mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case neg.inr
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
have h2γ : γ * n' = p' := by
    rw [← NNReal.coe_inj, ← inv_inj, hp', NNReal.coe_mul, h0γ, hn.coe.conjugate_eq]
    simp [field]

Goal state after:
case neg.inr
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
⊢ singularValues 0 = 0

Tactic:
ext1 i
  rw [Finsupp.zero_apply, singularValues_eq_zero_iff_le_finrank_range, range_zero]
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
r : 𝕜
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL (r • f) g = r • mapL f g

Tactic:
ext; simp [map_smul_left]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : r ≠ 0
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
⊢ (mkMetric' m) (s ∪ t) = (mkMetric' m) s + (mkMetric' m) t

Tactic:
refine tendsto_nhds_unique_of_eventuallyEq
    (mkMetric'.tendsto_pre _ _) ((mkMetric'.tendsto_pre _ _).add (mkMetric'.tendsto_pre _ _)) ?_

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : r ≠ 0
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
⊢ (fun r ↦ (mkMetric'.pre m r) (s ∪ t)) =ᶠ[𝓝[>] 0] fun x ↦ (mkMetric'.pre m x) s + (mkMetric'.pre m x) t
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hdet : ¬‖((toMatrix bu.toBasis bv.toBasis) f).det‖ = 0
h : IsUnit ((toMatrix bu.toBasis bv.toBasis) f).det
f' : U ≃ₗ[𝕜] V := LinearEquiv.ofIsUnitDet h
hf : f.range = ⊤
⊢ finrank 𝕜 U = finrank 𝕜 ↥f.range

Tactic:
rw [hf]

Goal state after:
case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hdet : ¬‖((toMatrix bu.toBasis bv.toBasis) f).det‖ = 0
h : IsUnit ((toMatrix bu.toBasis bv.toBasis) f).det
f' : U ≃ₗ[𝕜] V := LinearEquiv.ofIsUnitDet h
hf : f.range = ⊤
⊢ finrank 𝕜 U = finrank 𝕜 ↥⊤
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ Monotone m → Monotone fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s

Tactic:
intro h_mono s t hst

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
h_mono : Monotone m
s t : { s // s.Nonempty }
hst : s ≤ t
⊢ (fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s) s ≤ (fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s) t
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
h : (↑f).ker = ⊥
b : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(↑f).range
hf : f = (↑f).range.subtypeₗᵢ.toContinuousLinearMap ∘SL f.rangeRestrict
| (ContinuousLinearMap.adjoint f ∘SL f).det

Tactic:
rw [hf]

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
h : (↑f).ker = ⊥
b : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(↑f).range
hf : f = (↑f).range.subtypeₗᵢ.toContinuousLinearMap ∘SL f.rangeRestrict
| (ContinuousLinearMap.adjoint ((↑f).range.subtypeₗᵢ.toContinuousLinearMap ∘SL f.rangeRestrict) ∘SL
      (↑f).range.subtypeₗᵢ.toContinuousLinearMap ∘SL f.rangeRestrict).det
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
⊢ μHE[finrank ℝ U] (⇑f '' s) = ENNReal.ofReal f.normDet * volume s

Tactic:
rw [f.euclideanHausdorffMeasure_image, InnerProductSpace.euclideanHausdorffMeasure_eq_volume]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:

Tactic:
_ ≤ ‖K x x v‖ * ‖v‖ := by simp [norm_inner_le_norm]
    _ ≤ ‖K x x‖ * ‖v‖ * ‖v‖ := by simp [mul_le_mul_of_nonneg_right, le_opNorm]
    _ ≤ _ := by simp [mul_pow, mul_assoc, ← sq]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
hn : ‖v‖ ≠ 0
x : ℝ
⊢ ‖LinearMap.id x‖ * ‖‖v‖⁻¹ • v‖ = ‖x‖

Tactic:
rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel₀ hn, mul_one, LinearMap.id_apply]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
hn : ‖v‖ ≠ 0
iso_smul : Isometry ⇑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v))
⊢ ‖v‖₊ • μH[1] (⇑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s) = ‖v‖₊ • μH[1] s

Tactic:
iso_smul.hausdorffMeasure_image (Or.inl <| zero_le_one' ℝ)

Goal state after:
case inr
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
hn : ‖v‖ ≠ 0
iso_smul : Isometry ⇑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v))
⊢ ‖v‖₊ • μH[1] s = ‖v‖₊ • μH[1] s
Module: Mathlib.Analysis.LocallyConvex.PointwiseConvergence

Goal state before:
pp: α : Type u_1
R : Type u_2
𝕜₁ : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
inst✝¹² : NormedField 𝕜₁
inst✝¹¹ : NormedField 𝕜₂
inst✝¹⁰ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₃ →+* 𝕜₂
D : Type u_6
E : Type u_7
F : Type u_8
G : Type u_9
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜₁ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜₂ F
inst✝⁴ : AddCommGroup D
inst✝³ : TopologicalSpace D
inst✝² : Module 𝕜₃ D
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜₂ G
A : (E →SL[σ] F) →ₗ[𝕜₂] D →SL[τ] G
hbound : ∀ (f : D), ∃ s C, ∀ (B : E →SL[σ] F), ∃ g, ∃ (_ : g ∈ s), ‖(A B) f‖ ≤ C • ‖B g‖
⊢ Continuous
    (↑(toUniformConvergenceCLM τ G {s | Finite ↑s}) ∘ₗ A ∘ₗ ↑(toUniformConvergenceCLM σ F {s | Finite ↑s}).symm).toFun

Tactic:
apply PointwiseConvergenceCLM.withSeminorms.continuous_of_isBounded
      PointwiseConvergenceCLM.withSeminorms A
    intro f
    obtain ⟨s, C, h⟩ := hbound f
    use s, C
    rw [← Seminorm.finset_sup_smul]
    intro B
    obtain ⟨g, h₁, h₂⟩ := h ((toUniformConvergenceCLM _ _ _).symm B)
    refine le_trans ?_ (Seminorm.le_finset_sup_apply h₁)
    exact h₂

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ MellinConvergent (fun t ↦ f (t ^ a)) s ↔ MellinConvergent f (s / ↑a)

Tactic:
refine Iff.trans ?_ (integrableOn_Ioi_comp_rpow_iff' _ ha)

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ MellinConvergent (fun t ↦ f (t ^ a)) s ↔
    IntegrableOn (fun x ↦ x ^ (a - 1) • ↑(x ^ a) ^ (s / ↑a - 1) • f (x ^ a)) (Ioi 0) volume
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
σ : ℝ
f : ℂ → E
x : ℝ
hx : 0 < x
hx0 : ↑x ≠ 0
⊢ ↑x ^ (-↑σ) • |(2 * π)⁻¹| • ∫ (a : ℝ), ↑x ^ (-(↑a * I)) • f (↑σ + ↑a * I) =
    ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)

Tactic:
← Measure.integral_comp_mul_left

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
σ : ℝ
f : ℂ → E
x : ℝ
hx : 0 < x
hx0 : ↑x ≠ 0
⊢ ↑x ^ (-↑σ) • ∫ (x_1 : ℝ), ↑x ^ (-(↑(2 * π * x_1) * I)) • f (↑σ + ↑(2 * π * x_1) * I) =
    ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = (↑(t ^ (a - 1)) * ↑(t ^ a) ^ (s / ↑a - 1)) • f (t ^ a)

Tactic:
rw [ofReal_cpow (le_of_lt ht), ← cpow_mul_ofReal_nonneg (le_of_lt ht), ←
    cpow_add _ _ (ofReal_ne_zero.mpr <| ne_of_gt ht), ofReal_sub, ofReal_one, mul_sub,
    mul_div_cancel₀ _ (ofReal_ne_zero.mpr ha), add_comm, ← add_sub_assoc, mul_one, sub_add_cancel]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
n : ℕ
⊢ ‖log (1 + z) - logTaylor n z‖ ≤ max ‖log (1 + z)‖ (1 - ‖z‖)⁻¹ * ‖z‖ ^ n

Tactic:
cases n with
        | zero => simp [logTaylor_zero]
        | succ n =>
            refine (norm_log_sub_logTaylor_le n hz).trans ?_
            rw [mul_comm, ← div_one ((max _ _) * _)]
            gcongr
            · exact le_max_right ..
            · linarith

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ (cexp ∘ fun x ↦ ↑x * log (1 + g x)) =ᶠ[atTop] fun x ↦ (1 + g x) ^ ↑x

Tactic:
filter_upwards [hg0.eventually_ne (show 0 ≠ -1 by simp)] with x hg1

Goal state after:
g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
hg0 : Tendsto g atTop (𝓝 0)
x : ℝ
hg1 : g x ≠ -1
⊢ (cexp ∘ fun x ↦ ↑x * log (1 + g x)) x = (1 + g x) ^ ↑x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
x : ℝ
hx : x ∈ Ioi 0
⊢ ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) =
    ↑(rexp (-x)) * (↑x ^ (s + t - 1) * s.betaIntegral t)

Tactic:
← betaIntegral_scaled s t hx,

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
x : ℝ
hx : x ∈ Ioi 0
⊢ ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) =
    ↑(rexp (-x)) * ∫ (x_1 : ℝ) in 0..x, ↑x_1 ^ (s - 1) * (↑x - ↑x_1) ^ (t - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_2
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
this : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
⊢ x ≤ ↑⌈x - 1⌉₊ + 1

Tactic:
rw [← sub_le_iff_le_add]; exact Nat.le_ceil _

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

Goal state before:
pp: case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < (↑n.succ).re

Tactic:
rw [← ofReal_natCast, ofReal_re]; positivity

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: case inr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : CommRing k
inst✝³ : NoZeroDivisors k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
h✝ : Nontrivial k
w : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)
hw : ∀ (i : ↑Set.univ), ∑ j, w i j = 1
w' : Fin (2 + 1) → k
hw' : ∑ j, w' j = 1
h : ∀ (i : ↑Set.univ), ∃ r, ∀ (j : Fin (2 + 1)), r * {↑i}ᶜ.indicator (w i) j = {↑i}ᶜ.indicator w' j
hp' :
  ∀ (i : Fin 3),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
hp'w :
  ∀ (i : ↑Set.univ),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]
h' :
  ∀ (i : Fin 3), ∃ c, ∀ (j : Fin 3), j ≠ i → c * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
c : Fin 3 → k := fun i ↦ ⋯.choose
hc : ∀ (i j : Fin 3), j ≠ i → c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
hc1 : ∀ (i : Fin 3), c i * (1 - r i) = w' (i + 1)
hc2 : ∀ (i : Fin 3), c i * r i = w' (i + 2)
⊢ ∏ i, r i = ∏ i, (1 - r i)

Tactic:
have hcr : (∏ i, c i) * ∏ i, r i = (∏ i, c i) * ∏ i, (1 - r i) := by
    simp_rw [← Finset.prod_mul_distrib, Finset.prod_congr rfl (fun _ _ ↦ hc1 _),
      Finset.prod_congr rfl (fun _ _ ↦ hc2 _)]
    suffices ∏ i, (w' ∘ Equiv.addRight 2) i = ∏ i, (w' ∘ Equiv.addRight 1) i by
      simpa using this
    simp_rw [Finset.prod_comp_equiv]
    simp

Goal state after:
case inr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : CommRing k
inst✝³ : NoZeroDivisors k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
h✝ : Nontrivial k
w : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)
hw : ∀ (i : ↑Set.univ), ∑ j, w i j = 1
w' : Fin (2 + 1) → k
hw' : ∑ j, w' j = 1
h : ∀ (i : ↑Set.univ), ∃ r, ∀ (j : Fin (2 + 1)), r * {↑i}ᶜ.indicator (w i) j = {↑i}ᶜ.indicator w' j
hp' :
  ∀ (i : Fin 3),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
hp'w :
  ∀ (i : ↑Set.univ),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]
h' :
  ∀ (i : Fin 3), ∃ c, ∀ (j : Fin 3), j ≠ i → c * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
c : Fin 3 → k := fun i ↦ ⋯.choose
hc : ∀ (i j : Fin 3), j ≠ i → c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
hc1 : ∀ (i : Fin 3), c i * (1 - r i) = w' (i + 1)
hc2 : ∀ (i : Fin 3), c i * r i = w' (i + 2)
hcr : (∏ i, c i) * ∏ i, r i = (∏ i, c i) * ∏ i, (1 - r i)
⊢ ∏ i, r i = ∏ i, (1 - r i)
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹ : SeminormedRing R
inst✝ : SeminormedAddCommGroup M
x : tsze R M
⊢ ‖x‖₊ = ‖x.fst‖₊ + ‖x.snd‖₊

Tactic:
ext; simp [norm_def]

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: case h₁
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖r₁ * r₂‖ ≤ ‖r₁‖ * ‖r₂‖

Tactic:
apply norm_mul_le

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: case h₁
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖r₁ * r₂‖ ≤ ‖r₁‖ * ‖r₂‖

pp: case h₂
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖r₁ • m₂ + MulOpposite.op r₂ • m₁‖ ≤ ‖r₁‖ * ‖m₂‖ + ‖r₂‖ * ‖m₁‖

Tactic:
· apply norm_mul_le

Goal state after:
case h₂
𝕜 : Type u_1
S : Type u_2
R : Type u_3
M : Type u_4
inst✝¹¹ : SeminormedCommRing S
inst✝¹⁰ : SeminormedRing R
inst✝⁹ : SeminormedAddCommGroup M
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsBoundedSMul S R
inst✝⁵ : IsBoundedSMul S M
inst✝⁴ : Module R M
inst✝³ : IsBoundedSMul R M
inst✝² : Module Rᵐᵒᵖ M
inst✝¹ : IsBoundedSMul Rᵐᵒᵖ M
inst✝ : SMulCommClass R Rᵐᵒᵖ M
r₁ : R
m₁ : M
r₂ : R
m₂ : M
⊢ ‖r₁ • m₂ + MulOpposite.op r₂ • m₁‖ ≤ ‖r₁‖ * ‖m₂‖ + ‖r₂‖ * ‖m₁‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    ((z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! + (-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) / 2 +
      ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2

Tactic:
Fin.val_zero,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦ ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2)
    ((NormedSpace.exp (z * I) + NormedSpace.exp (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    ((z * I) ^ (2 * k + 0) / ↑(2 * k + 0)! + (-z * I) ^ (2 * k + 0) / ↑(2 * k + 0)!) / 2 +
      ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z * I) ^ (2 * k + 1) / ↑(2 * k + 1)! / I =
    ((-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! - (z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) * I / 2 +
      ((-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! - (z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) * I / 2

Tactic:
Fin.val_zero,

Goal state after:
case e'_6
z : ℂ
this :
  HasSum
    (fun x ↦
      ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2)
    ((NormedSpace.exp (-z * I) - NormedSpace.exp (z * I)) * I / 2)
k : ℕ
⊢ (z * I) ^ (2 * k + 1) / ↑(2 * k + 1)! / I =
    ((-z * I) ^ (2 * k + 0) / ↑(2 * k + 0)! - (z * I) ^ (2 * k + 0) / ↑(2 * k + 0)!) * I / 2 +
      ((-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! - (z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) * I / 2
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: α : Type u_1
L : SummationFilter α
f : α → ℝ
r : ℝ
h : HasSum f r L
⊢ HasSum (fun a ↦ ↑(f a)) (↑r) L

Tactic:
by simpa only using! h.map (algebraMap ℝ ℍ) (continuous_algebraMap _ _)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: x : ℝ
⊢ cosh x ≤ rexp (x ^ 2 / 2)

Tactic:
rw [cosh_eq_tsum, exp_eq_exp_ℝ, exp_eq_tsum ℝ]

Goal state after:
x : ℝ
⊢ ∑' (n : ℕ), x ^ (2 * n) / ↑(2 * n)! ≤ (fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) (x ^ 2 / 2)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
hμn : μ (x ^ n) = 0
⊢ 0 ^ (1 / ↑n) = 0

Tactic:
zero_rpow (one_div_cast_ne_zero (one_le_iff_ne_zero.mp hn))

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x : R
hx : μ x = 0
n : ℕ
hn : 1 ≤ n
hμn : μ (x ^ n) = 0
⊢ 0 = 0
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
⊢ ∃ f, IsPowMul ⇑f ∧ (∀ (x : K), f ((algebraMap K L) x) = ‖x‖) ∧ IsNonarchimedean ⇑f

Tactic:
have hem : Nonempty ι := B.index_nonempty

Goal state after:
K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
⊢ ∃ f, IsPowMul ⇑f ∧ (∀ (x : K), f ((algebraMap K L) x) = ‖x‖) ∧ IsNonarchimedean ⇑f