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

Goal state before:
pp: case refine_2
n : ℕ
inst✝⁷ : NeZero n
S T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑n
this✝¹ : NeZero ↑n
this✝ : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
this : (cyclotomic n K).degree ≠ 0
ζ : (cyclotomic n K).SplittingField
hζ : IsPrimitiveRoot ζ n
⊢ ∀ (x : CyclotomicField n K), x ∈ adjoin K {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}

Tactic:
rw [← Algebra.eq_top_iff, ← SplittingField.adjoin_rootSet, eq_comm]

Goal state after:
case refine_2
n : ℕ
inst✝⁷ : NeZero n
S T : Set ℕ
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : NeZero ↑n
this✝¹ : NeZero ↑n
this✝ : DecidableEq (CyclotomicField n K) := Classical.decEq (CyclotomicField n K)
this : (cyclotomic n K).degree ≠ 0
ζ : (cyclotomic n K).SplittingField
hζ : IsPrimitiveRoot ζ n
⊢ adjoin K ((cyclotomic n K).rootSet (cyclotomic n K).SplittingField) =
    adjoin K {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1}
Module: Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar

Goal state before:
pp: N : ℕ
inst✝ : NeZero N
j : ℤ
⊢ stdAddChar ↑j = cexp (2 * ↑π * I * ↑j / ↑N)

Tactic:
simp [stdAddChar, toCircle_intCast]

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

Goal state before:
pp: case pos
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

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, dif_pos]
          have hm : (hx.mul hy).unit = hx.unit * hy.unit := Units.ext rfl
          rw [hm, map_mul]
          norm_cast

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommMonoidWithZero R'
χ : MulChar R R'
⊢ χ * 1 = χ

Tactic:
ext
  simp only [mul_one, Pi.mul_apply, MulChar.coeToFun_mul, MulChar.one_apply_coe]

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 : MulChar R R'
ψ : AddChar R R'
a : Rˣ
⊢ gaussSum χ (ψ.mulShift ↑a) = χ⁻¹ ↑a * gaussSum χ ψ

Tactic:
← gaussSum_mulShift χ ψ a,

Goal state after:
R : Type u
inst✝² : CommRing R
inst✝¹ : Fintype R
R' : Type v
inst✝ : CommRing R'
χ : MulChar R R'
ψ : AddChar R R'
a : Rˣ
⊢ gaussSum χ (ψ.mulShift ↑a) = χ⁻¹ ↑a * (χ ↑a * gaussSum χ (ψ.mulShift ↑a))
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: case inr
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'
b : R
hb : b ≠ 0
x : R
⊢ χ x * ψ (b * (x - 1)) = χ (b * b⁻¹ * x) * ψ (b * x - b)

Tactic:
mul_inv_cancel₀ hb,

Goal state after:
case inr
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'
b : R
hb : b ≠ 0
x : R
⊢ χ x * ψ (b * (x - 1)) = χ (1 * x) * ψ (b * x - b)
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:
enter [1, 2, x, 2, y]
    rw [mul_mul_mul_comm, ← map_mul, ← map_add_eq_mul, ← sub_eq_add_neg]

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.Analysis.Fourier.ZMod

Goal state before:
pp: N : ℕ
inst✝² : NeZero N
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
c : ℂ
Φ : ZMod N → E
⊢ auxDFT (c • Φ) = (RingHom.id ℂ) c • auxDFT Φ

Tactic:
by
    ext; simp only [auxDFT, Pi.smul_apply, RingHom.id_apply, smul_sum, smul_comm c]

Goal state after:
Module: Mathlib.Analysis.Hofer

Goal state before:
pp: X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
H : ∀ ε' > 0, ∀ (x' : X), ε' ≤ ε → d x' x ≤ 2 * ε → ε * ϕ x ≤ ε' * ϕ x' → ∃ y, d x' y ≤ ε' ∧ 2 * ϕ x' < ϕ y
⊢ ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'

Tactic:
intro x'

Goal state after:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
H : ∀ ε' > 0, ∀ (x' : X), ε' ≤ ε → d x' x ≤ 2 * ε → ε * ϕ x ≤ ε' * ϕ x' → ∃ y, d x' y ≤ ε' ∧ 2 * ϕ x' < ϕ y
x' : X
⊢ ∀ (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
f : E →ₗ.[R] F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
hf : f.toFun.ker = ⊥
hf' : f.IsClosable
f' : F →ₗ.[R] E
h : f.inverse.graph.topologicalClosure = f'.graph
x : E
hx : x ∈ f.closure.domain
hx' : (x, 0) ∈ f.closure.graph
⊢ (0, x) ∈ (Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph).topologicalClosure

Tactic:
rw [← hf'.graph_closure_eq_closure_graph, ← SetLike.mem_coe,
        Submodule.topologicalClosure_coe] at hx'

Goal state after:
R : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module R E
inst✝⁷ : Module R F
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
f : E →ₗ.[R] F
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
hf : f.toFun.ker = ⊥
hf' : f.IsClosable
f' : F →ₗ.[R] E
h : f.inverse.graph.topologicalClosure = f'.graph
x : E
hx : x ∈ f.closure.domain
hx' : (x, 0) ∈ closure[instTopologicalSpaceProd] ↑f.graph
⊢ (0, x) ∈ (Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph).topologicalClosure
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≥0
hp : (↑(finrank ℝ E)).HolderConjugate p
h0p : 0 < ↑p
⊢ ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p ∂μ ≤ ↑(lintegralPowLePowLIntegralFDerivConst μ ↑p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂μ) ^ ↑p

Tactic:
exact lintegral_pow_le_pow_lintegral_fderiv μ hu h2u hp.coe

Goal state after:
Module: Mathlib.Dynamics.BirkhoffSum.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
E : Type u_3
inst✝³ : PseudoEMetricSpace X
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : X → X
g : X → E
hf : LipschitzWith 1 f
hg : UniformContinuous g
ε : ℝ
hε : 0 < ε
δ : ℝ≥0∞
hδ₀ : 0 < δ
hδε : ∀ (x y : X), (x, y) ∈ {p | edist p.1 p.2 ≤ δ} → (g x, g y) ∈ {p | dist p.1 p.2 ≤ ε}
x y : X
h : edist x y < δ
n : ℕ
⊢ ↑n * ε / ↑n ≤ ε

Tactic:
by
      rcases eq_or_ne n 0 with hn | hn <;> simp [hn, hε.le, mul_div_cancel_left₀]

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

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 : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
E' : Submodule 𝕜 E
F' : Submodule 𝕜 F
iE' : Module.Finite 𝕜 ↥E'
iF' : Module.Finite 𝕜 ↥F'
y : ↥E' ⊗[𝕜] ↥F'
hx : inner 𝕜 ((mapIncl E' F') y) ((mapIncl E' F') y) = 0
hz : {(mapIncl E' F') y} ⊆ ↑(mapIncl E' F').range
e : OrthonormalBasis (Fin (finrank 𝕜 ↥E')) 𝕜 ↥E'
f : OrthonormalBasis (Fin (finrank 𝕜 ↥F')) 𝕜 ↥F'
this✝ : ∀ (i : Fin (finrank 𝕜 ↥E')) (j : Fin (finrank 𝕜 ↥F')), ((e.toBasis.tensorProduct f.toBasis).repr y) (i, j) = 0
this : y = 0
⊢ (mapIncl E' F') y = 0

Tactic:
rw [this, map_zero]

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, ∑ i, inner 𝕜 (f (e i) ⊗ₜ[𝕜] g i) (f (e x) ⊗ₜ[𝕜] g x)) ≤
    ‖f‖ ^ 2 * RCLike.re (∑ x, ∑ i, inner 𝕜 (e i ⊗ₜ[𝕜] g i) (e x ⊗ₜ[𝕜] g x))

Tactic:
inner_tmul,

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, ∑ i, inner 𝕜 (f (e i)) (f (e x)) * inner 𝕜 (g i) (g x)) ≤
    ‖f‖ ^ 2 * RCLike.re (∑ x, ∑ i, inner 𝕜 (e i) (e x) * inner 𝕜 (g i) (g x))
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case pos
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) = ∞
⊢ μ (s \ t) ≤ μ s

Tactic:
exact μ.mono sdiff_subset

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

Goal state before:
pp: case e'_4
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ ↑t = (μ.restrict s) Set.univ ^ (1 / (↑q).toReal - 1 / (↑p').toReal)

Tactic:
rw [ENNReal.coe_rpow_of_nonneg]

Goal state after:
case e'_4
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ ↑(μ s).toNNReal ^ (1 / ↑q - 1 / ↑p') = (μ.restrict s) Set.univ ^ (1 / (↑q).toReal - 1 / (↑p').toReal)

case e'_4.h
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ 0 ≤ 1 / ↑q - 1 / ↑p'
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: 𝕜 : Type u_3
V : Type u_4
P : Type u_5
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedSpace 𝕜 V
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : BorelSpace P
d : ℕ
x : P
c : 𝕜
hc : c ≠ 0
s : Set P
⊢ μHE[d] (⇑(AffineMap.homothety x c) ⁻¹' s) = ‖c‖₊⁻¹ ^ d • μHE[d] s

Tactic:
by
  simp_rw [euclideanHausdorffMeasure_def, Measure.smul_apply]
  rw [hausdorffMeasure_homothety_preimage (by simp) x hc, smul_comm, NNReal.rpow_natCast]

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

Goal state before:
pp: 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 : Submodule ℝ V
p q : P
⊢ (s.measurableEquivProd p) q = (s.orthogonalProjectionOnto (q -ᵥ p), sᗮ.orthogonalProjectionOnto (q -ᵥ p))

Tactic:
simp [measurableEquivProd]

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

Goal state before:
pp: 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 : Submodule ℝ V
p : P
q : ↥s × ↥sᗮ
⊢ (s.measurableEquivProd p).symm q = (↑q.1 + ↑q.2) +ᵥ p

Tactic:
simp [measurableEquivProd]

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

Goal state before:
pp: 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
⊢ volume (⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    ∫⁻ (x : ↥s), μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑x) s.directionᗮ)) ∂μHE[finrank ℝ ↥s.direction]

Tactic:
volume_eq_prod,

Goal state after:
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
⊢ (volume.prod volume) (⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t) =
    ∫⁻ (x : ↥s), μHE[finrank ℝ ↥s.directionᗮ] (t ∩ ↑(mk' (↑x) s.directionᗮ)) ∂μHE[finrank ℝ ↥s.direction]
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: 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
⊢ finrank ℝ ↥s.directionᗮ = finrank ℝ ↥(mk' (↑(x +ᵥ p)) s.directionᗮ).direction

Tactic:
direction_mk'

Goal state after:
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
⊢ finrank ℝ ↥s.directionᗮ = finrank ℝ ↥s.directionᗮ
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup U
inst✝⁶ : InnerProductSpace 𝕜 U
inst✝⁵ : FiniteDimensional 𝕜 U
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
h : finrank 𝕜 U = finrank 𝕜 V
htop : f.range = ⊤
⊢ f.range = id.range

Tactic:
simp [htop]

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 ^ 2) = ↑(∏ i, f.singularValues ↑i ^ 2)

Tactic:
normDet_sq

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ LinearMap.det (adjoint f ∘ₗ f) = ↑(∏ i, f.singularValues ↑i ^ 2)
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case refine_3
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℝ
hd : 0 < d
x : X
ε : ℝ≥0∞
x✝ : 0 < ε
⊢ ∑' (n : ℕ), ⨆ (_ : ((fun x_1 ↦ {x}) n).Nonempty), ediam ((fun x_2 ↦ {x}) n) ^ d ≤ 0

Tactic:
· simp [hd]

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

Goal state before:
pp: 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
f : X → Y
d : ℝ
hf : Isometry f
hd : 0 ≤ d ∨ Surjective f
s : Set X
⊢ ((OuterMeasure.comap f) (OuterMeasure.mkMetric fun r ↦ r ^ d)) s = (OuterMeasure.mkMetric fun r ↦ r ^ d) s

Tactic:
rw [OuterMeasure.isometry_comap_mkMetric _ hf (hd.imp_left _)]

Goal state after:
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
f : X → Y
d : ℝ
hf : Isometry f
hd : 0 ≤ d ∨ Surjective f
s : Set X
⊢ 0 ≤ d → Monotone fun r ↦ r ^ d
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
⊢ ![x, y] = ![y, x] ∘ ⇑(Equiv.swap 0 1)

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
⊢ 0 ≠ 1

Tactic:
· ext i
    fin_cases i <;> rfl

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
⊢ 0 ≠ 1
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
this : ⟪o.rightAngleRotationAux₁ y, o.rightAngleRotationAux₁ x⟫ = ⟪y, x⟫
⊢ ⟪y, o.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x)⟫ = ⟪y, -x⟫

Tactic:
rw [o.inner_rightAngleRotationAux₁_right, ← o.inner_rightAngleRotationAux₁_left, this,
    inner_neg_right]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
f : StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ⇑(e_dual f) = ⇑(e.dualMap ↑f)
g : StrongDual 𝕜 E
⊢ ⇑g = ⇑(e_dual.symm g) ∘ ⇑e

Tactic:
have := he _ ▸ congr(⇑$(e_dual.apply_symm_apply g)).symm

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
f : StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ⇑(e_dual f) = ⇑(e.dualMap ↑f)
g : StrongDual 𝕜 E
this : ⇑g = ⇑(e.dualMap ↑(e_dual.symm g))
⊢ ⇑g = ⇑(e_dual.symm g) ∘ ⇑e
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:
← Complex.areaForm,

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 = (Complex.orientation.areaForm (f x)) (f y)
Module: Mathlib.Analysis.MellinInversion

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

Tactic:
mellinInv,

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