| import OAI.MeasureTheory.SelfSimilar.WordTypes |
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| namespace OAI |
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| universe uα uβ |
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| section |
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| open scoped BigOperators |
| namespace EntropyRateDimension |
| namespace DigitSumset |
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| noncomputable def counts {α : Type uα} {β : Type uβ} [Fintype α] [Fintype β] {k : ℕ} |
| (w : Fin k → β → α) : β → α → Fin (k+1) := by |
| classical |
| exact fun j a => ⟨(Finset.univ.filter (fun t => w t j=a)).card, |
| Nat.lt_succ_of_le ((Finset.card_le_card (Finset.filter_subset _ _)).trans_eq (Fintype.card_fin k))⟩ |
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| theorem counts_expand {α : Type uα} {β : Type uβ} [Fintype α] [Fintype β] {k : ℕ} |
| (v : β → α → ℝ) (w : Fin k → β → α) : |
| ∑ t, ∑ j, v j (w t j) = ∑ j, ∑ a, ((counts w j a).val : ℝ)*v j a := by |
| classical |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [← Finset.sum_fiberwise Finset.univ (fun t : Fin k => w t j)] |
| apply Finset.sum_congr rfl |
| intro a _ |
| calc |
| ∑ t ∈ Finset.univ.filter (fun t : Fin k => w t j=a), v j (w t j) = |
| ∑ _t ∈ Finset.univ.filter (fun t : Fin k => w t j=a), v j a := by |
| apply Finset.sum_congr rfl |
| intro t ht |
| rw [(Finset.mem_filter.mp ht).2] |
| _ = _ := by simp [counts] |
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| noncomputable def sumset (S : Finset ℝ) (k : ℕ) : Finset ℝ := by |
| classical |
| exact Finset.univ.image (fun w : Fin k → S => ∑ t, (w t : ℝ)) |
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| theorem card_sumset_le {α : Type uα} {β : Type uβ} [Fintype α] [Fintype β] |
| (S : Finset ℝ) (v : β → α → ℝ) |
| (hS : ∀ x ∈ S, ∃ w : β → α, x=∑ j, v j (w j)) (k : ℕ) : |
| (sumset S k).card ≤ (k+1)^(Fintype.card β*Fintype.card α) := by |
| classical |
| let F : (β → α → Fin (k+1)) → ℝ := fun c => ∑ j, ∑ a, ((c j a).val : ℝ)*v j a |
| have hin : sumset S k ⊆ Finset.univ.image F := by |
| intro x hx |
| obtain ⟨u,_,rfl⟩ := Finset.mem_image.mp hx |
| choose w hw using (fun t => hS (u t) (u t).property) |
| apply Finset.mem_image.mpr |
| refine ⟨counts w,Finset.mem_univ _,?_⟩ |
| change (∑ j, ∑ a, ((counts w j a).val : ℝ)*v j a)=∑ t, (u t : ℝ) |
| rw [← counts_expand] |
| exact Finset.sum_congr rfl (fun t _ => (hw t).symm) |
| calc |
| (sumset S k).card ≤ (Finset.univ.image F).card := Finset.card_le_card hin |
| _ ≤ (Finset.univ : Finset (β → α → Fin (k+1))).card := Finset.card_image_le |
| _ = _ := by simp only [Finset.card_univ,Fintype.card_fun,Fintype.card_fin,← pow_mul]; rw [Nat.mul_comm] |
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| end DigitSumset |
| end EntropyRateDimension |
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| end |
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| end OAI |
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