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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | 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
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