@startuml infinite-relations
!theme plain
title Infinite Type Relations: Unfold Pattern & Membership

abstract class InfiniteTypeRelation {
  + name: atom()
  + cardinality: aleph_zero | continuum
  + erlang_type: type()
  + membership_predicate: function()
  + unfold_generator: function()
  + constraints: constraints()
}

class Naturals {
  Erlang Type: integer()
  Membership: natural[X] :-
    erlang:is_integer(X),
    X >= 0.

  Unfold Generator:
  unfold(0, fun(N) -> N + 1 end)
  → 0, 1, 2, 3, ...

  Cardinality: aleph_zero
}

class Integers {
  Erlang Type: integer()
  Membership: integer[X] :-
    erlang:is_integer(X).

  Unfold Generator:
  unfold({0, 1}, fun({N, S}) ->
    case S of
      1 -> {N, -1};
      -1 -> {N+1, 1}
    end
  end)
  → 0, 1, -1, 2, -2, ...

  Cardinality: aleph_zero
}

class Rationals {
  Erlang Type: {integer(), integer()}
  Membership: rational[X] :-
    X is (P, Q),
    integer[P], integer[Q],
    Q ≠ 0.

  Unfold Generator:
  Stern-Brocot tree unfold
  → (0,1), (1,1), (1,2), ...

  Cardinality: aleph_zero
}

class Reals {
  Erlang Type: float()
  Membership: real[X] :-
    erlang:is_float(X).

  Unfold Generator:
  IEEE 754 float enumeration
  (impractical, continuum)

  Cardinality: continuum
}

class Strings {
  Erlang Type: string()
  Membership: string[X] :-
    erlang:is_list(X),
    all_chars(X).

  Unfold Generator:
  Length-ordered, lexicographic
  → "", "a", "b", ..., "aa", ...

  Cardinality: aleph_zero
}

class Plus {
  Erlang Type: {integer(), integer(), integer()}
  Membership: plus[A, B, Sum] :-
    integer[A], integer[B],
    Sum is A + B.

  Unfold Generator:
  Diagonalization of ℤ × ℤ
  → (0,0,0), (0,1,1), (1,0,1), ...

  Cardinality: continuum (ℤ × ℤ)
}

package "Constraint Application" {
  class ConstrainedGenerator {
    Pattern: Apply constraints during unfold
    base_generator: unfold()
    constraints: [constraint()]
    ..
    filter_unfold(Gen, Constraints) ->
      filter(Gen, satisfies_all(Constraints))
  }

  note bottom of ConstrainedGenerator
    **Example 1: Bounded Naturals**
    natural[X], X >= 0, X =< 100
    → Generates: 0, 1, 2, ..., 100 (finite!)

    **Example 2: Constrained Plus**
    plus[5, B, 12]
    → Solves: B = 7
    → Generates: single tuple (5, 7, 12)

    **Example 3: Range Query**
    integer[X], X >= -10, X =< 10
    → Generates: 0, 1, -1, 2, -2, ..., 10, -10
  end note
}

package "Composite Type Relations (Views)" {
  class Complex {
    Type Definition (Karuta):
    complex[X] :-
      X is (A, B),
      real[A], real[B].

    Implementation:
    Unfold real × real with
    product enumeration

    Not primitive: Defined as view
    over primitive real relation
  }
}

InfiniteTypeRelation <|-- Naturals
InfiniteTypeRelation <|-- Integers
InfiniteTypeRelation <|-- Rationals
InfiniteTypeRelation <|-- Reals
InfiniteTypeRelation <|-- Strings
InfiniteTypeRelation <|-- Plus

Naturals --> ConstrainedGenerator: apply\nconstraints
Integers --> ConstrainedGenerator
Rationals --> ConstrainedGenerator
Plus --> ConstrainedGenerator

Reals --> Complex: used in\nproduct
Complex : (view relation)

note right of Naturals
  **Core Pattern:** Infinite relations use unfold
  to generate members. Primitive types are
  infinite relations.

  **Membership:** Erlang type + constraints

  **Unfold Pattern:**
  unfold(Seed, Step) generates infinite sequence:
  Seed, Step(Seed), Step(Step(Seed)), ...

  This enables lazy evaluation and
  constraint-based pruning.
end note

note left of Plus
  **Function Relations:**
  Mathematical functions like plus, multiply, etc.
  are relations with continuum cardinality.

  Membership = evaluation
  Generation = enumeration
end note

@enduml