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

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommRing R'
χ : MulChar R R'
hχ : χ.IsQuadratic
x : Rˣ
h₂ : χ ↑x = -1
this : -1 = ↑(-1)
⊢ (χ ↑x)⁻¹ʳ = χ ↑x

Tactic:
h₂,

Goal state after:
case inr.inr
R : Type u_1
inst✝¹ : CommMonoid R
R' : Type u_2
inst✝ : CommRing R'
χ : MulChar R R'
hχ : χ.IsQuadratic
x : Rˣ
h₂ : χ ↑x = -1
this : -1 = ↑(-1)
⊢ (-1)⁻¹ʳ = -1
Module: Mathlib.NumberTheory.Cyclotomic.Basic

Goal state before:
pp: case a
n : ℕ
inst✝⁴ : NeZero 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
ζ : B
hζ : IsPrimitiveRoot ζ n
⊢ A[ζ] ≤ adjoin A {b | n ≠ 0 ∧ b ^ n = 1}

Tactic:
exact adjoin_mono <| Set.singleton_subset_iff.mpr ⟨NeZero.ne n, hζ.pow_eq_one⟩

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : F → E
hf : Differentiable ℝ f
x : F
p : ℝ≥0
hp : 1 < p
⊢ ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ ↑p) x‖ₑ ≤ ↑p * ‖f x‖ₑ ^ (↑p - 1) * ‖fderiv ℝ f x‖ₑ

Tactic:
simpa [enorm, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.2 <| NNReal.one_le_coe.2 hp.le),
    ← ENNReal.coe_mul] using nnnorm_fderiv_norm_rpow_le hf hp

Goal state after:
Module: Mathlib.NumberTheory.GaussSum

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' : Nat.Prime (ringChar F')
hc' : Fintype.card F' = ringChar F' ^ ↑n'
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' hch₁
⊢ (χ (-1) * ↑(Fintype.card F)) ^ (Fintype.card F' / 2) = χ ↑(Fintype.card F')

Tactic:
let FF' := CyclotomicField ψ.n F'

Goal state after:
F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' : Nat.Prime (ringChar F')
hc' : Fintype.card F' = ringChar F' ^ ↑n'
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' hch₁
FF' : Type u_2 := CyclotomicField (↑ψ.n) F'
⊢ (χ (-1) * ↑(Fintype.card F)) ^ (Fintype.card F' / 2) = χ ↑(Fintype.card F')
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
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
⊢ T.IsSymmetric →
    ((∀ (x : F), 0 ≤ re ⟪(↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv) x, x⟫) ↔ ∀ (x : E), 0 ≤ re ⟪T x, x⟫)

Tactic:
LinearIsometryEquiv.toLinearEquiv_symm,

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
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
⊢ T.IsSymmetric → ((∀ (x : F), 0 ≤ re ⟪(↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm) x, x⟫) ↔ ∀ (x : E), 0 ≤ re ⟪T x, x⟫)
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
inst✝ : U.HasOrthogonalProjection
⊢ True ∧ True

Tactic:
and_self

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ True ∧ True

Tactic:
and_self

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
p : E →L[𝕜] E
hp : IsIdempotentElem p
⊢ p.IsPositive ↔ IsSelfAdjoint p

Tactic:
rw [← isPositive_toLinearMap_iff, IsIdempotentElem.isPositive_iff_isSymmetric hp.toLinearMap]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
p : E →L[𝕜] E
hp : IsIdempotentElem p
⊢ (↑p).IsSymmetric ↔ IsSelfAdjoint p
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
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f g : E →ₗ.[R] F
hfg : g ≤ f
f' : E →ₗ.[R] F
hf : f.graph.topologicalClosure = f'.graph
⊢ g.graph.topologicalClosure ≤ f'.graph

Tactic:
← hf

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
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousAdd F
inst✝² : TopologicalSpace R
inst✝¹ : ContinuousSMul R E
inst✝ : ContinuousSMul R F
f g : E →ₗ.[R] F
hfg : g ≤ f
f' : E →ₗ.[R] F
hf : f.graph.topologicalClosure = f'.graph
⊢ g.graph.topologicalClosure ≤ f.graph.topologicalClosure
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
⊢ T.singularValues.support.card = finrank 𝕜 ↥T.range

Tactic:
have hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E := by
    grind [singularValues_of_finrank_le]

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E
⊢ T.singularValues.support.card = finrank 𝕜 ↥T.range
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: 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
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := EuclideanSpace ℝ (Fin (finrank ℝ F))
e : F ≃L[ℝ] F' := toEuclidean
C₁ : ℝ≥0 := ‖↑e.symm‖₊
C : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst μ ↑p
C₂ : ℝ≥0 := ‖↑e‖₊
v : E → F' := ⇑e ∘ u
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
h4v : ∀ (x : E), ‖fderiv ℝ v x‖ ≤ ↑C₂ * ‖fderiv ℝ u x‖
⊢ eLpNorm (⇑e.symm ∘ v) (↑p') μ ≤ C₁ • eLpNorm v (↑p') μ

Tactic:
apply eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul

Goal state after:
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
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := ⋯
e : F ≃L[ℝ] F' := ⋯
C₁ : ℝ≥0 := ⋯
C : ℝ≥0 := ⋯
C₂ : ℝ≥0 := ⋯
v : E → F' := ⋯
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
h4v : ∀ (x : E), ‖fderiv ℝ v x‖ ≤ ↑C₂ * ‖fderiv ℝ u x‖
⊢ ∀ᵐ (x : E) ∂μ, ‖(⇑e.symm ∘ v) x‖₊ ≤ C₁ * ‖v x‖₊
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case neg
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) = ∞
⊢ ∑' (k : ℕ), μ (S (2 * k + 1) \ S (2 * k)) + ∑' (k : ℕ), μ (S (2 * k + 1 + 1) \ S (2 * k + 1)) ≠ ∞

Tactic:
ENNReal.add_ne_top

Goal state after:
case neg
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) = ∞
⊢ ∑' (k : ℕ), μ (S (2 * k + 1) \ S (2 * k)) ≠ ∞ ∧ ∑' (k : ℕ), μ (S (2 * k + 1 + 1) \ S (2 * k + 1)) ≠ ∞
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case neg
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
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hrank : finrank 𝕜 U ≠ finrank 𝕜 ↥f.range
⊢ ‖((toMatrix bu.toBasis bv.toBasis) f).det‖ = 0

Tactic:
contrapose hrank with hdet

Goal state after:
case neg
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
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hdet : ¬‖((toMatrix bu.toBasis bv.toBasis) f).det‖ = 0
⊢ finrank 𝕜 U = finrank 𝕜 ↥f.range
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : (g ∘ₗ f).ker = ⊥
bw : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(g ∘ₗ f).range
bw' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(g.domRestrict f.range).range :=
  bw.map (LinearIsometryEquiv.ofEq (g ∘ₗ f).range (g.domRestrict f.range).range ⋯)
⊢ ‖((toMatrix bu.toBasis bw.toBasis) (g ∘ₗ f).rangeRestrict).det‖ =
    ‖((toMatrix bv.toBasis bw'.toBasis) (g.domRestrict f.range).rangeRestrict).det‖ *
      ‖((toMatrix bu.toBasis bv.toBasis) f.rangeRestrict).det‖

Tactic:
← norm_mul,

Goal state after:
case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : (g ∘ₗ f).ker = ⊥
bw : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(g ∘ₗ f).range
bw' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(g.domRestrict f.range).range :=
  bw.map (LinearIsometryEquiv.ofEq (g ∘ₗ f).range (g.domRestrict f.range).range ⋯)
⊢ ‖((toMatrix bu.toBasis bw.toBasis) (g ∘ₗ f).rangeRestrict).det‖ =
    ‖((toMatrix bv.toBasis bw'.toBasis) (g.domRestrict f.range).rangeRestrict).det *
        ((toMatrix bu.toBasis bv.toBasis) f.rangeRestrict).det‖
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
v : V
⊢ ‖⟪Finsupp.single (x, v) 1, Finsupp.single (x, v) 1⟫_𝕜‖ ≤ (√‖K x x‖ * ‖v‖) ^ 2

Tactic:
simp only [inner_H₀_def, RCLike.star_def, mul_zero, zero_mul,
    Finsupp.sum_single_index, mul_one, map_zero, map_one, one_mul]

Goal state after:
𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
v : V
⊢ ‖⟪(K x x) v, v⟫_𝕜‖ ≤ (√‖K x x‖ * ‖v‖) ^ 2
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case «0».«0»
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
h : ⟪x, y⟫ = 0
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ⟪![x, y] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0

Tactic:
simp_all

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

Goal state before:
pp: case «1».«1»
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
h : ⟪x, y⟫ = 0
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ⟪![x, y] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0

Tactic:
simp_all

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

Goal state before:
pp: case «0».«0»
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
hx : x ≠ 0
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0

Tactic:
simp_all

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

Goal state before:
pp: case «0».«1»
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
hx : x ≠ 0
hij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0

Tactic:
simp_all

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

Goal state before:
pp: case «1».«0»
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
hx : x ≠ 0
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0

Tactic:
simp_all

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

Goal state before:
pp: case «1».«1»
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
hx : x ≠ 0
hij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ⟪![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, o.rightAngleRotation x] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0

Tactic:
simp_all

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

Goal state before:
pp: case neg.mp
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 : ¬x = 0
a : ℝ := ((o.basisRightAngleRotation x hx).repr y) 0
b : ℝ := ((o.basisRightAngleRotation x hx).repr y) 1
⊢ 0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • o.rightAngleRotation x)

Tactic:
simp_all

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

Goal state before:
pp: z w : ℂ
⊢ ⟪Complex.orientation.rightAngleRotation z, w⟫ = ⟪I * z, w⟫

Tactic:
rw [Orientation.inner_rightAngleRotation_left]

Goal state after:
z w : ℂ
⊢ (Complex.orientation.areaForm z) w = ⟪I * z, w⟫
Module: Mathlib.Analysis.MellinTransform

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

Tactic:
rw [smul_eq_mul, ← rpow_add ht, ← sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel_left]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_1
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
j : ℝ
hj : b < j
hj' : j < a
w : ℝ
hw1 : b < w
hw2 : w < j
w' : ℝ
hw1' : j < w'
hw2' : w' < a
⊢ LocallyIntegrableOn (fun t ↦ |log t| * ‖f t‖) (Ioi 0) volume

Tactic:
simp_rw [mul_comm]

Goal state after:
case refine_1
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
j : ℝ
hj : b < j
hj' : j < a
w : ℝ
hw1 : b < w
hw2 : w < j
w' : ℝ
hw1' : j < w'
hw2' : w' < a
⊢ LocallyIntegrableOn (fun t ↦ ‖f t‖ * |log t|) (Ioi 0) volume
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case a
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
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 u : V
⊢ atTop • 𝓝 v +ᵥ pure u ≤ atTop • 𝓝 v

Tactic:
simp_rw [vadd_eq_add, add_pure, ← map₂_smul, map_map₂, ← map_prod_eq_map₂]

Goal state after:
case a
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
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 u : V
⊢ map (fun p ↦ p.1 • p.2 + u) (atTop ×ˢ 𝓝 v) ≤ map (fun p ↦ p.1 • p.2) (atTop ×ˢ 𝓝 v)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
hs : ∀ (c : k), 0 < c → ∀ x ∈ s, c • x ∈ s
⊢ asymptoticCone k s = closure[inst✝⁵] s

Tactic:
ext v
  rw [mem_closure_iff_frequently, ← map_snd_prod (atTop (α := k)) (𝓝 v), frequently_map,
    mem_asymptoticCone_iff, asymptoticNhds_eq_smul, ← map₂_smul, ← map_prod_eq_map₂, frequently_map]
  apply frequently_congr
  filter_upwards [tendsto_fst.eventually (eventually_gt_atTop 0)] with ⟨c, u⟩ hc
  refine ⟨fun hu => ?_, hs c hc u⟩
  specialize hs c⁻¹ (inv_pos_of_pos hc) (c • u) hu
  rwa [inv_smul_smul₀ hc.ne'] at hs

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

Goal state before:
pp: case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := ⋯
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun n ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))

Tactic:
simp_rw [mul_comm]

Goal state after:
case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun n ↦ ↑x ^ (s - 1) * ↑((1 - x / ↑n) ^ n)) atTop (𝓝 (↑x ^ (s - 1) * ↑(rexp (-x))))
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e ↦ dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e ↦ ((e.trans R).trans e.symm).trans R
e : PE ≃ᵢ PE
⊢ dist ((pointReflection ℝ z) (e.symm ((pointReflection ℝ z) (e z)))) z = 2 * dist (e z) z

Tactic:
rw [dist_pointReflection_fixed, ← e.dist_eq, e.apply_symm_apply,
      dist_pointReflection_self_real, dist_comm]

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

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

Tactic:
rw [tendsto_nhdsWithin_iff]

Goal state after:
s : ℂ
h1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
⊢ Tendsto ofReal (𝓝[≠] 1) (𝓝 1) ∧ ∀ᶠ (n : ℝ) in 𝓝[≠] 1, ↑n ∈ {1}ᶜ
Module: Mathlib.Analysis.Polynomial.Factorization

Goal state before:
pp: f : ℝ[X]
n : ℕ
hf : f.IsMonicOfDegree (n + 2)
⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 2 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂

Tactic:
obtain ⟨g₁, g₂, hd₁ | hd₂, h⟩ := hf.eq_isMonicOfDegree_one_or_two_mul

Goal state after:
case inl
f : ℝ[X]
n : ℕ
hf : f.IsMonicOfDegree (n + 2)
g₁ g₂ : ℝ[X]
h : f = g₁ * g₂
hd₁ : g₁.IsMonicOfDegree 1
⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 2 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂

case inr
f : ℝ[X]
n : ℕ
hf : f.IsMonicOfDegree (n + 2)
g₁ g₂ : ℝ[X]
h : f = g₁ * g₂
hd₂ : g₁.IsMonicOfDegree 2
⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 2 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂