-- Dedekind real numbers
-- Part of Cumulus, the Grace17 mathematics library.
-- Copyright (c) 2026 Grant Passmore.
-- Released under the Apache License 2.0. See LICENSE in the repository root.
-- Authors: Grant Passmore

module Grace.Cumulus.Real.Defs
import Rat.OrderedOperations

predicate ProperSubset (A B : Set) := (A ⊆ B) ∧ A ≠ B
notation infix 50 "⊂" => ProperSubset

-- A real is a lower Dedekind cut of the rational quotient set. The order used
-- here is the set-valued graph rationalLess.
predicate DedekindCut (A : Set) :=
  (A ⊆ ℚ) ∧ Nonempty(A) ∧ A ≠ ℚ ∧
  (∀ q ∈ A, ∀ p ∈ ℚ, p < q → p ∈ A) ∧
  (∀ q ∈ A, ∃ r ∈ A, relates(rationalLess, q, r))

def ℝ : Set := { A ∈ 𝒫(ℚ) | DedekindCut(A) }
class Real (x : Set) := x ∈ ℝ

-- Strict real order is proper inclusion of lower cuts. The predicate-level
-- view remains useful alongside the instance-selected `<` relation graph.
predicate RealLess (x y : Set) := Real(x) ∧ Real(y) ∧ (x ⊂ y)

-- The set-valued graph used by algebraic structures and relation theory.
-- RealLess remains the convenient proposition-level view.
def realLess : Set :=
  { p ∈ ℝ × ℝ |
    ∃ x ∈ ℝ, ∃ y ∈ ℝ, p = (x, y) ∧ (x ⊂ y) }

-- The rational q determines its lower initial segment. Once rationalLess is
-- proved dense and unbounded, rationalCut(q) is the canonical Q → R map.
def rationalCut (q : Set) : Set :=
  { p ∈ ℚ | relates(rationalLess, p, q) }