mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
s : Set A := {a | IsStrictlyPositive a}
f : ℝ → A → A := fun p a ↦ if a ∈ s then cfc (fun x ↦ p⁻¹ * (x ^ p - 1)) a else 0
g : A → A := fun a ↦ if a ∈ s then log a else 0
hg : Set.EqOn g log s
h₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0 < p ∧ p < 1
p : ℝ
hp : 0 < p
hp' : p < 1
a : A
ha : a ∈ {a | IsStrictlyPositive a}
b : A
hb : b ∈ {a | IsStrictlyPositive a}
hab : a ≤ b
⊢ p⁻¹ • (a ^ p - 1) ≤ p⁻¹ • (b ^ p - 1)

Tactic:
gcongr

Goal state after:
case hp
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
s : Set A := ⋯
f : ℝ → A → A := ⋯
g : A → A := ⋯
hg : Set.EqOn g log s
h₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0 < p ∧ p < 1
p : ℝ
hp : 0 < p
hp' : p < 1
a : A
ha : a ∈ {a | IsStrictlyPositive a}
b : A
hb : b ∈ {a | IsStrictlyPositive a}
hab : a ≤ b
⊢ p ∈ Set.Icc 0 1
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)
pbound_pos : 0 < pbound
h₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0 < p ∧ p < pbound
h₂ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
hcont : ContinuousOn (fun x ↦ ‖log x‖ ^ 2) s
hcont' : ContinuousOn (fun x ↦ ‖log x‖) s
p : ℝ
hp₁ : 0 < p
hp₂ : p < pbound
hp₃ : p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
x : ℝ
hx : x ∈ s
hx' : ‖p * log x‖ ≤ 1
pinv_nonneg : 0 ≤ p⁻¹
⊢ p * ‖log x‖ ^ 2 ≤ p * sSup ((fun x ↦ ‖log x‖ ^ 2) '' s)

Tactic:
gcongr

Goal state after:
case hbc
s : Set ℝ
hs : s ⊆ Ioi 0
hs' : IsCompact s
ε : ℝ
hε : 0 < ε
pbound : ℝ := ⋯
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
h₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0 < p ∧ p < pbound
h₂ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
hcont : ContinuousOn (fun x ↦ ‖log x‖ ^ 2) s
hcont' : ContinuousOn (fun x ↦ ‖log x‖) s
p : ℝ
hp₁ : 0 < p
hp₂ : p < pbound
hp₃ : p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
x : ℝ
hx : x ∈ s
hx' : ‖p * log x‖ ≤ 1
pinv_nonneg : 0 ≤ p⁻¹
⊢ ‖log x‖ ^ 2 ≤ sSup ((fun x ↦ ‖log x‖ ^ 2) '' s)
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)
pbound_pos : 0 < pbound
h₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0 < p ∧ p < pbound
h₂ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
hcont : ContinuousOn (fun x ↦ ‖log x‖ ^ 2) s
hcont' : ContinuousOn (fun x ↦ ‖log x‖) s
p : ℝ
hp₁ : 0 < p
hp₂ : p < pbound
hp₃ : p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
x : ℝ
hx : x ∈ s
hx' : ‖p * log x‖ ≤ 1
pinv_nonneg : 0 ≤ p⁻¹
⊢ p * sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) ≤ pbound * (sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) + 1)

Tactic:
gcongr

Goal state after:
case h₂
s : Set ℝ
hs : s ⊆ Ioi 0
hs' : IsCompact s
ε : ℝ
hε : 0 < ε
pbound : ℝ := ⋯
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
h₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0 < p ∧ p < pbound
h₂ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
hcont : ContinuousOn (fun x ↦ ‖log x‖ ^ 2) s
hcont' : ContinuousOn (fun x ↦ ‖log x‖) s
p : ℝ
hp₁ : 0 < p
hp₂ : p < pbound
hp₃ : p ≤ 1 / (sSup ((fun x ↦ ‖log x‖) '' s) + 1)
x : ℝ
hx : x ∈ s
hx' : ‖p * log x‖ ≤ 1
pinv_nonneg : 0 ≤ p⁻¹
⊢ sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) ≤ sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) + 1
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
x : ℝ
hx : x ∈ uIoc a b
⊢ ‖x⁻¹ • (f x - V)‖ ≤ x⁻¹ * δ

Tactic:
by
            have hx_pos : 0 < x :=
              lt_of_lt_of_le (lt_min ha hb) (uIoc_subset_uIcc hx).1
            rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hx_pos)]
            exact mul_le_mul_of_nonneg_left (h x hx) (inv_nonneg.2 hx_pos.le)

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
⊢ (conjSqrt x⁻¹ʳ) ((a • 1 + b • z) ^ (-1)) ≤ (conjSqrt x⁻¹ʳ) (a • 1 + b • z ^ (-1))

Tactic:
gcongr

Goal state after:
case h
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 := ⋯
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
⊢ (a • 1 + b • z) ^ (-1) ≤ a • 1 + b • z ^ (-1)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: case inr
p t x : ℝ
hp : 0 < p
ht : 0 ≤ t
hx : 0 ≤ x
ht_pos : 0 < t
⊢ (t + x)⁻¹ ≤ t⁻¹

Tactic:
gcongr

Goal state after:
case hba
p t x : ℝ
hp : 0 < p
ht : 0 ≤ t
hx : 0 ≤ x
ht_pos : 0 < t
⊢ t ≤ t + x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 ≤ t
x y : ℝ
hy : y ∈ Ici 0
hxy : x ≤ y
h : ¬x = 0
hx : 0 ≤ x
⊢ t ^ p * (t⁻¹ - (t + x)⁻¹) ≤ t ^ p * (t⁻¹ - (t + y)⁻¹)

Tactic:
gcongr

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

Goal state before:
pp: case refine_2
⊢ Differentiable ℂ fun s ↦ (Gamma (s / 2))⁻¹

Tactic:
exact differentiable_one_div_Gamma.comp (differentiable_id.div_const _)

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

Goal state before:
pp: p t x : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
hx : 0 ≤ x
⊢ t ^ (p - 1) * x / (t + x) ≤ t ^ (p - 1) * x / t

Tactic:
gcongr

Goal state after:
case hdb
p t x : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
hx : 0 ≤ x
⊢ t ≤ t + x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p t x : ℝ
hp : p ∈ Ioo 0 1
ht : 0 ≤ t
hx : 0 ≤ x
hx_zero : ¬x = 0
⊢ t ^ (p - 1) * x / (t + x) ≤ t ^ (p - 1) * x / x

Tactic:
gcongr

Goal state after:
case hdb
p t x : ℝ
hp : p ∈ Ioo 0 1
ht : 0 ≤ t
hx : 0 ≤ x
hx_zero : ¬x = 0
⊢ x ≤ t + x
Module: Mathlib.NumberTheory.Harmonic.EulerMascheroni

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

Tactic:
sub_pos,

Goal state after:
n : ℕ
⊢ log (1 + 1 / (↑n + 1)) < ↑(↑n + 1)⁻¹
Module: Mathlib.NumberTheory.Harmonic.EulerMascheroni

Goal state before:
pp: case inr
n : ℕ
hn : n > 0
⊢ 0 < log ↑(n + 1) - log ↑n - ↑(↑(n + 1))⁻¹

Tactic:
sub_pos,

Goal state after:
case inr
n : ℕ
hn : n > 0
⊢ ↑(↑(n + 1))⁻¹ < log ↑(n + 1) - log ↑n
Module: Mathlib.NumberTheory.Harmonic.EulerMascheroni

Goal state before:
pp: case inr
m n r : ℕ
hr : 0 < r
⊢ ↑(harmonic r) - log (↑r + 1) < ↑(harmonic r) - log ↑r

Tactic:
gcongr

Goal state after:
case h
m n r : ℕ
hr : 0 < r
⊢ ↑r < ↑r + 1
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
this :
  ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =
    x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
⊢ ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x = x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1

Tactic:
rwa [integral_comp_mul_deriv_Ioi (by fun_prop), mul_zero] at this

Goal state after:
case hft
p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
this :
  ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =
    x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
⊢ Tendsto (fun x_1 ↦ x * x_1) atTop atTop

case hff'
p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
this :
  ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =
    x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
⊢ ∀ x_1 ∈ Ioi 0, HasDerivWithinAt (fun x_2 ↦ x * x_2) x (Ioi x_1) x_1

case hg_cont
p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
this :
  ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =
    x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
⊢ ContinuousOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ((fun x_1 ↦ x * x_1) '' Ioi 0)

case hg1
p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
this :
  ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =
    x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
⊢ IntegrableOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ((fun x_1 ↦ x * x_1) '' Ici 0) volume

case hg2
p x : ℝ
hp : p ∈ Ioo 0 1
hx✝ : 0 ≤ x
hx : 0 < x
this :
  ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =
    x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1
⊢ IntegrableOn (fun x_1 ↦ ((fun x_2 ↦ p.rpowIntegrand₀₁ x_2 x) ∘ fun x_2 ↦ x * x_2) x_1 * x) (Ici 0) volume
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deligne

Goal state before:
pp: case neg
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
h : s - 1 ≠ 0
⊢ (2 - s).Gammaℝ⁻¹ = s.Gammaℂ * sin (↑π * s / 2) * (s + 1).Gammaℝ⁻¹

Tactic:
have h' (n : ℕ) : s - 1 ≠ -n := by
    rcases n with - | m
    · rwa [Nat.cast_zero, neg_zero]
    · rw [Ne, sub_eq_iff_eq_add]
      convert! hs m using 2
      push_cast
      ring

Goal state after:
case neg
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
h : s - 1 ≠ 0
h' : ∀ (n : ℕ), s - 1 ≠ -↑n
⊢ (2 - s).Gammaℝ⁻¹ = s.Gammaℂ * sin (↑π * s / 2) * (s + 1).Gammaℝ⁻¹
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: case refine_2
p : ℝ
hp : p ∈ Ioo 0 1
C : ℝ≥0 := NNReal.mk (∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1)⁻¹ ⋯
x : ℝ
hx : x ∈ Ici 0
⊢ (∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1)⁻¹ * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x =
    C • ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x ∂volume

Tactic:
NNReal.smul_def,

Goal state after:
case refine_2
p : ℝ
hp : p ∈ Ioo 0 1
C : ℝ≥0 := NNReal.mk (∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1)⁻¹ ⋯
x : ℝ
hx : x ∈ Ici 0
⊢ (∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1)⁻¹ * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x =
    ↑C • ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x ∂volume
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: ⊢ γ = -deriv Gamma 1

Tactic:
show (1 : ℝ) = ↑(0 : ℕ) + 1 by simp,

Goal state after:
⊢ γ = -deriv Gamma (↑0 + 1)
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ deriv Gamma (1 / 2) =
    deriv Gamma (1 / 2) * Gamma (1 / 2 + 1 / 2) + (√π * deriv (fun s ↦ Gamma (s + 1 / 2)) (1 / 2) + √π * γ)

pp: case hc
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ Gamma (1 / 2)

pp: case hd
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ (fun s ↦ Gamma (s + 1 / 2)) (1 / 2)

Tactic:
← mul_add,

Goal state after:
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ deriv Gamma (1 / 2) =
    deriv Gamma (1 / 2) * Gamma (1 / 2 + 1 / 2) + √π * (deriv (fun s ↦ Gamma (s + 1 / 2)) (1 / 2) + γ)

case hc
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ Gamma (1 / 2)

case hd
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ (fun s ↦ Gamma (s + 1 / 2)) (1 / 2)
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: case refine_1
m : ℕ
⊢ ¬↑(1 / 2) = ↑(-↑m)

Tactic:
ofReal_inj

Goal state after:
case refine_1
m : ℕ
⊢ ¬1 / 2 = -↑m
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a) ≤ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)

Tactic:
gcongr

Goal state after:
case hb
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a) ≤ cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ t⁻¹ • a ≤ t⁻¹ • b

Tactic:
gcongr

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

Goal state before:
pp: x : ℝ
hex : x ∈ Ici (rexp 1)
y : ℝ
hey : y ∈ Ici (rexp 1)
hxy : x ≤ y
x_pos : 0 < x
y_pos : 0 < y
hlogx : 1 ≤ log x
hyx : 0 ≤ y / x - 1
⊢ log y - log x ≤ (fun x ↦ log x / x) x * y - log x

Tactic:
calc
    log y - log x = log (y / x) := by rw [log_div y_pos.ne' x_pos.ne']
    _ ≤ y / x - 1 := log_le_sub_one_of_pos (div_pos y_pos x_pos)
    _ ≤ log x * (y / x - 1) := le_mul_of_one_le_left hyx hlogx
    _ = log x / x * y - log x := by ring

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

Goal state before:
pp: a : ℝ
ha : 0 < a
x : ℝ
hex : x ∈ Ici (rexp a⁻¹)
y : ℝ
x✝ : y ∈ Ici (rexp a⁻¹)
hxy : x ≤ y
x_pos : 0 < x
y_pos : 0 < y
z : ℝ
hz : z ∈ Ici (rexp a⁻¹)
⊢ rexp 1 = rexp a⁻¹ ^ a

pp: a : ℝ
ha : 0 < a
x : ℝ
hex : x ∈ Ici (rexp a⁻¹)
y : ℝ
x✝ : y ∈ Ici (rexp a⁻¹)
hxy : x ≤ y
x_pos : 0 < x
y_pos : 0 < y
z : ℝ
hz : z ∈ Ici (rexp a⁻¹)
⊢ 0 ≤ rexp a⁻¹

Tactic:
· simp only [← exp_mul, Real.exp_eq_exp, field]

Goal state after:
a : ℝ
ha : 0 < a
x : ℝ
hex : x ∈ Ici (rexp a⁻¹)
y : ℝ
x✝ : y ∈ Ici (rexp a⁻¹)
hxy : x ≤ y
x_pos : 0 < x
y_pos : 0 < y
z : ℝ
hz : z ∈ Ici (rexp a⁻¹)
⊢ 0 ≤ rexp a⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Log.Monotone

Goal state before:
pp: a : ℝ
ha : 0 < a
x : ℝ
hex : x ∈ Ici (rexp a⁻¹)
y : ℝ
x✝ : y ∈ Ici (rexp a⁻¹)
hxy : x ≤ y
x_pos : 0 < x
y_pos : 0 < y
hbound : ∀ {z : ℝ}, z ∈ Ici (rexp a⁻¹) → z ^ a ∈ {b | rexp 1 ≤ b}
⊢ x ^ a ≤ y ^ a

Tactic:
gcongr

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

Goal state before:
pp: case neg.inl
b : ℕ
r : ℝ
hr✝ : 0 ≤ r
hr : 0 < r
h✝ : b = 0
⊢ ⌈logb (↑b) r⌉ = Int.clog b r

Tactic:
simp_all only [CharP.cast_eq_zero, logb_zero_left, Int.ceil_zero, Int.clog_zero_left]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.BoundedContinuousFunction

Goal state before:
pp: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
μ : Measure X
E : Type u_2
inst✝ : NormedAddCommGroup E
f : X →ᵇ E
⊢ ∫⁻ (x : X), ↑‖f x‖₊ ∂μ ≤ ∫⁻ (x : X), ↑‖f‖₊ ∂μ

Tactic:
gcongr

Goal state after:
case _a
X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
μ : Measure X
E : Type u_2
inst✝ : NormedAddCommGroup E
f : X →ᵇ E
x : X
⊢ ‖f x‖₊ ≤ ‖f‖₊
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: case pos
b : ℝ
h : b = -1 ∨ b = 0 ∨ b = 1
⊢ logb b =O[⊤] log

Tactic:
obtain rfl | rfl | rfl := h

Goal state after:
case pos.inl
⊢ logb (-1) =O[⊤] log

case pos.inr.inl
⊢ logb 0 =O[⊤] log

case pos.inr.inr
⊢ logb 1 =O[⊤] log
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
⊢ μ.toSphere univ < ∞

Tactic:
by
    rw [toSphere_apply_univ']
    exact ENNReal.mul_lt_top (ENNReal.natCast_lt_top _) <|
      measure_ball_lt_top.trans_le' <| measure_mono sdiff_subset

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
⊢ dist y ((1 - ε / 4) • x) + ‖(1 - ε / 4) • x‖ < ε / 4 + ‖(1 - ε / 4) • x‖

Tactic:
gcongr

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
⊢ dist y ((1 - ε / 4) • x) + dist ((1 - ε / 4) • x) x < ε / 4 + dist ((1 - ε / 4) • x) x

Tactic:
gcongr

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: r : ℝ
hr : 0 < r
s : ℂ
hs : ‖s‖ < r
l : ℂ
h : 2 * r ≤ ‖l‖
this✝ : s ≠ l
this : 0 < ‖l‖
⊢ ‖l ^ 2 - (s - l) ^ 2‖ / (‖s - l‖ ^ 2 * ‖l‖ ^ 2) ≤ ‖l ^ 2 - (s - l) ^ 2‖ / ((‖l‖ / 2) ^ 2 * ‖l‖ ^ 2)

Tactic:
gcongr

Goal state after:
case hab
r : ℝ
hr : 0 < r
s : ℂ
hs : ‖s‖ < r
l : ℂ
h : 2 * r ≤ ‖l‖
this✝ : s ≠ l
this : 0 < ‖l‖
⊢ ‖l‖ / 2 ≤ ‖s - l‖
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a b : ℕ
hb : b ≠ 0
c : ℕ
hc1 : c ^ (n + 1) ≤ a
hc2 : a < (c + 1) ^ (n + 1)
⊢ (c + 1) ^ (n + 1) ≤ ((c ^ (n + 1) / b ^ n + n * b) / (n + 1) + 1) ^ (n + 1)

Tactic:
gcongr

Goal state after:
case h₁
n a b : ℕ
hb : b ≠ 0
c : ℕ
hc1 : c ^ (n + 1) ≤ a
hc2 : a < (c + 1) ^ (n + 1)
⊢ c ≤ (c ^ (n + 1) / b ^ n + n * b) / (n + 1)
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
ε : ℝ
hε : 0 < ε
x : ↑(sphere 0 1)
⊢ ↑(toSphereBallBound (dim E) ε) * μ (ball 0 1) ≤ μ.toSphere (ball x ε)

Tactic:
have : Nontrivial E := ⟨⟨x, 0, ne_of_apply_ne Norm.norm (by simp)⟩⟩

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
ε : ℝ
hε : 0 < ε
x : ↑(sphere 0 1)
this : Nontrivial E
⊢ ↑(toSphereBallBound (dim E) ε) * μ (ball 0 1) ≤ μ.toSphere (ball x ε)
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a b : ℕ
hb : b ≠ 0
c : ℕ
hc1 : c ^ (n + 1) ≤ a
hc2 : a < (c + 1) ^ (n + 1)
⊢ ((c ^ (n + 1) / b ^ n + n * b) / (n + 1) + 1) ^ (n + 1) ≤ ((a / b ^ n + n * b) / (n + 1) + 1) ^ (n + 1)

Tactic:
gcongr

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
ε : ℝ
hε : 0 < ε
x : ↑(sphere 0 1)
this✝ : Nontrivial E
this : ∀ {ε : ℝ}, 0 < ε → ε ≤ 2 → ↑(toSphereBallBound (dim E) ε) * μ (ball 0 1) ≤ μ.toSphere (ball x ε)
hε₂ : ¬ε ≤ 2
⊢ μ.toSphere (ball x (min ε 2)) ≤ μ.toSphere (ball x ε)

Tactic:
gcongr

Goal state after:
case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
ε : ℝ
hε : 0 < ε
x : ↑(sphere 0 1)
this✝ : Nontrivial E
this : ∀ {ε : ℝ}, 0 < ε → ε ≤ 2 → ↑(toSphereBallBound (dim E) ε) * μ (ball 0 1) ≤ μ.toSphere (ball x ε)
hε₂ : ¬ε ≤ 2
⊢ min ε 2 ≤ ε
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case inl
n a b : ℕ
hn : n ≠ 0
hle : a ≤ n.nthRoot b
⊢ a ^ n ≤ n.nthRoot b ^ n

Tactic:
gcongr

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case inr
n a b : ℕ
hn : n ≠ 0
hlt : n.nthRoot b < a
⊢ (n.nthRoot b + 1) ^ n ≤ a ^ n

Tactic:
gcongr

Goal state after:
case hab
n a b : ℕ
hn : n ≠ 0
hlt : n.nthRoot b < a
⊢ n.nthRoot b + 1 ≤ a
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n : ℕ
hn : n ≠ 0
a b : ℕ
⊢ b ≤ n.nthRoot (a ^ n) ↔ b ≤ a

Tactic:
rw [le_nthRoot_iff hn]

Goal state after:
n : ℕ
hn : n ≠ 0
a b : ℕ
⊢ b ^ n ≤ a ^ n ↔ b ≤ a
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case neg
L : PeriodPair
l₀ : ℂ
r : ℝ
hr : r > 0
s : ℂ
hs : ‖s‖ < r
l : ↥L.lattice
h : 2 * r ≤ ‖↑l‖
h✝ : ¬↑l = l₀
this✝ : s ≠ ↑l
this : l ≠ 0
⊢ 2 * ‖l‖ ^ 3 ≤ 2 * (2 * ‖↑l - s‖) ^ 3

Tactic:
gcongr

Goal state after:
case hab
L : PeriodPair
l₀ : ℂ
r : ℝ
hr : r > 0
s : ℂ
hs : ‖s‖ < r
l : ↥L.lattice
h : 2 * r ≤ ‖↑l‖
h✝ : ¬↑l = l₀
this✝ : s ≠ ↑l
this : l ≠ 0
⊢ ‖l‖ ≤ 2 * ‖↑l - s‖
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ (fun x ↦ (ENNReal.ofReal (↑x ^ (dim E - 1))).toReal • f ↑x) =ᵐ[Measure.comap Subtype.val volume] fun x ↦
    ↑x ^ (dim E - 1) • f ↑x

Tactic:
refine .of_forall ?_

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ ∀ (x : ↑(Ioi 0)), (fun x ↦ (ENNReal.ofReal (↑x ^ (dim E - 1))).toReal • f ↑x) x = (fun x ↦ ↑x ^ (dim E - 1) • f ↑x) x