| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
| namespace ControlledBoundary.FormalDivisor |
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| variable {Prime : Type*} |
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| def positivePart (D : Prime →₀ ℚ) : Prime →₀ ℚ := |
| D.mapRange (fun x => max x 0) (by simp) |
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| def negativePart (D : Prime →₀ ℚ) : Prime →₀ ℚ := |
| D.mapRange (fun x => max (-x) 0) (by simp) |
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| lemma positivePart_nonneg (D : Prime →₀ ℚ) : 0 ≤ positivePart D := |
| fun _ => le_max_right _ _ |
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| lemma negativePart_nonneg (D : Prime →₀ ℚ) : 0 ≤ negativePart D := |
| fun _ => le_max_right _ _ |
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| lemma signed_decomposition (D : Prime →₀ ℚ) : |
| positivePart D - negativePart D = D := by |
| ext p |
| change max (D p) 0 - max (-D p) 0 = D p |
| rcases le_total 0 (D p) with h | h |
| · rw [max_eq_left h, max_eq_right (neg_nonpos.mpr h), sub_zero] |
| · rw [max_eq_right h, max_eq_left (neg_nonneg.mpr h)] |
| ring |
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| lemma positivePart_coeff_lt_one (D : Prime →₀ ℚ) (h : ∀ p, D p < 1) : |
| ∀ p, positivePart D p < 1 := by |
| intro p |
| exact max_lt (h p) zero_lt_one |
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| |
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| lemma negativePart_support_subset (D : Prime →₀ ℚ) (S : Finset Prime) |
| (h : ∀ p, p ∉ S → 0 ≤ D p) : (negativePart D).support ⊆ S := by |
| intro p hp |
| by_contra hn |
| have hz : negativePart D p = 0 := max_eq_right (neg_nonpos.mpr (h p hn)) |
| exact (Finsupp.mem_support_iff.mp hp) hz |
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| |
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| lemma negativePart_signed_support [DecidableEq Prime] (G J K : Prime →₀ ℚ) |
| (s : ℚ) (hs : 0 ≤ s) (hG : 0 ≤ G) : |
| (negativePart (s • G - J - K)).support ⊆ J.support ∪ K.support := by |
| classical |
| apply negativePart_support_subset |
| intro p hp |
| have hJ : J p = 0 := Finsupp.notMem_support_iff.mp (fun h => |
| hp (Finset.mem_union_left _ h)) |
| have hK : K p = 0 := Finsupp.notMem_support_iff.mp (fun h => |
| hp (Finset.mem_union_right _ h)) |
| simpa only [Finsupp.sub_apply, Finsupp.smul_apply, smul_eq_mul, hJ, hK, sub_zero] |
| using mul_nonneg hs (hG p) |
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| |
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| theorem normalize_correction {n : ℕ} |
| (M F E C : Prime →₀ ℚ) (P : Fin n → Prime →₀ ℚ) (a : Fin n → ℚ) |
| (t : ℚ) (ht : 0 < t) (hF : 0 ≤ F) (hE : 0 ≤ E) |
| (hSupp : F.support ⊆ C.support) |
| (hrep : E = t • M + F + ∑ j, a j • P j) : |
| ∃ C₀ E₀ : Prime →₀ ℚ, |
| 0 ≤ C₀ ∧ C₀.support ⊆ C.support ∧ 0 ≤ E₀ ∧ |
| E₀ = M + C₀ + ∑ j, (t⁻¹ * a j) • P j := by |
| refine ⟨t⁻¹ • F, t⁻¹ • E, ?_, ?_, ?_, ?_⟩ |
| · intro p |
| exact mul_nonneg (inv_nonneg.mpr ht.le) (hF p) |
| · exact (Finsupp.support_smul).trans hSupp |
| · intro p |
| exact mul_nonneg (inv_nonneg.mpr ht.le) (hE p) |
| · rw [hrep, smul_add, smul_add, smul_smul, inv_mul_cancel₀ (ne_of_gt ht), |
| one_smul, Finset.smul_sum] |
| simp only [smul_smul] |
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| end ControlledBoundary.FormalDivisor |
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| end |
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| end OAI |
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