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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | import OAI.Probability.InvariantIsing.Spectral.CompactResolventTests
namespace OAI
/-! 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)
end InvariantIsing
end
end OAI
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