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a2ee19a | 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 | import OAI.Probability.SATComputability.EffectiveLog
namespace OAI
namespace PeriodicLattice.CertifiedReal
open Encodable Filter
open scoped Topology BigOperators
local instance effectiveSumsRatPrimcodable : Primcodable ℚ := RecursiveArithmetic.ratPrimcodable
variable {A : Type*} [Primcodable A]
theorem Effective.sum_range {f : A → ℕ → ℝ}
(hf : Effective (fun p : A × ℕ => f p.1 p.2))
{m : A → ℕ} (hm : Computable m) :
Effective (fun a => ∑ i ∈ Finset.range (m a), f a i) := by
obtain ⟨q, hq, herror⟩ := hf
apply of_enclosures
(q := fun an => ∑ i ∈ Finset.range (m an.1), q ((an.1,i),an.2))
(e := fun an => (m an.1 : ℚ)*qerror an.2)
· apply computable_sum (hm.comp Computable.fst)
unfold Computable₂
fun_prop
· fun_prop
· intro a n
simp only [Rat.cast_sum, Rat.cast_mul, Rat.cast_natCast, cast_qerror]
rw [← Finset.sum_sub_distrib]
calc
_ ≤ ∑ i ∈ Finset.range (m a), |f a i - (q ((a,i),n) : ℝ)| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _i ∈ Finset.range (m a), error n :=
Finset.sum_le_sum (fun i _ => herror (a,i) n)
_ = _ := by simp
· intro a
simpa only [Rat.cast_mul, Rat.cast_natCast, cast_qerror, mul_zero] using
error_tendsto.const_mul (m a : ℝ)
theorem effective_limit {f : A → ℝ} {g : A → ℕ → ℝ}
(hg : Effective (fun p : A × ℕ => g p.1 p.2))
(tail : A × ℕ → ℚ) (htail : Computable tail)
(hbound : ∀ a n, |f a - g a n| ≤ (tail (a,n) : ℝ))
(hzero : ∀ a, Tendsto (fun n => (tail (a,n) : ℝ)) atTop (nhds 0)) :
Effective f := by
obtain ⟨q, hq, he⟩ := hg
apply of_enclosures (q := fun an => q ((an.1,an.2),an.2))
(e := fun an => tail an + qerror an.2)
· fun_prop
· fun_prop
· intro a n
simp only [Rat.cast_add, cast_qerror]
exact (abs_sub_le _ _ _).trans (add_le_add (hbound a n) (he (a,n) n))
· intro a
simpa only [Rat.cast_add, cast_qerror, add_zero] using (hzero a).add error_tendsto
end PeriodicLattice.CertifiedReal
end OAI
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