| import OAI.MathematicalPhysics.AlternatingFlow.EffectiveProfiles |
| import OAI.MathematicalPhysics.AlternatingFlow.Names |
|
|
| namespace OAI |
|
|
| section ExpNamesDevelopment |
|
|
| open scoped BigOperators ContDiff Topology |
| open Filter |
|
|
| namespace AlternatingNS.Effective |
| attribute [local instance] Arithmetic.rationalCoding |
|
|
| lemma rat_sum_range {A : Type*} [Primcodable A] {f : A → ℕ → ℚ} |
| (hf : Computable₂ f) : Computable₂ (fun a n => ∑ i ∈ Finset.range n, f a i) := by |
| have hs : Computable₂ (fun a : A => fun z : ℕ × ℚ => z.2 + f a z.1) := |
| Arithmetic.rat_add.to_comp.comp (Computable.snd.comp Computable.snd) |
| (hf.comp Computable.fst (Computable.fst.comp Computable.snd)) |
| have h := Computable.nat_rec (Computable.snd : Computable (fun z : A × ℕ => z.2)) |
| (Computable.const (0 : ℚ)) |
| (hs.comp (Computable.fst.comp Computable.fst) Computable.snd).to₂ |
| refine h.to₂.of_eq ?_ |
| rintro ⟨a,n⟩ |
| induction n with |
| | zero => simp |
| | succ n ih => simp only [Finset.sum_range_succ, ih] |
|
|
| lemma abs_rat_cast_le_num (q : ℚ) : |(q : ℝ)| ≤ q.num.natAbs := by |
| have h : (q.den : ℝ) ≥ 1 := by exact_mod_cast q.den_pos |
| rw [Rat.cast_def, abs_div, ← Int.cast_abs, ← Int.natCast_natAbs, Int.cast_natCast, |
| abs_of_nonneg (Nat.cast_nonneg _)] |
| exact div_le_self (Nat.cast_nonneg _) h |
|
|
| lemma exp_named : Named (fun q : ℚ => Real.exp (q : ℝ)) := by |
| let N (z : ℚ × ℕ) : ℕ := 2 * z.1.num.natAbs + z.2 + 1 |
| have hN : Computable N := (Primrec.nat_add.comp |
| (Primrec.nat_add.comp (Primrec.nat_mul.comp (Primrec.const 2) |
| (Arithmetic.int_abs.comp (Arithmetic.rat_num.comp Primrec.fst))) Primrec.snd) |
| (Primrec.const 1)).to_comp |
| let a (z : ℚ × ℕ) : ℚ := ∑ i ∈ Finset.range (N z), z.1 ^ i / i.factorial |
| let err (z : ℚ × ℕ) : ℚ := |z.1| ^ N z / (N z).factorial * 2 |
| have ht : Computable₂ (fun q : ℚ => fun i : ℕ => q ^ i / i.factorial) := |
| Arithmetic.rat_div.to_comp.comp Arithmetic.rat_pow.to_comp |
| (Arithmetic.rat_natCast.to_comp.comp (factorial.to_comp.comp Computable.snd)) |
| have ha : Computable a := (rat_sum_range ht).comp Computable.fst hN |
| have he : Computable err := Arithmetic.rat_mul.to_comp.comp |
| (Arithmetic.rat_div.to_comp.comp |
| (Arithmetic.rat_pow.to_comp.comp (Arithmetic.rat_abs.to_comp.comp Computable.fst) hN) |
| (Arithmetic.rat_natCast.to_comp.comp (factorial.to_comp.comp hN))) (Computable.const 2) |
| apply Named.of_certificate _ a err ha he |
| · intro q n |
| have hl : ‖(q : ℂ)‖ / (N (q,n)).succ ≤ 1 / 2 := by |
| rw [div_le_iff₀ (by positivity : (0 : ℝ) < (N (q,n)).succ)] |
| rw [show (q : ℂ) = ((q : ℝ) : ℂ) by simp, Complex.norm_real, Real.norm_eq_abs] |
| have hh := abs_rat_cast_le_num q |
| dsimp [N]; push_cast; linarith |
| have h := Complex.exp_bound' hl |
| have hc : (∑ i ∈ Finset.range (N (q,n)), (q : ℂ) ^ i / i.factorial) = (a (q,n) : ℂ) := by |
| simp [a] |
| rw [hc, show (q : ℂ) = ((q : ℝ) : ℂ) by simp, ← Complex.ofReal_exp, |
| show (a (q,n) : ℂ) = ((a (q,n) : ℝ) : ℂ) by simp, |
| ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs, |
| Complex.norm_real, Real.norm_eq_abs] at h |
| simpa only [err, Rat.cast_mul, Rat.cast_div, Rat.cast_pow, Rat.cast_abs, |
| Rat.cast_natCast, Rat.cast_ofNat] using h |
| · intro q |
| have hlim := (FloorSemiring.tendsto_pow_div_factorial_atTop |(q : ℝ)|).mul_const 2 |
| have hshift : Tendsto (fun n => N (q,n)) atTop atTop := by |
| apply tendsto_atTop_mono (f := fun n : ℕ => n) _ tendsto_id |
| intro n; dsimp [N]; omega |
| simpa only [err, Function.comp_def, Rat.cast_mul, Rat.cast_div, Rat.cast_pow, |
| Rat.cast_abs, Rat.cast_natCast, Rat.cast_ofNat, zero_mul] using hlim.comp hshift |
| |
| end AlternatingNS.Effective |
| |
| end ExpNamesDevelopment |
| |
| end OAI |
| |
