| import Mathlib |
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| namespace OAI |
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|
| noncomputable section |
| open scoped BigOperators Classical |
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| namespace BinaryCoordinateSweeps |
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| lemma exists_bit_blocks (r : ℕ) (hr : 0<r) : ∀d : ℕ, r≤d → |
| ∃b : ℕ, 1≤b ∧ ∃bits : Fin b → ℕ, |
| (∀j,r≤bits j ∧ bits j≤2*r) ∧ ∑j,bits j=d := by |
| intro d |
| induction d using Nat.strong_induction_on with |
| | h d ih => |
| intro hd |
| by_cases hsmall : d≤2*r |
| · refine ⟨1,by omega,fun _ => d,fun _ => ⟨hd,hsmall⟩,?_⟩ |
| simp |
| obtain ⟨b,hb,bits,hbits,hsum⟩ := ih (d-r) (by omega) (by omega) |
| refine ⟨b+1,by omega,Fin.cons r bits,?_,?_⟩ |
| · intro j |
| refine Fin.cases ?_ (fun i => ?_) j |
| · simp only [Fin.cons_zero]; omega |
| · simpa using hbits i |
| · rw [Fin.sum_univ_succ] |
| simp only [Fin.cons_zero,Fin.cons_succ,hsum,Nat.add_sub_of_le hd] |
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| end BinaryCoordinateSweeps |
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| end |
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| end OAI |
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|
