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

Goal state before:
pp: case right
S : Set ℕ
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
i : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})
f : AlgebraicClosure K →ₐ[K] AlgebraicClosure K
x : AlgebraicClosure K
hx : x ∈ (IntermediateField.map f (IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})).toSubalgebra
⊢ x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}

Tactic:
IntermediateField.toSubalgebra_map,

Goal state after:
case right
S : Set ℕ
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension S K L
i : L ≃ₐ[K] ↥(IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1})
f : AlgebraicClosure K →ₐ[K] AlgebraicClosure K
x : AlgebraicClosure K
hx : x ∈ Subalgebra.map f (IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}).toSubalgebra
⊢ x ∈ IntermediateField.adjoin K {x | ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1}
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
inst✝⁴ : NeZero n
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K ((X ^ n - 1).rootSet L) = ⊤

Tactic:
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]

Goal state after:
n : ℕ
inst✝⁴ : NeZero n
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K ((X ^ n - 1).rootSet L) = adjoin K {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: n : ℕ
inst✝⁴ : NeZero n
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K ((cyclotomic n K).rootSet L) = ⊤

Tactic:
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]

Goal state after:
n : ℕ
inst✝⁴ : NeZero n
K : Type w
L : Type z
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {n} K L
⊢ adjoin K ((cyclotomic n K).rootSet L) = adjoin K {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}
Module: Mathlib.Data.Int.Range

Goal state before:
pp: m n r : ℤ
⊢ r ∈ m.range n ↔ m ≤ r ∧ r < n

Tactic:
by
  simp only [range, List.mem_map, List.mem_range, lt_toNat, lt_sub_iff_add_lt, add_comm]
  exact ⟨fun ⟨a, ha⟩ => ha.2 ▸ ⟨le_add_of_nonneg_right (Int.natCast_nonneg _), ha.1⟩,
    fun h => ⟨toNat (r - m), by simp [toNat_of_nonneg (sub_nonneg.2 h.1), h.2] ⟩⟩

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
⊢ (if hx : IsUnit 1 then ↑(f hx.unit) else 0) = 1

Tactic:
have h1 : (isUnit_one.unit : Rˣ) = 1 := Units.ext rfl
    simp only [h1, dif_pos, Units.val_eq_one, map_one, isUnit_one]

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
f : Rˣ →* R'ˣ
x y : R
hx : IsUnit x
hy : ¬IsUnit y
⊢ (if h : IsUnit y then ↑(f ⋯.unit) else 0) = ↑(f ⋯.unit) * if hx : IsUnit y then ↑(f hx.unit) else 0

Tactic:
simp only [hy, not_false_iff, dif_neg, mul_zero]

Goal state after:
Module: Mathlib.NumberTheory.MulChar.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝² : CommMonoid R
R' : Type u_2
inst✝¹ : CommRing R'
R'' : Type u_3
inst✝ : CommRing R''
χ : MulChar R R'
f : R' →+* R''
n : ℕ
ih : χ.ringHomComp f ^ n = (χ ^ n).ringHomComp f
⊢ χ.ringHomComp f ^ (n + 1) = (χ ^ (n + 1)).ringHomComp f

Tactic:
simp only [pow_succ, ih, ringHomComp_mul]

Goal state after:
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
χ φ : MulChar R R'
ψ : AddChar R R'
| ∑ i, ∑ j, χ i * ψ i * (φ j * ψ j) = ∑ t, ∑ x, χ x * φ (t - x) * ψ t

Tactic:
conv =>

Goal state after:
R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
χ φ : MulChar R R'
ψ : AddChar R R'
x x_1 : R
| χ x * φ x_1 * (ψ x * ψ x_1)
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: R : Type u
inst✝³ : Field R
inst✝² : Fintype R
R' : Type v
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ ≠ 1
ψ : AddChar R R'
hψ : ψ.IsPrimitive
| ∑ x, ∑ i, χ i * ψ i * (χ x⁻¹ * ψ (-x)) = ↑(Fintype.card R)

Tactic:
conv =>

Goal state after:
R : Type u
inst✝³ : Field R
inst✝² : Fintype R
R' : Type v
inst✝¹ : CommRing R'
inst✝ : IsDomain R'
χ : MulChar R R'
hχ : χ ≠ 1
ψ : AddChar R R'
hψ : ψ.IsPrimitive
x y : R
| χ (y * x⁻¹) * ψ (y - x)
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: case succ
R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
p : ℕ
fp : Fact (Nat.Prime p)
hch : CharP R' p
hp : IsUnit ↑p
χ : MulChar R R'
hχ : χ.IsQuadratic
ψ : AddChar R R'
n : ℕ
ih : gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ
⊢ (gaussSum χ ψ ^ p ^ n) ^ p = χ (↑p ^ (n + 1)) * gaussSum χ ψ

Tactic:
ih,

Goal state after:
case succ
R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
p : ℕ
fp : Fact (Nat.Prime p)
hch : CharP R' p
hp : IsUnit ↑p
χ : MulChar R R'
hχ : χ.IsQuadratic
ψ : AddChar R R'
n : ℕ
ih : gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ
⊢ (χ (↑p ^ n) * gaussSum χ ψ) ^ p = χ (↑p ^ (n + 1)) * gaussSum χ ψ
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: F : Type u_1
inst✝¹ : Fintype F
inst✝ : Field F
hF : ringChar F ≠ 2
hp2 : ∀ (n : ℕ), 2 ^ n ≠ 0
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
FF : Type u_1 := CyclotomicField 8 F
hchar : ringChar F = ringChar FF
FFp : Nat.Prime (ringChar FF)
this : Fact (Nat.Prime (ringChar FF))
hFF : ringChar FF ≠ 2
hu : IsUnit ↑(ringChar FF)
ψ₈ : PrimitiveAddChar (ZMod ↑8) F := primitiveZModChar 8 F ⋯
ψ₈char : AddChar (ZMod 8) FF := ψ₈.char
τ : FF := ψ₈char 1
⊢ τ ^ 4 = -1

Tactic:
rw [show τ = ψ₈.char 1 from rfl]

Goal state after:
F : Type u_1
inst✝¹ : Fintype F
inst✝ : Field F
hF : ringChar F ≠ 2
hp2 : ∀ (n : ℕ), 2 ^ n ≠ 0
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
FF : Type u_1 := CyclotomicField 8 F
hchar : ringChar F = ringChar FF
FFp : Nat.Prime (ringChar FF)
this : Fact (Nat.Prime (ringChar FF))
hFF : ringChar FF ≠ 2
hu : IsUnit ↑(ringChar FF)
ψ₈ : PrimitiveAddChar (ZMod ↑8) F := primitiveZModChar 8 F ⋯
ψ₈char : AddChar (ZMod 8) FF := ψ₈.char
τ : FF := ψ₈char 1
⊢ ψ₈.char 1 ^ 4 = -1
Module: Mathlib.Data.Finset.Interval

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Preorder β
f : Finset α → β
⊢ Monotone f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f s ≤ f (cons a s ha)

Tactic:
simp [monotone_iff_forall_covBy, covBy_iff_exists_cons]

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
⊢ (AffineMap.lineMap a b) t₁ -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)

Tactic:
AffineMap.lineMap_apply,

Goal state after:
V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
⊢ (t₁ • (b -ᵥ a) +ᵥ a) -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)
Module: Mathlib.Analysis.InnerProductSpace.Affine

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
⊢ (t₁ • (b -ᵥ a) +ᵥ a) -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)

Tactic:
AffineMap.lineMap_apply,

Goal state after:
V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
⊢ (t₁ • (b -ᵥ a) +ᵥ a) -ᵥ (t₂ • (c -ᵥ a) +ᵥ a) = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)
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
⊢ { toFun := ⇑(adjoint counit), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r * x =
    x * { toFun := ⇑(adjoint counit), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } r

Tactic:
by
    dsimp
    simp_rw [← rTensor_tmul, ← lTensor_tmul, ← adjoint_lTensor, ← adjoint_rTensor,
      ← comp_apply, ← adjoint_comp, rTensor_counit_comp_comul, lTensor_counit_comp_comul,
      ← toLinearMap_symm_rid, ← toLinearMap_symm_lid, ← comm_trans_lid,
      ← toLinearEquiv_commIsometry, ← toLinearEquiv_lidIsometry, ← toLinearEquiv_trans,
      ← toLinearEquiv_symm, adjoint_toLinearMap_eq_symm]
    simp

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

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
⊢ innerProductForm.flip = innerProductForm

Tactic:
by
  apply linearMap_ext
  ext
  simp only [LinearMap.compAlternatingMap_apply, LinearMap.flip_apply,
    innerProductForm_ιMulti_ιMulti]
  rw [← Matrix.det_transpose]
  congr 1
  ext
  exact real_inner_comm _ _

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

Goal state before:
pp: ι : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : Fintype ι
p : ℝ
hp : (↑#ι).HolderConjugate p
u : (ι → ℝ) → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
this : 1 ≤ ↑#ι - 1
⊢ ∫⁻ (x : ι → ℝ), (‖u x‖ₑ ^ (1 / (↑#ι - 1))) ^ ↑#ι = ∫⁻ (x : ι → ℝ), ∏ _i, ‖u x‖ₑ ^ (1 / (↑#ι - 1))

Tactic:
congr! 2 with x
        simp_rw [prod_const]
        norm_cast

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

Goal state before:
pp: F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this
⊢ (↑#ι).HolderConjugate p

Tactic:
hιcard

Goal state after:
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this
⊢ (↑(finrank ℝ E)).HolderConjugate p
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
⊢ dimH s = ⨆ d, ⨆ (_ : μH[↑d] s = ∞), ↑d

Tactic:
by
  borelize X; rw [dimH]

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

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
⊢ mkMetric' m = ⨆ n, pre m (↑n)⁻¹

Tactic:
ext1 s

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s : Set X
⊢ (mkMetric' m) s = (⨆ n, pre m (↑n)⁻¹) s
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
K : ℝ≥0
f : X → Y
hf : AntilipschitzWith K f
s : Set Y
⊢ dimH (f ⁻¹' s) ≤ dimH s

Tactic:
by
  borelize X Y
  refine dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top ?_
  have := hf.hausdorffMeasure_preimage_le d.coe_nonneg s
  rw [hd, top_le_iff] at this
  contrapose! this
  exact ENNReal.mul_ne_top (by simp) this

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ ^ 2 =
    4 / (4 * 4 + 0 * 0) *
        (‖x + x‖ * ‖x + x‖ - 0 * 0 - (‖x - x‖ * ‖x - x‖ - 0 * 0) +
            ((0 * ‖I • x + x‖ - im I * 0) * ‖I • x + x‖ - (0 * 0 + im I * ‖I • x + x‖) * 0) -
          ((0 * ‖I • x - x‖ - im I * 0) * ‖I • x - x‖ - (0 * 0 + im I * ‖I • x - x‖) * 0)) -
      -0 / (4 * 4 + 0 * 0) *
        (‖x + x‖ * 0 + 0 * ‖x + x‖ - (‖x - x‖ * 0 + 0 * ‖x - x‖) +
            ((0 * ‖I • x + x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x + x‖) * ‖I • x + x‖) -
          ((0 * ‖I • x - x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x - x‖) * ‖I • x - x‖))

Tactic:
have h₁ : ‖x - x‖ = 0 := by simp

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h₁ : ‖x - x‖ = 0
⊢ ‖x‖ ^ 2 =
    4 / (4 * 4 + 0 * 0) *
        (‖x + x‖ * ‖x + x‖ - 0 * 0 - (‖x - x‖ * ‖x - x‖ - 0 * 0) +
            ((0 * ‖I • x + x‖ - im I * 0) * ‖I • x + x‖ - (0 * 0 + im I * ‖I • x + x‖) * 0) -
          ((0 * ‖I • x - x‖ - im I * 0) * ‖I • x - x‖ - (0 * 0 + im I * ‖I • x - x‖) * 0)) -
      -0 / (4 * 4 + 0 * 0) *
        (‖x + x‖ * 0 + 0 * ‖x + x‖ - (‖x - x‖ * 0 + 0 * ‖x - x‖) +
            ((0 * ‖I • x + x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x + x‖) * ‖I • x + x‖) -
          ((0 * ‖I • x - x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x - x‖) * ‖I • x - x‖))
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 * μHE[finrank ℝ U] s

Tactic:
by
  simp_rw [euclideanHausdorffMeasure_def, Measure.smul_apply, nnreal_smul_coe_apply,
    hausdorffMeasure_image]
  exact mul_left_comm _ _ _

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

Goal state before:
pp: case inl.inr.inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
d : ℝ
hd : 0 ≤ d
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
h'd : 0 < d
⊢ μH[d] {f x} ≤ ↑0 ^ d * μH[↑r * d] s

Tactic:
have := nullSingletonClass_hausdorff Y h'd
      simp only [zero_le, measure_singleton]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
⊢ innerProp' E r

Tactic:
intro x y
  rw [← re_add_im r, add_smul, add_left, real_prop _ x, ← smul_smul, real_prop _ _ y, I_prop,
    map_add, map_mul, conj_ofReal, conj_ofReal, conj_I]
  ring

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 : ι
⊢ 0 ≤ ↑(b i) - ↑(a i)

Tactic:
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
a✝ : ℕ
hn : ∀ (i : γ a✝), ediam (t a✝ i) ≤ 1 / ↑a✝
i : γ a✝
x✝ : i ∈ Finset.univ
⊢ ediam (t a✝ i) ^ Fintype.card ι ≤ (1 / ↑a✝) ^ Fintype.card ι

Tactic:
exact pow_le_pow_left' (hn i) _

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

Goal state before:
pp: case refine_2
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
K : Submodule ℝ E := ℝ ∙ x
this : Nontrivial ↥Kᗮ
w : ↥Kᗮ
hw₀ : w ≠ 0
hw' : ⟪x, ↑w⟫ = 0
hw : ↑w ≠ 0
⊢ |(o.areaForm x) ↑w| ≤ ‖o.rightAngleRotationAux₁ x‖ * ‖↑w‖

Tactic:
rw [← o.inner_rightAngleRotationAux₁_left x w]

Goal state after:
case refine_2
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
K : Submodule ℝ E := ℝ ∙ x
this : Nontrivial ↥Kᗮ
w : ↥Kᗮ
hw₀ : w ≠ 0
hw' : ⟪x, ↑w⟫ = 0
hw : ↑w ≠ 0
⊢ |⟪o.rightAngleRotationAux₁ x, ↑w⟫| ≤ ‖o.rightAngleRotationAux₁ x‖ * ‖↑w‖
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).rightAngleRotation x = -o.rightAngleRotation x

Tactic:
by
  apply ext_inner_right ℝ
  intro y
  rw [inner_rightAngleRotation_left]
  simp

Goal state after:
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: case neg
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
i j : n
hij : ¬i = j
⊢ 0 = lowerInv hS i ⬝ᵥ S *ᵥ (lowerInv hS)ᴴᵀ j

Tactic:
conjTranspose,

Goal state after:
case neg
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
i j : n
hij : ¬i = j
⊢ 0 = lowerInv hS i ⬝ᵥ S *ᵥ ((lowerInv hS)ᵀ.map star)ᵀ j
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
⊢ ↑⟪y, x⟫ + (o.areaForm x) y • Complex.I = (starRingEnd ℂ) (↑⟪y, x⟫ + (o.areaForm y) x • Complex.I)

Tactic:
areaForm_swap

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
⊢ ↑⟪y, x⟫ + -(o.areaForm y) x • Complex.I = (starRingEnd ℂ) (↑⟪y, x⟫ + (o.areaForm y) x • Complex.I)
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case inl
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
hx : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖x‖ = 0
⊢ x = 0 ∨ y = 0

Tactic:
left
    simpa using h

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

Goal state before:
pp: 𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
| Continuous[instTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun a ↦ z ((g.comp a) x)

Tactic:
conv =>

Goal state after:
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| σ_li (σ_li.symm (z (g.toCLM (a x))))
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)) = cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))

Tactic:
conv =>

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.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 1 - t * ‖z‖ = 1 - ‖↑t * z‖

Tactic:
rw [norm_mul, Complex.norm_of_nonneg ht.1]

Goal state after:
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
⊢ Γ (a * s + b * t) ≤ Γ s ^ a * Γ t ^ b

Tactic:
have posf' : ∀ c u : ℝ, ∀ᵐ x : ℝ ∂volume.restrict (Ioi 0), 0 ≤ f c u x := fun c u =>
    (ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ (posf c u))

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
⊢ Γ (a * s + b * t) ≤ Γ s ^ a * Γ t ^ b
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
n : ℕ
hn : n > 0
| ‖z‖ ^ n / ↑n ≤ ‖z‖ ^ n

Tactic:
conv =>

Goal state after:
z : ℂ
hz : ‖z‖ < 1
n : ℕ
hn : n > 0
| ‖z‖ ^ n / 1
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
| Tendsto (fun x ↦ ∑ i ∈ Finset.range x, (-1) ^ (i + 1) * z ^ i / ↑i - log (1 + z)) atTop (nhds 0)

Tactic:
conv =>

Goal state after:
z : ℂ
hz : ‖z‖ < 1
x : ℕ
| (logTaylor x z - log (1 + z)) / 1
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
hab : a < b
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-a)
t : ℝ
ht : t ∈ Ioi 0
⊢ t ^ (-(b - a)) = t ^ (a - b)

Tactic:
neg_sub

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

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
| HasSum (fun n ↦ z ^ n / ↑n) (-log (1 - z))

Tactic:
conv =>

Goal state after:
z : ℂ
hz : ‖z‖ < 1
n : ℕ
| - -(z ^ n / ↑n)