mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case refine_4
n : ℕ
inst✝¹⁰ : NeZero n
S T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑n
x : CyclotomicField n K
this : NeZero ↑n
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
  y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
    (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
  z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
    (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ ∃ x,
    y * z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑x.2 =
      (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) x.1

Tactic:
refine ⟨⟨a.1 * b.1, a.2 * b.2, mul_mem_nonZeroDivisors.2 ⟨a.2.2, b.2.2⟩⟩, ?_⟩

Goal state after:
case refine_4
n : ℕ
inst✝¹⁰ : NeZero n
S T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : Algebra A B
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Algebra K L
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : NeZero ↑n
x : CyclotomicField n K
this : NeZero ↑n
y z : CyclotomicField n K
a : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
ha :
  y * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑a.2 =
    (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) a.1
b : CyclotomicRing n A K × ↥(nonZeroDivisors (CyclotomicRing n A K))
hb :
  z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑b.2 =
    (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) b.1
⊢ y * z * (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) ↑(a.1 * b.1, ⟨↑a.2 * ↑b.2, ⋯⟩).2 =
    (algebraMap (CyclotomicRing n A K) (CyclotomicField n K)) (a.1 * b.1, ⟨↑a.2 * ↑b.2, ⋯⟩).1
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst✝ : NeZero n
ψ : DirichletCharacter R m
h : (changeLevel ⋯) χ = (changeLevel ⋯) ψ
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
⊢ ∃ z, ↑z = ↑x ∧ ↑z = 1

Tactic:
suffices x.val.val ≡ 1 [MOD n.gcd m] by
      obtain ⟨z, hz₁, hz₂⟩ := Nat.chineseRemainder' this
      refine ⟨z, ?_, ?_⟩
      · simpa [← ZMod.natCast_eq_natCast_iff] using hz₁
      · rwa [← ZMod.natCast_eq_natCast_iff, Nat.cast_one] at hz₂

Goal state after:
R : Type u_1
inst✝¹ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
m : ℕ
inst✝ : NeZero n
ψ : DirichletCharacter R m
h : (changeLevel ⋯) χ = (changeLevel ⋯) ψ
x : (ZMod n)ˣ
hx : x ∈ (ZMod.unitsMap ⋯).ker
⊢ (↑x).val ≡ 1 [MOD n.gcd m]
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: S : Set ℕ
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C : Subalgebra A B
h : IsCyclotomicExtension S A ↥C
m : ℕ
ζ : B
h₁ : m ∈ S
h₂ : m ≠ 0
hζ : ζ ^ m = 1
η : ↥C
hη : IsPrimitiveRoot ((algebraMap (↥C) B) η) m
this : NeZero m
⊢ ζ ∈ C

Tactic:
obtain ⟨k, _, rfl⟩ := hη.eq_pow_of_pow_eq_one hζ

Goal state after:
S : Set ℕ
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
C : Subalgebra A B
h : IsCyclotomicExtension S A ↥C
m : ℕ
h₁ : m ∈ S
h₂ : m ≠ 0
η : ↥C
hη : IsPrimitiveRoot ((algebraMap (↥C) B) η) m
this : NeZero m
k : ℕ
left✝ : k < m
hζ : ((algebraMap (↥C) B) η ^ k) ^ m = 1
⊢ (algebraMap (↥C) B) η ^ k ∈ C
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ ≠ 1
b : Rˣ
hb : χ ↑b ≠ 1
⊢ ∑ a, χ a = 0

Tactic:
refine eq_zero_of_mul_eq_self_left hb ?_

Goal state after:
R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ ≠ 1
b : Rˣ
hb : χ ↑b ≠ 1
⊢ χ ↑b * ∑ a, χ a = ∑ a, χ a
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommMonoidWithZero R
n : ℕ
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m
hm : n ∣ m
χ : DirichletCharacter R n
a : ℤ
ha : IsCoprime a ↑χ.conductor
this : (changeLevel ⋯) ((changeLevel hm) χ).primitiveCharacter = χ.primitiveCharacter
⊢ ((changeLevel hm) χ).primitiveCharacter ↑a = χ.primitiveCharacter ↑a

Tactic:
have := DFunLike.congr_fun this (a : ZMod _)
      rwa [changeLevel_eq_cast_of_dvd' _ _ ha] at this

Goal state after:
Module: Mathlib.Order.Grade

Goal state before:
pp: case zero
𝕆 : Type u_1
ℙ : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : Preorder 𝕆
inst✝³ : Preorder ℙ
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : GradeMinOrder (Fin 0) α
a : Fin 0
h : IsMin a
⊢ IsMin ↑a

Tactic:
exact a.elim0

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case e'_5
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t ↦ (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t ↦ f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
⊢ (∫⁻ (t : A i), f (X t) ∂μ i) ^ (1 - k * p) = (∫⋯∫⁻_{i}, f ∂μ) x ^ (1 - k * p)

Tactic:
rw [lmarginal_singleton]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : Coalgebra 𝕜 E
x y z : E
⊢ x * (y + z) = x * y + x * z

Tactic:
simp [tmul_add]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : Coalgebra 𝕜 E
r : 𝕜
x : E
⊢ r • x = { toFun := ⇑(adjoint counit), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r * x

Tactic:
by
    dsimp
    simp_rw [← rTensor_tmul, ← adjoint_rTensor, ← comp_apply, ← adjoint_comp,
      rTensor_counit_comp_comul, ← toLinearMap_symm_lid, ← toLinearEquiv_lidIsometry,
      ← toLinearEquiv_symm, adjoint_toLinearMap_eq_symm]
    simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace 𝕜 F
f₁ f₂ : E →ₗ[𝕜] E
h₁ : (f₂ - f₁).IsPositive
h₂ : (f₁ - f₂).IsPositive
x : E
hba2 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
rw [← neg_le_neg_iff, ← map_neg, ← inner_neg_left, ← neg_apply, neg_sub, neg_zero] at hba2

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace 𝕜 F
f₁ f₂ : E →ₗ[𝕜] E
h₁ : (f₂ - f₁).IsPositive
h₂ : (f₁ - f₂).IsPositive
x : E
hba2 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : f.IsPositive
h : ∃ s, Nonempty (Module.Basis (↥s) 𝕜 E)
⊢ 0 ≤ (traceAux 𝕜 ⋯.some) f

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : f.IsPositive
h : ¬∃ s, Nonempty (Module.Basis (↥s) 𝕜 E)
⊢ 0 ≤ 0 f

Tactic:
· have : FiniteDimensional 𝕜 E := Module.Finite.of_basis h.choose_spec.some
    simp_rw [traceAux_eq 𝕜 _ (stdOrthonormalBasis 𝕜 E).toBasis]
    exact posSemidef_toMatrix_iff (stdOrthonormalBasis 𝕜 E) |>.mpr hf |>.trace_nonneg

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : f.IsPositive
h : ¬∃ s, Nonempty (Module.Basis (↥s) 𝕜 E)
⊢ 0 ≤ 0 f
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : H →L[𝕜] H
c : ℝ≥0
hc : 0 < c
h : ∀ (x : H), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖
⊢ AntilipschitzWith c⁻¹ ⇑f

Tactic:
refine f.antilipschitz_of_bound (K := c⁻¹) fun x ↦ ?_
  rw [NNReal.coe_inv, inv_mul_eq_div, le_div_iff₀ (by exact_mod_cast hc)]
  simp_rw [sq, mul_assoc] at h
  by_cases hx0 : x = 0
  · simp [hx0]
  · apply (map_le_map_iff <| OrderIso.mulLeft₀ ‖x‖ (norm_pos_iff.mpr hx0)).mp
    exact (h x).trans <| (norm_inner_le_norm _ _).trans <| (mul_comm _ _).le

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: 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
this : 0 < ↑p⁻¹ - (↑n)⁻¹
⊢ ↑p < ↑n

Tactic:
rwa [NNReal.coe_inv, sub_pos,
      inv_lt_inv₀ _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this

Goal state after:
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
this : (↑n)⁻¹ < (↑p)⁻¹
⊢ 0 < ↑n
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case neg
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'
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
have h1n : 1 ≤ (n : ℝ≥0) := hn.lt.le

Goal state after:
case neg
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
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case pos
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'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
rw [eLpNorm_nnreal_eq_lintegral hp'0, h3u, ENNReal.zero_rpow_of_pos] <;> positivity

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: 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'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑p') ≤ ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p)

Tactic:
rwa [← h2q, ENNReal.rpow_sub _ _ h3u h4u, ENNReal.div_le_iff h5u h6u]

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
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ RCLike.re (∑ x, ∑ x_1, ‖f‖ ^ 2 • inner 𝕜 (e x_1) (e x) * inner 𝕜 (g x_1) (g x)) -
      RCLike.re (∑ x, ∑ x_1, (∑ i, Matrix.vecMulVec (A i) (star (A i))) x_1 x * inner 𝕜 (g x_1) (g x)) ≤
    RCLike.re (∑ x, ∑ x_1, ‖f‖ ^ 2 • (inner 𝕜 (e x_1) (e x) * inner 𝕜 (g x_1) (g x)))

Tactic:
smul_mul_assoc,

Goal state after:
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
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ RCLike.re (∑ x, ∑ x_1, ‖f‖ ^ 2 • (inner 𝕜 (e x_1) (e x) * inner 𝕜 (g x_1) (g x))) -
      RCLike.re (∑ x, ∑ x_1, (∑ i, Matrix.vecMulVec (A i) (star (A i))) x_1 x * inner 𝕜 (g x_1) (g x)) ≤
    RCLike.re (∑ x, ∑ x_1, ‖f‖ ^ 2 • (inner 𝕜 (e x_1) (e x) * inner 𝕜 (g x_1) (g x)))
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
e : X ≃ᵢ Y
⊢ dimH univ = dimH univ

Tactic:
← e.dimH_preimage univ,

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
e : X ≃ᵢ Y
⊢ dimH univ = dimH (⇑e ⁻¹' univ)
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: case e_μ
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
p : P
v : V
hv : v ≠ 0
t : Set P
ht : MeasurableSet t
hrank : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction = 1
hrank' : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).directionᗮ = finrank ℝ V - 1
f : ℝ ≃L[ℝ] ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction :=
  (ContinuousLinearEquiv.toSpanNonzeroSingleton ℝ v hv).trans
    (ContinuousLinearEquiv.ofEq (ℝ ∙ v) (AffineSubspace.mk' p (ℝ ∙ v)).direction ⋯)
hf : MeasurableEmbedding ⇑f
p' : ↥(AffineSubspace.mk' p (ℝ ∙ v)) := ⟨p, ⋯⟩
g : ℝ → ↥(AffineSubspace.mk' p (ℝ ∙ v)) := ⇑(IsometryEquiv.vaddConst p') ∘ ⇑f
hadd : MeasurableEmbedding ⇑(IsometryEquiv.vaddConst p')
hg : MeasurableEmbedding g
⊢ volume = ‖v‖ₑ • map (⇑f) volume

Tactic:
let v' : (AffineSubspace.mk' p (ℝ ∙ v)).direction := ⟨v, by simp⟩

Goal state after:
case e_μ
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
p : P
v : V
hv : v ≠ 0
t : Set P
ht : MeasurableSet t
hrank : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction = 1
hrank' : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).directionᗮ = finrank ℝ V - 1
f : ℝ ≃L[ℝ] ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction :=
  (ContinuousLinearEquiv.toSpanNonzeroSingleton ℝ v hv).trans
    (ContinuousLinearEquiv.ofEq (ℝ ∙ v) (AffineSubspace.mk' p (ℝ ∙ v)).direction ⋯)
hf : MeasurableEmbedding ⇑f
p' : ↥(AffineSubspace.mk' p (ℝ ∙ v)) := ⟨p, ⋯⟩
g : ℝ → ↥(AffineSubspace.mk' p (ℝ ∙ v)) := ⇑(IsometryEquiv.vaddConst p') ∘ ⇑f
hadd : MeasurableEmbedding ⇑(IsometryEquiv.vaddConst p')
hg : MeasurableEmbedding g
v' : ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction := ⟨v, ⋯⟩
⊢ volume = ‖v‖ₑ • map (⇑f) volume
Module: Mathlib.Analysis.InnerProductSpace.StarOrder

Goal state before:
pp: case mpr
𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f p : H →L[𝕜] H
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ p.IsPositive

Tactic:
induction hp using AddSubmonoid.closure_induction with

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f g : H →L[𝕜] H
⊢ (∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p) → f ≤ g

Tactic:
rintro ⟨p, hp, rfl⟩
      rw [le_def, add_sub_cancel_left]
      induction hp using AddSubmonoid.closure_induction with
      | mem _ hf =>
        obtain ⟨f, rfl⟩ := hf
        simpa using! ContinuousLinearMap.IsPositive.adjoint_conj isPositive_one f
      | zero => exact isPositive_zero
      | add f g _ _ hf hg => exact hf.add hg

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

Goal state before:
pp: E : Type u_5
P : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : BorelSpace P
x y : P
⊢ nndist x y • volume (Icc 0 1) = edist x y

Tactic:
Real.volume_Icc,

Goal state after:
E : Type u_5
P : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor E P
inst✝ : BorelSpace P
x y : P
⊢ nndist x y • ENNReal.ofReal (1 - 0) = edist x y
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ

Tactic:
let := (Sᵀ.toInnerProductSpace hS.transpose.posSemidef)

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case succ
z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
this : HasDerivAt (fun x ↦ x ^ (n + 1) / (↑n + 1)) (z ^ n) z
⊢ HasDerivAt (fun z ↦ (-1) ^ (n + 1 + 1) * (z ^ (n + 1) / (↑n + 1))) ((-1) ^ n * z ^ n) z

Tactic:
convert! HasDerivAt.const_mul _ this using 2

Goal state after:
case e'_9
z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
this : HasDerivAt (fun x ↦ x ^ (n + 1) / (↑n + 1)) (z ^ n) z
⊢ (-1) ^ n = (-1) ^ (n + 1 + 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ ∀ (m : ℕ), 1 ≠ -↑m

Tactic:
by norm_cast; simp

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

Goal state before:
pp: t : ℝ
⊢ Tendsto (fun x ↦ (1 + t / x) ^ x) atTop (𝓝 (rexp t))

Tactic:
apply tendsto_one_add_rpow_exp_of_tendsto

Goal state after:
t : ℝ
⊢ Tendsto (fun x ↦ x * (t / x)) atTop (𝓝 t)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
⊢ ↑n ≠ 0

Tactic:
contrapose! hn

Goal state after:
z : ℂ
n : ℕ
hz : z ≠ 0
hn : ↑n = 0
⊢ n < 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
⊢ Tendsto
    (fun n ↦ (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)
    atTop (𝓝 (f 0))

Tactic:
have zero_mem : (0 : ℝ) ∈ closure (interior (Icc 0 (π / 2))) := by
    rw [interior_Icc, closure_Ioo pi_div_two_pos.ne, left_mem_Icc]
    exact pi_div_two_pos.le

Goal state after:
f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
zero_mem : 0 ∈ closure (interior (Icc 0 (π / 2)))
⊢ Tendsto
    (fun n ↦ (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)
    atTop (𝓝 (f 0))