mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 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
b : OrthonormalBasis ι 𝕜 U
⊢ ↑(f.normDet ^ 2) = (Matrix.gram 𝕜 fun x ↦ f (b x)).det

Tactic:
suffices ↑(f.normDet ^ 2) = (Matrix.gram 𝕜 (f.rangeRestrict <| b ·)).det by
    simpa
  by_cases h : f.ker = ⊥
  · let bv := orthonormalBasis_range h b
    rw [Matrix.gram_eq_conjTranspose_mul bv, Matrix.det_mul, Matrix.det_conjTranspose]
    rw [RCLike.star_def, RCLike.conj_mul, f.normDet_eq_norm_det_toMatrix_rangeRestrict b bv]
    simp only [map_pow]
    congr
    ext i j
    simp [LinearMap.toMatrix_apply]
  · trans 0
    · simp [show f.normDet = 0 from (f.normDet_eq_zero_tfae.out 1 0).mp h]
    have hrank := (f.normDet_eq_zero_tfae.out 1 3).mp h
    symm
    contrapose! hrank with h0
    rw [finrank_eq_card_basis b.toBasis]
    exact (Matrix.linearIndependent_of_det_gram_ne_zero h0).fintype_card_le_finrank

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
x : X
v : V
f : H
⊢ ⟪f, (kerFun H x) v⟫_𝕜 = ⟪f x, v⟫_𝕜

Tactic:
simp [kerFun, ← adjoint_inner_left]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
x : X
⊢ ‖kernel H x x‖ = ‖kerFun H x‖ ^ 2

Tactic:
sq,

Goal state after:
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
x : X
⊢ ‖kernel H x x‖ = ‖kerFun H x‖ * ‖kerFun H x‖
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
x : X
⊢ ‖kerFun H x‖ = √‖kernel H x x‖

Tactic:
by
  rw [norm_kernel_eq_norm_kerFun_sq, Real.sqrt_sq (norm_nonneg _)]

Goal state after:
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✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ f.normDet = ∏ i ∈ Finset.range (finrank 𝕜 U), f.singularValues i

Tactic:
by
  rw [← sq_eq_sq₀ f.normDet_nonneg (Finset.prod_nonneg fun i _ ↦ f.singularValues_nonneg i),
    ← RCLike.ofReal_inj (K := 𝕜), ← Finset.prod_pow, ← Fin.prod_univ_eq_prod_range, normDet_sq]
  simp_rw [sq_singularValues_fin]
  push_cast
  rw [← LinearMap.IsSymmetric.det_eq_prod_eigenvalues]

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
n : ℕ
hn : 1 ≤ n
⊢ (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i

Tactic:
have npos : (0 : ℝ) < n := Nat.cast_pos.2 hn

Goal state after:
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
n : ℕ
hn : 1 ≤ n
npos : 0 < ↑n
⊢ (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case refine_1
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
i : ι
x✝ : i ∈ Finset.univ
⊢ Tendsto (fun n ↦ ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) atTop (𝓝 (ENNReal.ofReal (↑(b i) - ↑(a i))))

pp: case refine_2
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
i : ι
x✝ : i ∈ Finset.univ
⊢ ENNReal.ofReal (↑(b i) - ↑(a i)) ≠ ∞

Tactic:
· 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]

Goal state after:
case refine_2
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
i : ι
x✝ : i ∈ Finset.univ
⊢ ENNReal.ofReal (↑(b i) - ↑(a i)) ≠ ∞
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✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
bE' : (𝓝 0).HasBasis (fun x ↦ True) fun n ↦ c ^ n • b n
V : Set F
hV : V ∈ 𝓝 0
u : ℕ → E
hu : ∀ (i : ℕ), u i ∈ c ^ i • b i
hu' : ∀ (i : ℕ), f (u i) ∉ V
h_tendsto : Tendsto (fun n ↦ (c ^ n)⁻¹ • u n) atTop (𝓝 0)
h_bounded : IsVonNBounded 𝕜 (Set.range fun n ↦ (c ^ n)⁻¹ • u n)
hf : IsVonNBounded 𝕜' (Set.range fun x ↦ (σ c ^ x)⁻¹ • f (u x))
this : Tendsto (⇑f ∘ u) atTop (𝓝 0)
⊢ ∀ᶠ (x : ℕ) in atTop, False

Tactic:
filter_upwards [this.eventually_mem hV] using hu'

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

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹³ : NormedField 𝕜₁
inst✝¹² : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : IsTopologicalAddGroup E
inst✝⁷ : ContinuousSMul 𝕜₁ E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₂ F
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₂ F
inst✝¹ : T1Space E
inst✝ : MontelSpace 𝕜₁ E
⊢ Tendsto (⇑(LinearEquiv.toCompactConvergenceCLM σ E F).symm) (𝓝 0)
    (𝓝 ((LinearEquiv.toCompactConvergenceCLM σ E F).symm 0))

Tactic:
_root_.map_zero,

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹³ : NormedField 𝕜₁
inst✝¹² : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : IsTopologicalAddGroup E
inst✝⁷ : ContinuousSMul 𝕜₁ E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₂ F
inst✝⁴ : TopologicalSpace F
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousSMul 𝕜₂ F
inst✝¹ : T1Space E
inst✝ : MontelSpace 𝕜₁ E
⊢ Tendsto (⇑(LinearEquiv.toCompactConvergenceCLM σ E F).symm) (𝓝 0) (𝓝 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)
x y : E
⊢ (o.areaForm x) (o.rightAngleRotation y) = ⟪x, y⟫

Tactic:
by
  rw [← o.inner_rightAngleRotation_left, o.inner_comp_rightAngleRotation]

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ (((map (Fin 2) φ.toLinearEquiv) o).areaForm x) y = ⟪o.rightAngleRotation (φ.symm x), φ.symm y⟫

Tactic:
simp [o.areaForm_map]

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
x : ℝ
s : ℂ
f : E
⊢ rexp (-x) • cexp (-↑x) ^ (s - 1) • f = cexp (-s * ↑x) • f

Tactic:
by
  change (rexp (-x) : ℂ) • _ = _ • f
  rw [← smul_assoc, smul_eq_mul]
  push_cast
  conv in cexp _ * _ => lhs; rw [← cpow_one (cexp _)]
  rw [← cpow_add _ _ (Complex.exp_ne_zero _), cpow_def_of_ne_zero (Complex.exp_ne_zero _),
    Complex.log_exp (by simp [pi_pos]) (by simpa using pi_nonneg)]
  ring_nf

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
| cexp (-s * ↑u) • f (rexp (-u))

Tactic:
← re_add_im s

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
| cexp (-(↑s.re + ↑s.im * I) * ↑u) • f (rexp (-u))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hE : ¬CompleteSpace E
⊢ mellin (fun t ↦ f (t ^ 0)) s = |0|⁻¹ • mellin f (s / ↑0)

Tactic:
simp [integral, mellin, hE]

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 : 0 < a
⊢ mellin (fun t ↦ f (a * t)) s = ↑a ^ (-s) • mellin f s

Tactic:
by
  simp_rw [mellin]
  have : EqOn (fun t : ℝ => (t : ℂ) ^ (s - 1) • f (a * t))
      (fun t : ℝ => (a : ℂ) ^ (1 - s) • (fun u : ℝ => (u : ℂ) ^ (s - 1) • f u) (a * t))
        (Ioi 0) := fun t ht ↦ by
    dsimp only
    rw [ofReal_mul, mul_cpow_ofReal_nonneg ha.le (le_of_lt ht), ← mul_smul,
      (by ring : 1 - s = -(s - 1)), cpow_neg, inv_mul_cancel_left₀]
    rw [Ne, cpow_eq_zero_iff, ofReal_eq_zero, not_and_or]
    exact Or.inl ha.ne'
  rw [setIntegral_congr_fun measurableSet_Ioi this, integral_smul,
    integral_comp_mul_left_Ioi (fun u ↦ (u : ℂ) ^ (s - 1) • f u) _ ha,
    mul_zero, ← Complex.coe_smul, ← mul_smul, sub_eq_add_neg,
    cpow_add _ _ (ofReal_ne_zero.mpr ha.ne'), cpow_one, ofReal_inv,
    mul_assoc, mul_comm, inv_mul_cancel_right₀ (ofReal_ne_zero.mpr ha.ne')]

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

Goal state before:
pp: case h₁
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ ‖z‖ ^ 2 ≤ ‖z‖ / 2

Tactic:
sq,

Goal state after:
case h₁
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ ‖z‖ * ‖z‖ ≤ ‖z‖ / 2
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
A : rexp (-x) = rexp (-a * x) * rexp (-b * x)
⊢ x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1))

Tactic:
← rpow_add hx,

Goal state after:
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
f_mem_Lp : ∀ {c u : ℝ}, 0 < c → 0 < u → MemLp (f c u) (ENNReal.ofReal (1 / c)) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
A : rexp (-x) = rexp (-a * x) * rexp (-b * x)
⊢ x ^ (a * s + b * t - 1) = x ^ (a * (s - 1) + b * (t - 1))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖(↑t ^ (z - 1) * ↑(log t)) • f t‖ ≤ bound t

Tactic:
bound,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖(↑t ^ (z - 1) * ↑(log t)) • f t‖ ≤ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℝ
t : ℝ
hg : Tendsto (fun x ↦ x * g x) atTop (𝓝 t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ Tendsto (fun x ↦ x * log (1 + g x)) atTop (𝓝 t)

Tactic:
rw [← tendsto_ofReal_iff] at hg ⊢

Goal state after:
g : ℝ → ℝ
t : ℝ
hg : Tendsto (fun x ↦ ↑(x * g x)) atTop (𝓝 ↑t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ Tendsto (fun x ↦ ↑(x * log (1 + g x))) atTop (𝓝 ↑t)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℝ
t : ℝ
hg : Tendsto (fun x ↦ x * g x) atTop (𝓝 t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ Tendsto (fun x ↦ (1 + g x) ^ x) atTop (𝓝 (rexp t))

Tactic:
rw [← tendsto_ofReal_iff] at hg ⊢

Goal state after:
g : ℝ → ℝ
t : ℝ
hg : Tendsto (fun x ↦ ↑(x * g x)) atTop (𝓝 ↑t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ Tendsto (fun x ↦ ↑((1 + g x) ^ x)) atTop (𝓝 ↑(rexp t))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ (1 - 0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u

Tactic:
zero_cpow,

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ (1 - 0) / (u - 1 + 1) = 1 / u

u : ℂ
hu : 0 < u.re
⊢ u - 1 + 1 ≠ 0
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
h4 : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), ∀ z ∈ Metric.ball s v, ‖F' z t‖ ≤ bound t
⊢ IntegrableOn bound (Ioi 0) volume

Tactic:
bound,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
h4 : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), ∀ z ∈ Metric.ball s v, ‖F' z t‖ ≤ bound t
⊢ IntegrableOn (fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖) (Ioi 0) volume
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
⊢ ↑π * z * A * B / ↑C =
    (↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * (n + 1))) /
      ↑((2 * ↑n + 1) / (2 * ↑n + 2) * C)

Tactic:
change
      π * z * A * B / C =
        (π * z * (A * ((1 : ℂ) - z ^ 2 / ((n : ℂ) + 1) ^ 2)) *
            ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) /
          ((2 * n + 1) / (2 * n + 2) * C : ℝ)

Goal state after:
case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
⊢ ↑π * z * A * B / ↑C =
    (↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ)) /
      ↑((2 * ↑n + 1) / (2 * ↑n + 2) * C)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case refine_1
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
hx' : x ≤ 1
⊢ Tendsto (fun n ↦ f x - f 1 - x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f x - f 1))

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
hx' : x ≤ 1
⊢ ∀ᶠ (b : ℕ) in atTop, f x - f 1 - x * (log (↑b + 1) - log ↑b) ≤ logGammaSeq x b

pp: case refine_3
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
hx' : x ≤ 1
⊢ ∀ᶠ (b : ℕ) in atTop, logGammaSeq x b ≤ f x - f 1

Tactic:
· have : f x - f 1 = f x - f 1 - x * 0 := by ring
    nth_rw 2 [this]
    exact Tendsto.sub tendsto_const_nhds (tendsto_log_nat_add_one_sub_log.const_mul _)

Goal state after:
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
hx' : x ≤ 1
⊢ ∀ᶠ (b : ℕ) in atTop, f x - f 1 - x * (log (↑b + 1) - log ↑b) ≤ logGammaSeq x b

case refine_3
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
hx' : x ≤ 1
⊢ ∀ᶠ (b : ℕ) in atTop, logGammaSeq x b ≤ f x - f 1
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ asymptoticNhds ℝ P v ≤ cobounded P

Tactic:
rw [← tendsto_id', ← Metric.tendsto_dist_right_atTop_iff p,
    asymptoticNhds_eq_smul_vadd v p, vadd_pure, ← map₂_smul, ← map_prod_eq_map₂, map_map,
    tendsto_map'_iff]

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ Tendsto ((fun x ↦ dist (id x) p) ∘ (fun x ↦ x +ᵥ p) ∘ fun p ↦ p.1 • p.2) (atTop ×ˢ 𝓝 v) atTop
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

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

Tactic:
div_div

Goal state after:
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n ^ (s + 1) * ↑n ! / ((∏ x ∈ Finset.range n, (s + 1 + ↑x)) * (s + 1 + ↑n) * s)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
⊢ asymptoticNhds k P v ≤ (𝓝 v).bind (asymptoticNhds k P)

Tactic:
rw [← pure_bind v (asymptoticNhds k P)]

Goal state after:
case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
⊢ (pure v).bind (asymptoticNhds k P) ≤ (𝓝 v).bind (asymptoticNhds k P)
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f (g 0)‖ = ‖f.contLinear (g 0) + f 0‖

Tactic:
f.decomp

Goal state after:
𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖(⇑f.contLinear + Function.const V (f 0)) (g 0)‖ = ‖f.contLinear (g 0) + (⇑f.contLinear + Function.const V (f 0)) 0‖