66 lines
2.6 KiB
Plaintext
66 lines
2.6 KiB
Plaintext
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import Mathlib.Analysis.SpecialFunctions.Integrals
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import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
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import Mathlib.MeasureTheory.Integral.CircleIntegral
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import Mathlib.MeasureTheory.Measure.Restrict
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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-- The following theorem was suggested by Gareth Ma on Zulip
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theorem logInt
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{t : ℝ}
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(ht : 0 < t) :
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∫ x in (0 : ℝ)..t, log x = t * log t - t := by
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rw [← integral_add_adjacent_intervals (b := 1)]
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trans (-1) + (t * log t - t + 1)
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· congr
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· -- ∫ x in 0..1, log x = -1, same proof as before
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rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
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· simp
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· simp
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· intro x hx
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norm_num at hx
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convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
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norm_num
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· rw [← neg_neg log]
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apply IntervalIntegrable.neg
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apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
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· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
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· intro x hx
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norm_num at hx
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convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
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norm_num
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· intro x hx
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norm_num at hx
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rw [Pi.neg_apply, Left.nonneg_neg_iff]
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exact (log_nonpos_iff hx.left).mpr hx.right.le
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· have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
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simp_rw [rpow_one, mul_comm] at this
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-- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
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convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
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norm_num
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· rw [(by simp : -1 = 1 * log 1 - 1)]
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apply tendsto_nhdsWithin_of_tendsto_nhds
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exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
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· -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
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rw [integral_log_of_pos zero_lt_one ht]
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norm_num
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· abel
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· -- log is integrable on [[0, 1]]
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rw [← neg_neg log]
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apply IntervalIntegrable.neg
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apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
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· exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
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· intro x hx
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norm_num at hx
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convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
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norm_num
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· intro x hx
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norm_num at hx
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rw [Pi.neg_apply, Left.nonneg_neg_iff]
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exact (log_nonpos_iff hx.left).mpr hx.right.le
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· -- log is integrable on [[0, t]]
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simp [Set.mem_uIcc, ht]
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