This project is to prove several theorems in transcendental number theory:
- Countability argument: abstract existence of transcendental number;
- Liouvielle theorem and hence
is transcendental;
-
is transcendental;
-
is transcendental.
The main theorem is in algebraic_countable_over_Z.lean
theorem transcendental_number_exists : ∃ x : real, ¬ (is_algebraic ℤ x)
The other version is in algebraic_countable_over_Q.lean
theorem transcendental_number_exists : ∃ x : real, ¬ (is_algebraic ℚ x)
Definition of the explicit Liouville number is in liouville_theorem.lean
def α := ∑' n, ten_pow_n_fact_inverse n
The main theorem is in liouville_theorem.lean:
theorem liouville_numbers_transcendental : ∀ x : real, liouville_number x -> ¬(is_algebraic ℤ x)
theorem transcendental_α : transcendental α := liouville_numbers_transcendental α liouville_α
We defined e in e_transcendental.lean as :
def e : ℝ := real.exp 1
The main theorem is at e_transcendental.lean:
theorem e_transcendental : ¬ is_algebraic ℤ e :=
Almost immediately, we can prove
theorem e_irrational : irrational e
theorem e_pow_transcendental (n : ℕ) (hn : n ≥ 1) : transcendental (e^n)
theorem e_pow_n_irrational (n : ℕ) (hn : n ≥ 1) : irrational (e ^ n)
Please see this for an explanation of the proof of transcendence of
I haven't finished documentation (not even close), but you can click around the proves I documented so far at e_trans_helpers2.lean ande_transcendental.lean.