| import OAI.Probability.SATComputability.EffectiveLog |
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| namespace OAI |
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| namespace PeriodicLattice.CertifiedReal |
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| open Encodable Filter |
| open scoped Topology BigOperators |
| local instance effectiveSumsRatPrimcodable : Primcodable ℚ := RecursiveArithmetic.ratPrimcodable |
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| variable {A : Type*} [Primcodable A] |
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| 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) |
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| 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 |
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| end PeriodicLattice.CertifiedReal |
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| end OAI |
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