| import Mathlib |
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| namespace OAI |
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| /-! Calabi Cutoff Algebra. -/ |
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| section |
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| namespace CalabiEstimate |
| lemma cross_absorb {x y r : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) |
| (hr : r^2 ≤ 8*x*y) : -(y/2+4*x) ≤ r := by |
| have hab : |r| ≤ y/2+4*x := by |
| apply (sq_le_sq₀ (abs_nonneg _) (by positivity)).mp |
| rw [sq_abs] |
| nlinarith [sq_nonneg (y/2-4*x)] |
| exact (neg_le_neg hab).trans (neg_abs_le r) |
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| lemma cutoff_max_bound {S Q η C δ LS LT Lη cross : ℝ} |
| (hS : 0 ≤ S) (hQ : 0 ≤ Q) (hη : 0 ≤ η) (hη1 : η ≤ 1) |
| (hC : 0 ≤ C) (hδ : 0 < δ) |
| (hLS : Q-C*(1+S) ≤ LS) (hLT : δ*S-C ≤ LT) |
| (hLη : -C ≤ Lη) (hcross : cross^2 ≤ 8*C*η*S*Q) |
| (hmax : η*LS+S*Lη+cross+((6*C+1)/δ)*LT ≤ 0) : |
| S ≤ C*(1+(6*C+1)/δ) := by |
| let A := (6*C+1)/δ |
| have hA : 0 ≤ A := div_nonneg (by positivity) hδ.le |
| have hAd : A*δ = 6*C+1 := by dsimp [A]; field_simp |
| have hx : -(η*Q/2+4*(C*S)) ≤ cross := by |
| apply cross_absorb (mul_nonneg hC hS) (mul_nonneg hη hQ) |
| nlinarith [hcross] |
| have h1 := mul_le_mul_of_nonneg_left hLS hη |
| have h2 := mul_le_mul_of_nonneg_left hLη hS |
| have h3 := mul_le_mul_of_nonneg_left hLT hA |
| have h4 := mul_le_mul_of_nonneg_right hη1 (mul_nonneg hC (by linarith : 0 ≤ 1+S)) |
| have h5 := mul_nonneg hη hQ |
| have he : A*(δ*S-C) = (6*C+1)*S-A*C := by rw [mul_sub,← mul_assoc,hAd] |
| rw [he] at h3 |
| change η*LS+S*Lη+cross+A*LT ≤ 0 at hmax |
| change S ≤ C*(1+A) |
| nlinarith |
| end CalabiEstimate |
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| end |
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| end OAI |
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