128 lines
2.9 KiB
Plaintext
128 lines
2.9 KiB
Plaintext
import Mathlib.Analysis.SpecialFunctions.Log.Basic
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open Real
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noncomputable def logpos : ℝ → ℝ := fun r ↦ max 0 (log r)
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notation "log⁺" => logpos
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theorem loglogpos {r : ℝ} : log r = log⁺ r - log⁺ r⁻¹ := by
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unfold logpos
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rw [log_inv]
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by_cases h : 0 ≤ log r
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· simp [h]
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· simp at h
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have : 0 ≤ -log r := Left.nonneg_neg_iff.2 (le_of_lt h)
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simp [h, this]
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exact neg_nonneg.mp this
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theorem logpos_norm {r : ℝ} : log⁺ r = 2⁻¹ * (log r + ‖log r‖) := by
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by_cases hr : 0 ≤ log r
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· rw [norm_of_nonneg hr]
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have : logpos r = log r := by
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unfold logpos
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simp [hr]
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rw [this]
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ring
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· rw [norm_of_nonpos (le_of_not_ge hr)]
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have : logpos r = 0 := by
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unfold logpos
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simp
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exact le_of_not_ge hr
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rw [this]
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ring
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theorem logpos_abs
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{x : ℝ} :
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log⁺ x = log⁺ |x| := by
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unfold logpos
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simp
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-- Warning: This should be fixed in Mathlib
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theorem log_pos_iff' (hx : 0 ≤ x) : 0 < log x ↔ 1 < x := by
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by_cases h₁x : x = 0
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· rw [h₁x]; simp
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· exact log_pos_iff (lt_of_le_of_ne hx (Ne.symm h₁x))
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theorem Real.monotoneOn_logpos :
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MonotoneOn logpos (Set.Ici 0) := by
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intro x hx y hy hxy
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unfold logpos
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simp
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by_cases h : log x ≤ 0
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· tauto
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· simp [h]
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simp at h
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have : log x ≤ log y := by
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apply log_le_log
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--
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rw [log_pos_iff' hx] at h
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have : (0 : ℝ) < 1 := by exact Real.zero_lt_one
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exact lt_trans this h
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assumption
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constructor
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· linarith
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· linarith
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theorem logpos_add_le_add_logpos_add_log2₀
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{a b : ℝ}
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(h : |a| ≤ |b|) :
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log⁺ (a + b) ≤ log⁺ a + log⁺ b + log 2 := by
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nth_rw 1 [logpos_abs]
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nth_rw 2 [logpos_abs]
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nth_rw 3 [logpos_abs]
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calc log⁺ |a + b|
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_ ≤ log⁺ (|a| + |b|) := by
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apply Real.monotoneOn_logpos
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simp [abs_nonneg]; simp [abs_nonneg]
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apply add_nonneg
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simp [abs_nonneg]; simp [abs_nonneg]
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exact abs_add_le a b
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_ ≤ log⁺ (|b| + |b|) := by
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apply Real.monotoneOn_logpos
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simp [abs_nonneg]
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apply add_nonneg
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simp [abs_nonneg]; simp [abs_nonneg]
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simp [h]
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linarith
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_ = log⁺ (2 * |b|) := by
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congr; ring
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_ ≤ log⁺ |b| + log 2 := by
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unfold logpos; simp
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constructor
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· apply add_nonneg
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simp
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exact log_nonneg one_le_two
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· by_cases hb: b = 0
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· rw [hb]; simp
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exact log_nonneg one_le_two
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· rw [log_mul, log_abs, add_comm]
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simp
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exact Ne.symm (NeZero.ne' 2)
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exact abs_ne_zero.mpr hb
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_ ≤ log⁺ |a| + log⁺ |b| + log 2 := by
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unfold logpos; simp
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theorem logpos_add_le_add_logpos_add_log2
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{a b : ℝ} :
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log⁺ (a + b) ≤ log⁺ a + log⁺ b + log 2 := by
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by_cases h : |a| ≤ |b|
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· exact logpos_add_le_add_logpos_add_log2₀ h
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· rw [add_comm a b, add_comm (log⁺ a) (log⁺ b)]
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apply logpos_add_le_add_logpos_add_log2₀
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exact le_of_not_ge h
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