite (K₃, IsAbsorptiveSequence), converging to ℵ₀ without reaching it on ℕ's proxy — periodic on a wrapping one.
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Public Member Functions | |
| constexpr | SuccessorOrbit (N seed) |
| Public Member Functions inherited from dedekind::sequences::Path< N > | |
| constexpr Codomain | operator() (std::size_t i) const |
| constexpr N | at (std::size_t i) const |
| constexpr std::size_t | size () const noexcept |
| constexpr auto | begin () const noexcept |
| constexpr auto | end () const noexcept |
| constexpr auto | cardinality () const noexcept |
| The Path is generally viewed as a mapping from N. | |
Additional Inherited Members | |
| Public Types inherited from dedekind::sequences::Path< N > | |
| using | Domain |
| using | Codomain |
| using | cardinality_type |
| using | index_type |
| Public Attributes inherited from dedekind::sequences::Path< N > | |
| std::function< N(std::size_t)> | generator |
| The underlying mapping f(n). | |
| std::size_t | extent |
Detailed Description
struct dedekind::sequences::SuccessorOrbit< N >
The orbit of a seed under the NNO step, \(n \mapsto S^{n}(a)\): the unique morphism the universal property gives from ℕ into the carrier ( \(f(0) = a,\; f(n+1) = S(f(n))\)), as a sequence — the unfold iterate(a, Successor<N>{}) named, so the sequence-shape traits can attach to it. Its shape classifies the carrier's posture at the bound: an injective infinite stream on an unbounded chain (the NNO proper), monotone and bounded on a saturating one — eventually constant at ⊤ when the chain is finite (K₃, IsAbsorptiveSequence), converging to ℵ₀ without reaching it on ℕ's proxy — periodic on a wrapping one.
at(n) is \(O(1)\) where the carrier's + is the iterated step (Peano addition, \(S^{n}(a) = a + n\): the integrals, the ℕ proxy), and the \(O(n)\) walk otherwise (K₃).
- Template Parameters
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N the carrier, with the NNO step.
The documentation for this struct was generated from the following file:
- src/main/modules/dedekind/sequences/convergence.cppm
