| import OAI.Probability.InvariantIsing.Spectral.CompactResolventTests |
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| namespace OAI |
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|
| /-! The exact uniform modulus used when clipping nonnegative spectra. -/ |
| noncomputable section |
| namespace InvariantIsing |
|
|
| theorem positiveResolventTest_sub_le {t : ℝ} (ht : 0 < t) (x y : ℝ) : |
| |positiveResolventTest t x-positiveResolventTest t y| ≤ |x-y|/t^2 := by |
| have hx := le_max_right x 0 |
| have hy := le_max_right y 0 |
| have htx : 0 < t+max x 0 := add_pos_of_pos_of_nonneg ht hx |
| have hty : 0 < t+max y 0 := add_pos_of_pos_of_nonneg ht hy |
| have hd : t^2 ≤ (t+max x 0)*(t+max y 0) := by |
| nlinarith [mul_nonneg hx hy] |
| have hn : |max y 0-max x 0| ≤ |x-y| := by |
| simpa only [Real.dist_eq,NNReal.coe_one,one_mul,abs_sub_comm,id_eq] using |
| (LipschitzWith.id.max_const (0 : ℝ)).dist_le_mul y x |
| have he : positiveResolventTest t x-positiveResolventTest t y = |
| (max y 0-max x 0)/((t+max x 0)*(t+max y 0)) := by |
| unfold positiveResolventTest |
| field_simp [htx.ne',hty.ne'] |
| ring |
| rw [he,abs_div,abs_of_pos (mul_pos htx hty)] |
| exact (div_le_div_of_nonneg_right hn (mul_pos htx hty).le).trans |
| (div_le_div_of_nonneg_left (abs_nonneg _) (sq_pos_of_pos ht) hd) |
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|
| end InvariantIsing |
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| end |
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| end OAI |
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|
