mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: p : ℝ
hp : p < 0
this : -p * log p < (1 - p) * log (1 - p)
⊢ p * -log p + (1 - p) * -log (1 - p) < 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: p : ℝ
hp : p < 0
hp' : p < -1
⊢ -p < 1 - p

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: p : ℝ
hp : p < 0
hp' : ¬p < -1
this : -p * log p ≤ 0
⊢ 1 < 1 - p

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
p : ℝ
⊢ deriv^[2] (qaryEntropy q) p = -1 / (p * (1 - p))

Tactic:
by
  simp only [Function.iterate_succ, Function.iterate_zero, Function.id_comp, Function.comp_apply]
  by_cases is_x_where_nondiff : p ≠ 0 ∧ p ≠ 1  -- normal case
  · obtain ⟨xne0, xne1⟩ := is_x_where_nondiff
    suffices ∀ᶠ y in (𝓝 p),
        deriv (fun p ↦ (qaryEntropy q) p) y = log (q - 1) + log (1 - y) - log y by
      refine (Filter.EventuallyEq.deriv_eq this).trans ?_
      rw [deriv_fun_sub ?_ (differentiableAt_log xne0)]
      · rw [deriv.log differentiableAt_fun_id xne0]
        simp only [deriv_id'', one_div]
        · have {q : ℝ} (p : ℝ) : DifferentiableAt ℝ (fun p => q - p) p := by fun_prop
          simp [field, sub_ne_zero_of_ne xne1.symm, this]
          ring
      · apply DifferentiableAt.add
        · simp only [differentiableAt_const]
        exact DifferentiableAt.log (by fun_prop) (sub_ne_zero.mpr xne1.symm)
    filter_upwards [eventually_ne_nhds xne0, eventually_ne_nhds xne1]
      with y xne0 h2 using deriv_qaryEntropy xne0 h2
  -- Pathological case where we use junk value (because function not differentiable)
  · have : p = 0 ∨ p = 1 := Decidable.or_iff_not_not_and_not.mpr is_x_where_nondiff
    rw [deriv_zero_of_not_differentiableAt]
    · simp_all only [ne_eq, not_and, Decidable.not_not]
      cases this <;>
        simp_all only [mul_zero, one_ne_zero, zero_ne_one, sub_zero, mul_one, div_zero, sub_self]
    · intro h
      have contAt := h.continuousAt
      cases this <;>
        simp_all [not_continuousAt_deriv_qaryEntropy_zero, not_continuousAt_deriv_qaryEntropy_one]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ p ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ ↑q - 1 ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ 1 - p ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
hp : p ∈ interior (Icc (1 - 1 / ↑q) 1)
this : 2 ≤ ↑q
⊢ 1 < ↑q

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ p ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ ↑q - 1 ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ 1 - p ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ a ∈ Ici 0,
    IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
      a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ
h₃' : EqOn (fun a ↦ a ^ p) (fun a ↦ ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ) (Ici 0)
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), MonotoneOn (cfcₙ ((↑p).rpowIntegrand₀₁ x)) (Ici 0)

Tactic:
filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ a ∈ Ici 0,
    IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
      a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ
h₃' : EqOn (fun a ↦ a ^ p) (fun a ↦ ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ) (Ici 0)
t : ℝ
ht : t ∈ Ioi 0
⊢ MonotoneOn (cfcₙ ((↑p).rpowIntegrand₀₁ t)) (Ici 0)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ a ∈ Ici 0,
    IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
      a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ
h₃' : EqOn (fun a ↦ a ^ p) (fun a ↦ ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ) (Ici 0)
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), ConcaveOn ℝ (Ici 0) (cfcₙ ((↑p).rpowIntegrand₀₁ x))

Tactic:
filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ a ∈ Ici 0,
    IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
      a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ
h₃' : EqOn (fun a ↦ a ^ p) (fun a ↦ ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ) (Ici 0)
t : ℝ
ht : t ∈ Ioi 0
⊢ ConcaveOn ℝ (Ici 0) (cfcₙ ((↑p).rpowIntegrand₀₁ t))
Module: Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto

Goal state before:
pp: s : Set ℝ
hs : s ⊆ Ioi 0
hs' : IsCompact s
ε : ℝ
hε : 0 < ε
pbound : ℝ := ε / (sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) + 1)
hxs : ∀ x ∈ s, x ≠ 0
sSup_nonneg : 0 ≤ sSup ((fun x ↦ ‖log x‖ ^ 2) '' s)
sSup_nonneg' : 0 ≤ sSup ((fun x ↦ ‖log x‖) '' s)
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ x ∈ s, (log x, n⁻¹ * (x ^ n - 1)) ∈ {p | dist p.1 p.2 ≤ ε}

Tactic:
have pbound_pos : 0 < pbound := by positivity

Goal state after:
s : Set ℝ
hs : s ⊆ Ioi 0
hs' : IsCompact s
ε : ℝ
hε : 0 < ε
pbound : ℝ := ε / (sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) + 1)
hxs : ∀ x ∈ s, x ≠ 0
sSup_nonneg : 0 ≤ sSup ((fun x ↦ ‖log x‖ ^ 2) '' s)
sSup_nonneg' : 0 ≤ sSup ((fun x ↦ ‖log x‖) '' s)
pbound_pos : 0 < pbound
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ x ∈ s, (log x, n⁻¹ * (x ^ n - 1)) ∈ {p | dist p.1 p.2 ≤ ε}
Module: Mathlib.Analysis.SpecialFunctions.FrullaniIntegral

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ |δ| * |log (b / a)| = δ * |log (b / a)|

Tactic:
abs_of_nonneg hδ

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ δ * |log (b / a)| = δ * |log (b / a)|
Module: Mathlib.Analysis.SpecialFunctions.FrullaniIntegral

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ |δ * log (b / a)| = δ * |log (b / a)|

Tactic:
exact (abs_mul δ (log (b / a))).trans (by rw [abs_of_nonneg hδ])

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
r : ℝ
hr : r ∈ spectrum ℝ z
this : (a • 1 + b • r) ^ (-1) ≤ a • 1 ^ (-1) + b • r ^ (-1)
⊢ (a + b * r) ^ (-1) ≤ a + b * r ^ (-1)

Tactic:
← Real.rpow_intCast

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
r : ℝ
hr : r ∈ spectrum ℝ z
this : (a • 1 + b • r) ^ ↑(-1) ≤ a • 1 ^ ↑(-1) + b • r ^ ↑(-1)
⊢ (a + b * r) ^ (-1) ≤ a + b * r ^ (-1)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p t x : ℝ
hp : 0 < p ∧ p < 1
ht : 0 ≤ t
hx : 0 ≤ x
hxt : 0 ≤ x * t
hx_zero : x = 0
⊢ p - 1 ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p t : ℝ
hp : p ∈ Ioo 0 1
ht : 1 ≤ t
this : p - 2 = p - 1 - 1
⊢ 0 < t

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p t : ℝ
hp : p ∈ Ioo 0 1
ht : 1 ≤ t
⊢ 0 ≤ t

Tactic:
by linarith

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.EulerMascheroni

Goal state before:
pp: n : ℕ
⊢ 0 < ↑(↑n + 1)⁻¹ - log ((↑n + 1) / (↑n + 1) + 1 / (↑n + 1))

Tactic:
div_self (by positivity),

Goal state after:
n : ℕ
⊢ 0 < ↑(↑n + 1)⁻¹ - log (1 + 1 / (↑n + 1))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p x : ℝ
hp : p ∈ Ioo 0 1
hx : 0 ≤ x
⊢ AEStronglyMeasurable (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) (volume.restrict (Ioi 1))

Tactic:
refine ContinuousOn.aestronglyMeasurable ?_ measurableSet_Ioi

Goal state after:
p x : ℝ
hp : p ∈ Ioo 0 1
hx : 0 ≤ x
⊢ ContinuousOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) (Ioi 1)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p x : ℝ
hp : p ∈ Ioo 0 1
hx : 0 ≤ x
⊢ ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x = x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1

Tactic:
obtain (rfl | hx) := hx.eq_or_lt

Goal state after:
case inl
p : ℝ
hp : p ∈ Ioo 0 1
hx : 0 ≤ 0
⊢ ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 0 = 0 ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1

case inr
p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
⊢ ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x = x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: n : ℕ
f : ℝ → ℝ := log ∘ Gamma
this : deriv (fun x ↦ log (Gamma x)) (↑n + 1) = -γ + ↑(harmonic n)
m : ℕ
⊢ ↑n + 1 ≠ -↑m

Tactic:
by linarith

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
x : ℝ
hx : 0 < x
m : ℕ
⊢ -↑m < x

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deligne

Goal state before:
pp: case pos
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
h : s = 1
⊢ 1⁻¹ = 1

Tactic:
inv_one

Goal state after:
case pos
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
h : s = 1
⊢ 1 = 1
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p : ℝ
hp : 0 < p ∧ p < 1
⊢ p - 2 < -1

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p x : ℝ
hx : 0 ≤ x
hx_zero : x = 0
hp : 0 < p ∧ p < 1
⊢ ?m.78 ≠ 0

Tactic:
by linarith

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
n : ℕ
hn : 0 < n
⊢ ↑n < ↑n + 1

Tactic:
by linarith

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
n : ℕ
⊢ ↑n + 1 < ↑n + 2

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
⊢ ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t

Tactic:
filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
t : ℝ
ht : t ∈ Ioi 0
⊢ ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: case a
n : ℕ
f : ℝ → ℝ := ⋯
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ γ ≤ -deriv f 1

Tactic:
· apply le_of_tendsto tendsto_harmonic_sub_log_add_one
    filter_upwards with n
    rw [sub_le_iff_le_add', ← sub_eq_add_neg, le_sub_iff_add_le', ← hder_nat]
    exact derivUB n

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: n m : ℕ
⊢ ↑n + 1 ≠ -↑m

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation

Goal state before:
pp: a : ℂ
ha : a.re < 0
b : ℂ
s : ℝ
⊢ (fun x ↦ cexp (a * ↑x ^ 2 + b * ↑x)) =o[atBot ⊔ atTop] fun x ↦ |x| ^ s

Tactic:
isLittleO_sup

Goal state after:
a : ℂ
ha : a.re < 0
b : ℂ
s : ℝ
⊢ ((fun x ↦ cexp (a * ↑x ^ 2 + b * ↑x)) =o[atBot] fun x ↦ |x| ^ s) ∧
    (fun x ↦ cexp (a * ↑x ^ 2 + b * ↑x)) =o[atTop] fun x ↦ |x| ^ s
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
⊢ ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t

Tactic:
filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
t : ℝ
ht : t ∈ Ioi 0
⊢ ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p).rpowIntegrand₁₂ x 0 = 0

Tactic:
filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
⊢ (↑p).rpowIntegrand₁₂ t 0 = 0
Module: Mathlib.Data.Int.Log

Goal state before:
pp: case inl
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hr : 1 ≤ 1
⊢ ↑0 = -↑0

Tactic:
Int.ofNat_zero,

Goal state after:
case inl
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hr : 1 ≤ 1
⊢ 0 = -0
Module: Mathlib.Data.Int.Log

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hb : b ≤ 1
r : R
h : r ≤ 1
⊢ -↑0 = 0

Tactic:
Int.ofNat_zero,

Goal state after:
case inr
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hb : b ≤ 1
r : R
h : r ≤ 1
⊢ -0 = 0
Module: Mathlib.Data.Int.Log

Goal state before:
pp: R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
r : R
hr : r ≤ 0
⊢ -↑0 = 0

Tactic:
Int.ofNat_zero,

Goal state after:
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
r : R
hr : r ≤ 0
⊢ -0 = 0
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b : ℝ
hb : 1 < b
⊢ 0 < b

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b : ℝ
hb : 1 < b
⊢ b ≠ 1

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b x y : ℝ
hb : 1 < b
h : 0 < x
hxy : x ≤ y
⊢ 0 < y

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Monotone

Goal state before:
pp: x : ℝ
hx : x ∈ interior (Ici (rexp (-1)))
hlt : rexp (-1) < x
⊢ 0 < deriv (fun x ↦ x * log x) x

Tactic:
have hpos : 0 < x := by grind [Real.exp_pos]

Goal state after:
x : ℝ
hx : x ∈ interior (Ici (rexp (-1)))
hlt : rexp (-1) < x
hpos : 0 < x
⊢ 0 < deriv (fun x ↦ x * log x) x
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b : ℝ
b_lt_one : b < 1
⊢ b ≠ 1

Tactic:
by linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral

Goal state before:
pp: E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
ε : ℝ
g : E →ᵇ ℝ
hε : 0 < ε
N : ℕ
hgN : ε * (‖g‖ * ‖g‖) < ↑N
n : ℕ
hn : N ≤ n
hnpos : 0 < ↑n
x : E
⊢ -1 ≤ 0

Tactic:
by linarith

Goal state after: