mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets = dimH s

Tactic:
refine le_antisymm (iSup₂_le fun x _ => ?_) ?_

Goal state after:
case refine_1
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
x : X
x✝ : x ∈ s
⊢ limsup dimH (𝓝[s] x).smallSets ≤ dimH s

case refine_2
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ dimH s ≤ ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
r n i j : ℕ
hj : i < j
A : (↑(2 * j + r))⁻¹ < (↑(2 * i + 1 + r))⁻¹
⊢ (↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹ ≠ 0

Tactic:
simpa [tsub_eq_zero_iff_le] using A

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: case e_f
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
u : Set ↥(mk' (↑(x +ᵥ p)) s.directionᗮ) := Subtype.val ⁻¹' (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
hu : MeasurableSet u
hinter : t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ) = Subtype.val '' u
hxp : ↑(x +ᵥ p) ∈ mk' (↑(x +ᵥ p)) s.directionᗮ
hrank : finrank ℝ ↥s.directionᗮ = finrank ℝ ↥(mk' (↑(x +ᵥ p)) s.directionᗮ).direction
⊢ volume (Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑((IsometryEquiv.vaddConst p) x)) s.directionᗮ))

Tactic:
IsometryEquiv.vaddConst_apply,

Goal state after:
case e_f
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
u : Set ↥(mk' (↑(x +ᵥ p)) s.directionᗮ) := Subtype.val ⁻¹' (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
hu : MeasurableSet u
hinter : t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ) = Subtype.val '' u
hxp : ↑(x +ᵥ p) ∈ mk' (↑(x +ᵥ p)) s.directionᗮ
hrank : finrank ℝ ↥s.directionᗮ = finrank ℝ ↥(mk' (↑(x +ᵥ p)) s.directionᗮ).direction
⊢ volume (Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup U
inst✝³ : InnerProductSpace 𝕜 U
inst✝² : FiniteDimensional 𝕜 U
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace 𝕜 V
f : U →ₗ[𝕜] V
b : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
⊢ finrank 𝕜 ↥f.range = finrank 𝕜 U

Tactic:
simpa using finrank_eq_card_basis b.toBasis

Goal state after:
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
h_mono : Monotone m
⊢ Monotone fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s

Tactic:
s

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 : { s // s.Nonempty }
⊢ ∀ ⦃b : { s // s.Nonempty }⦄,
    s ≤ b → (fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s) s ≤ (fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s) b
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
⊢ ⨆ i, (mkMetric'.pre (fun s ↦ m (ediam s)) (↑i)⁻¹).trim = ⨆ n, mkMetric'.pre (fun s ↦ m (ediam s)) (↑n)⁻¹

Tactic:
congr 1 with n : 1

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
n : ℕ
⊢ (mkMetric'.pre (fun s ↦ m (ediam s)) (↑n)⁻¹).trim = mkMetric'.pre (fun s ↦ m (ediam s)) (↑n)⁻¹
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case pos
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
⊢ ↑((↑f).normDet ^ 2) =
    ((toMatrix bu.toBasis bu.toBasis) (↑(ContinuousLinearMap.adjoint f.rangeRestrict) ∘ₗ ↑f.rangeRestrict)).det

Tactic:
toMatrix_comp bu.toBasis b.toBasis bu.toBasis,

Goal state after:
case pos
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
⊢ ↑((↑f).normDet ^ 2) =
    ((toMatrix b.toBasis bu.toBasis) ↑(ContinuousLinearMap.adjoint f.rangeRestrict) *
        (toMatrix bu.toBasis b.toBasis) ↑f.rangeRestrict).det
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case pos
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
x✝ : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
htr : ∀ (n : ℕ), ediam (t n) ≤ r
⊢ ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)) =
    ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n))

Tactic:
congr 1 with n : 1

Goal state after:
case pos.e_f
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
r : ℝ≥0∞
x✝ : r > 0
t : ℕ → Set X
ht : s ⊆ iUnion t
htr : ∀ (n : ℕ), ediam (t n) ≤ r
n : ℕ
⊢ ⨆ (_ : (t n).Nonempty), ⨅ (_ : ediam (t n) ≤ r), m (ediam (t n)) = ⨆ (_ : (t n).Nonempty), m (ediam (t n))
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
⊢ (volume.addHaarScalarFactor μH[↑(finrank ℝ U)] • μH[↑(finrank ℝ U)]) (⇑f '' s) =
    ENNReal.ofReal f.normDet * (volume.addHaarScalarFactor μH[↑(finrank ℝ U)] • μH[↑(finrank ℝ U)]) s

Tactic:
Measure.smul_apply,

Goal state after:
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
⊢ volume.addHaarScalarFactor μH[↑(finrank ℝ U)] • μH[↑(finrank ℝ U)] (⇑f '' s) =
    ENNReal.ofReal f.normDet * volume.addHaarScalarFactor μH[↑(finrank ℝ U)] • μH[↑(finrank ℝ U)] s
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
i : ι
⊢ 0 ≤ ↑(b i) - ↑(a i)

Tactic:
by
    simpa only [sub_nonneg, Rat.cast_le] using (H i).le

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

Goal state before:
pp: ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
⊢ liminf (fun n ↦ ∏ i, ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) atTop = ∏ i, volume (Ioo ↑(a i) ↑(b i))

Tactic:
by
      simp only [Real.volume_Ioo]
      apply Tendsto.liminf_eq
      refine ENNReal.tendsto_finsetProd_of_ne_top _ (fun i _ => ?_) fun i _ => ?_
      · apply
          Tendsto.congr' _
            ((ENNReal.continuous_ofReal.tendsto _).comp
              ((tendsto_nat_ceil_mul_div_atTop (I i)).comp tendsto_natCast_atTop_atTop))
        apply eventually_atTop.2 ⟨1, fun n hn => _⟩
        intro n hn
        simp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply,
          ENNReal.ofReal_natCast]
      · simp only [ENNReal.ofReal_ne_top, Ne, not_false_iff]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : IsTopologicalAddGroup F
f : E →ₛₗ[σ] F
U : Set F
hU : U ∈ 𝓝 0
V : Set E
hV : V ∈ 𝓝 0
h✝ : IsVonNBounded 𝕜' (⇑f '' V)
r : ℝ
hr : r > 0
x : 𝕜
h : ⇑f '' V ⊆ σ x • U
hx : r < ‖x‖
x_ne : x ≠ 0
⊢ x⁻¹ • ⇑f ⁻¹' (σ x • U) = ⇑f ⁻¹' U

Tactic:
by rw [x_ne.isUnit.preimage_smul_setₛₗ _, inv_smul_smul₀ x_ne _]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
⊢ o.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x) = -x

Tactic:
by
  apply ext_inner_left ℝ
  intro y
  have : ⟪o.rightAngleRotationAux₁ y, o.rightAngleRotationAux₁ x⟫ = ⟪y, x⟫ :=
    LinearIsometry.inner_map_map o.rightAngleRotationAux₂ y x
  rw [o.inner_rightAngleRotationAux₁_right, ← o.inner_rightAngleRotationAux₁_left, this,
    inner_neg_right]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ (o.kahler x) y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0

Tactic:
refine ⟨?_, fun h => o.kahler_ne_zero h.1 h.2⟩

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ (o.kahler x) y ≠ 0 → x ≠ 0 ∧ y ≠ 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
f : E ≃ₗᵢ[ℝ] ℂ
hf : (map (Fin 2) f.toLinearEquiv) o = Complex.orientation
x y : E
⊢ (o.areaForm x) y = ((starRingEnd ℂ) (f x) * f y).im

Tactic:
rw [← Complex.areaForm, ← hf, areaForm_map]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
f : E ≃ₗᵢ[ℝ] ℂ
hf : (map (Fin 2) f.toLinearEquiv) o = Complex.orientation
x y : E
⊢ (o.areaForm x) y = (o.areaForm (f.symm (f x))) (f.symm (f y))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
n : ℕ
⊢ ∀ (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s

Tactic:
s

Goal state after:
s✝ : ℂ
hs : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
s : ℂ
⊢ -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ ¬ContinuousAt Gamma (-↑0)

Tactic:
Nat.cast_zero,

Goal state after:
⊢ ¬ContinuousAt Gamma (-0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: x : ℝ
n : ℕ
| x * log ↑(n + 1) + log ↑((n + 1) * n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log (x + ↑0)) + log x -
    (x + 1) * (log (↑n + 1) - log ↑n)

Tactic:
Nat.cast_zero,

Goal state after:
x : ℝ
n : ℕ
| x * log ↑(n + 1) + log ↑((n + 1) * n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log (x + 0)) + log x -
    (x + 1) * (log (↑n + 1) - log ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

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

Tactic:
Nat.cast_zero,

Goal state after:
case zero
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
x : ℝ
hx : 0 < x
⊢ 0 < x → x ≤ 0 + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
⊢ Tendsto (fun n ↦ ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))

Tactic:
have A :
    Tendsto
      (fun n : ℕ =>
        ((π * z * ∏ j ∈ Finset.range n, ((1 : ℂ) - z ^ 2 / ((j : ℂ) + 1) ^ 2)) *
            ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n)) /
          (∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n) : ℝ))
      atTop (𝓝 <| _) :=
    Tendsto.congr (fun n => sin_pi_mul_eq z n) tendsto_const_nhds

Goal state after:
z : ℂ
A :
  Tendsto
    (fun n ↦
      ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
          ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
        ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)))
    atTop (𝓝 (Complex.sin (↑π * z)))
⊢ Tendsto (fun n ↦ ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case zero
u : ℂ
hu : 0 < u.re
⊢ u.betaIntegral (↑0 + 1) = ↑0! / ∏ j ∈ Finset.range (0 + 1), (u + ↑j)

Tactic:
Nat.cast_zero,

Goal state after:
case zero
u : ℂ
hu : 0 < u.re
⊢ u.betaIntegral (0 + 1) = ↑0! / ∏ j ∈ Finset.range (0 + 1), (u + ↑j)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case inr.inr
x : ℝ
hx : x ∈ Icc 1 2
hmin : IsMinOn Γ (Icc 1 2) x
h1 : Γ (3 / 2) < Γ 1
h2 : Γ (3 / 2) < Γ 2
h3half : Γ x ≤ Γ (3 / 2)
y : ℝ
x✝ : y ∈ Ioi 0
hy : y ∈ Ici 2
⊢ y ∈ {x_1 | (fun x_2 ↦ Γ x ≤ Γ x_2) x_1}

Tactic:
exact h3half.trans h2.le |>.trans <| Gamma_strictMonoOn_Ici.monotoneOn (by simp) hy hy

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

Goal state before:
pp: case succ.refine_2
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
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =
    (↑n + 1) * ↑n ! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + ↑0))

Tactic:
Nat.cast_zero,

Goal state after:
case succ.refine_2
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
this : u.betaIntegral (↑n.succ + 1) = ↑n.succ * (u + 1).betaIntegral ↑n.succ / u
⊢ (↑n + 1) * (↑n ! / ∏ j ∈ Finset.range (n + 1), (u + 1 + ↑j)) / u =
    (↑n + 1) * ↑n ! / ((∏ k ∈ Finset.range (n + 1), (u + ↑(k + 1))) * (u + 0))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n / (↑n + 1 + s) * (↑n ^ s * ↑n ! / ((∏ k ∈ Finset.range n, (s + ↑(k + 1))) * (s + ↑0)))

Tactic:
Nat.cast_zero,

Goal state after:
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n / (↑n + 1 + s) * (↑n ^ s * ↑n ! / ((∏ k ∈ Finset.range n, (s + ↑(k + 1))) * (s + 0)))
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
f : V →ᴬ[𝕜] W
h : f 0 = 0
⊢ ‖f‖ = max ‖f 0‖ ‖f.contLinear‖

Tactic:
rw [norm_def]

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
⊢ ↑n * ↑n ! ^ 2 / ((∏ x ∈ Finset.range n, (z + ↑(x + 1)) * (1 - z + ↑x)) * ((z + ↑0) * (1 - z + ↑n))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))

Tactic:
Nat.cast_zero,

Goal state after:
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
⊢ ↑n * ↑n ! ^ 2 / ((∏ x ∈ Finset.range n, (z + ↑(x + 1)) * (1 - z + ↑x)) * ((z + 0) * (1 - z + ↑n))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  ((f.trans (pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv
hx : e x = x
hy : e y = y
hm :
  f.symm ((pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv (f (midpoint ℝ x y))) =
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv.symm (midpoint ℝ x y)
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)

Tactic:
← toIsometryEquiv_symm,

Goal state after:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  ((f.trans (pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv
hx : e x = x
hy : e y = y
hm :
  f.symm ((pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv (f (midpoint ℝ x y))) =
    (pointReflection ℝ (midpoint ℝ x y)).symm.toIsometryEquiv (midpoint ℝ x y)
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case zero
s : ℂ
hs : -s.re < ↑0
⊢ DifferentiableAt ℂ (fun s ↦ (Gamma s)⁻¹) s

Tactic:
Nat.cast_zero,

Goal state after:
case zero
s : ℂ
hs : -s.re < 0
⊢ DifferentiableAt ℂ (fun s ↦ (Gamma s)⁻¹) s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
⊢ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π

Tactic:
have h3 : Tendsto ((↑) : ℝ → ℂ) (𝓝[≠] 1) (𝓝[≠] 1) := by
    rw [tendsto_nhdsWithin_iff]; constructor
    · exact tendsto_nhdsWithin_of_tendsto_nhds continuous_ofReal.continuousAt
    · exact eventually_nhdsWithin_iff.mpr (Eventually.of_forall fun t ht => ofReal_ne_one.mpr ht)

Goal state after:
s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
⊢ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ (Gamma ↑(2 * t))⁻¹ * (↑(2 ^ (1 - 2 * t)))⁻¹ * (↑√π)⁻¹ = (Gamma (2 * ↑t))⁻¹ * 2 ^ (2 * ↑t - 1) / ↑√π

Tactic:
(by simp : 2 * (t : ℂ) = ↑(2 * t)),

Goal state after:
s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
t : ℝ
ht : 0 < t
⊢ (Gamma ↑(2 * t))⁻¹ * (↑(2 ^ (1 - 2 * t)))⁻¹ * (↑√π)⁻¹ = (Gamma ↑(2 * t))⁻¹ * 2 ^ (↑(2 * t) - 1) / ↑√π
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁵ : Field 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
n : ℕ
⊢ ((expSeries 𝕜 (tsze R M) n) fun x_1 ↦ x).fst = (expSeries 𝕜 R n) fun x_1 ↦ x.fst

Tactic:
simp [expSeries_apply_eq]

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : ProperSpace 𝕜
inst✝² : SeminormedRing F
inst✝¹ : NormedAlgebra 𝕜 F
inst✝ : NormOneClass F
x : F
⊢ ∃ z, IsMinOn (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) univ z

Tactic:
by
  have : Tendsto (‖x - algebraMap 𝕜 F ·‖) (cobounded 𝕜) atTop := by
    exact tendsto_norm_cobounded_atTop |>.comp <| tendsto_const_sub_cobounded x |>.comp <| by simp
  simp only [isMinOn_univ_iff]
  refine (show Continuous fun z : 𝕜 ↦ ‖x - algebraMap 𝕜 F z‖ by fun_prop)
    |>.exists_forall_le_of_isBounded 0 ?_
  simpa [isBounded_def, compl_ofPred, Ioi] using this (Ioi_mem_atTop ‖x - (0 : 𝕜) • 1‖)

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: case h
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ^ m ≤ 1
i : ℕ
hi : i ∈ Finset.range (m + 1)
⊢ ‖x‖ ^ i ≤ 1

Tactic:
rcases eq_or_ne m 0 with rfl | hm

Goal state after:
case h.inl
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
i : ℕ
hm : max 1 (‖x‖ ^ 0) = 1
hx : ‖x‖ ^ 0 ≤ 1
hi : i ∈ Finset.range (0 + 1)
⊢ ‖x‖ ^ i ≤ 1

case h.inr
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
hm✝ : max 1 (‖x‖ ^ m) = 1
hx : ‖x‖ ^ m ≤ 1
i : ℕ
hi : i ∈ Finset.range (m + 1)
hm : m ≠ 0
⊢ ‖x‖ ^ i ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
⊢ ∃ c, ∃ (_ : 0 < c), ∀ (x y : L), B.norm (x * y) ≤ c * B.norm x * B.norm y

Tactic:
use B.norm (B c.1 * B c.2)

Goal state after:
case h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
⊢ ∃ (_ : 0 < B.norm (B c.1 * B c.2)), ∀ (x y : L), B.norm (x * y) ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y