import Erdos1135.Tao.Syracuse.Defs

/-!
# Basic Deterministic Facts About The Syracuse Map

This module records small facts about `syracuse := Terras.oddOnly` using Mathlib's `ordCompl`
factorization API.  It does not state or prove any Collatz/Syracuse transfer theorem.
-/

namespace Erdos1135
namespace Tao

/-- The exponent stripped by one Syracuse step. -/
def syracuseExponent (N : ℕ) : ℕ :=
  Terras.twoAdicExponent (3 * N + 1)

theorem syracuse_eq_ordCompl_two (N : ℕ) :
    syracuse N = ordCompl[2] (3 * N + 1) :=
  rfl

theorem syracuse_odd (N : ℕ) :
    Odd (syracuse N) := by
  rw [syracuse_eq_ordCompl_two]
  exact Nat.not_even_iff_odd.mp (by
    intro heven
    have hdiv : (2 : ℕ) ∣ ordCompl[2] (3 * N + 1) := even_iff_two_dvd.mp heven
    exact Nat.not_dvd_ordCompl Nat.prime_two (by omega : 3 * N + 1 ≠ 0) hdiv)

theorem syracuse_pos (N : ℕ) :
    0 < syracuse N := by
  rw [syracuse_eq_ordCompl_two]
  exact Nat.ordCompl_pos 2 (by omega : 3 * N + 1 ≠ 0)

theorem two_pow_syracuseExponent_mul_syracuse (N : ℕ) :
    (2 : ℕ) ^ syracuseExponent N * syracuse N = 3 * N + 1 := by
  simpa [syracuse, Terras.oddOnly, syracuseExponent, Terras.twoAdicExponent]
    using Nat.ordProj_mul_ordCompl_eq_self (3 * N + 1) 2

theorem syracuseExponent_pos_of_odd {N : ℕ} (hN : Odd N) :
    0 < syracuseExponent N := by
  have hnum_ne : 3 * N + 1 ≠ 0 := by omega
  have hdiv : (2 : ℕ) ∣ 3 * N + 1 := by
    rcases hN with ⟨k, hk⟩
    use 3 * k + 2
    omega
  unfold syracuseExponent Terras.twoAdicExponent
  exact Nat.Prime.factorization_pos_of_dvd Nat.prime_two hnum_ne hdiv

end Tao
end Erdos1135