mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
⊢ 1 - x ≠ 0

Tactic:
refine tsub_pos_of_lt ?_ |>.ne'

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
⊢ x < 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
f : C(↑s, 𝕜)₀
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) f

Tactic:
have : CompactSpace s := by rwa [isCompact_iff_compactSpace] at hs

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
f : C(↑s, 𝕜)₀
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) f
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
⊢ ‖z - y * z‖ ≤ c

Tactic:
by
  rw [← norm_inr (𝕜 := ℂ), inr_sub, inr_mul]
  refine norm_sub_mul_self_le _ ?_ ?_ hc h
  · rwa [inr_nonneg_iff]
  · have hy := hx₀.trans hxy
    rw [Set.mem_Icc, inr_le_iff _ _ hx₀.isSelfAdjoint hy.isSelfAdjoint,
      ← norm_le_one_iff_of_nonneg _, norm_inr]
    exact ⟨hxy, hy₁⟩

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: case id
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra 𝕜 A
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
⊢ Commute ((cfcHom ha) (ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜))) b

Tactic:
rwa [cfcHom_id ha]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: case pos
A : Type u_2
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : IsTopologicalRing A
inst✝³ : T2Space A
inst✝² : PartialOrder A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : StarOrderedRing A
a b : A
hb : Commute a b
f : ℝ≥0 → ℝ≥0
ha : 0 ≤ a
⊢ Commute (cfcₙ f a) b

Tactic:
rw [cfcₙ_nnreal_eq_real ..]
    exact hb.cfcₙ_real _

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra 𝕜 A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule 𝕜 A
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : ContinuousStar A
a : A
ha : p a
⊢ ⇑(cfcHom ha) '' {ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜)} = {a}

Tactic:
simpa using cfcHom_id ha

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
⊢ (fun x ↦ f x ↑z) =ᵐ[μ] fun x ↦ (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) z

Tactic:
filter_upwards [hf₁, hf₂] with x cont_x zero_x

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(quasispectrum 𝕜 a)
x : X
cont_x : ContinuousOn (f x) (quasispectrum 𝕜 a)
zero_x : f x 0 = 0
⊢ f x ↑z = (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) z
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
⊢ ↑((toDualMap 𝕜 E) x) (b i) = ↑((toDualMap 𝕜 E) y) (b i)

Tactic:
simp only [ContinuousLinearMap.coe_coe, toDualMap_apply_apply]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
x y : E
b : Basis ι 𝕜 E
h : ∀ (i : ι), ⟪b i, x⟫ = ⟪b i, y⟫
i : ι
⊢ ⟪x, b i⟫ = ⟪y, b i⟫
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Ring 𝕜
inst✝⁵ : LinearOrder 𝕜
inst✝⁴ : IsStrictOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : IsTorsionFree 𝕜 E
A : Set E
x : E
hA : Convex 𝕜 A
⊢ x ∈ extremePoints 𝕜 A ↔ x ∈ A ∧ Convex 𝕜 (A \ {x})

Tactic:
use fun hx ↦ ⟨hx.1, (isExtreme_singleton.2 hx).convex_sdiff hA⟩
  rintro ⟨hxA, hAx⟩
  refine mem_extremePoints_iff_forall_segment.2 ⟨hxA, fun x₁ hx₁ x₂ hx₂ hx ↦ ?_⟩
  rw [convex_iff_segment_subset] at hAx
  by_contra! h
  exact (hAx ⟨hx₁, fun hx₁ ↦ h.1 (mem_singleton_iff.2 hx₁)⟩
      ⟨hx₂, fun hx₂ ↦ h.2 (mem_singleton_iff.2 hx₂)⟩ hx).2 rfl

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

Goal state before:
pp: case a
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
i j : n
x : ℝ
a : n
h_a : a ∈ Finset.univ.val ∧ ‖U i a‖ ^ 2 = x
⊢ 0 ≤ x

Tactic:
rw [← h_a.2]

Goal state after:
case a
𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
i j : n
x : ℝ
a : n
h_a : a ∈ Finset.univ.val ∧ ‖U i a‖ ^ 2 = x
⊢ 0 ≤ ‖U i a‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.Matrix

Goal state before:

Tactic:
_ ≤ _ := Finset.sum_le_sum fun i _ ↦ by grw [mul_pow, norm_le_pi_norm v i]
      _ = _ := by simp [mul_pow, EuclideanSpace.norm_sq_eq x, Finset.mul_sum]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Opens X
hs : Nonempty ↥s
⊢ (e.subtypeRestr hs).target ⊆ e.target

Tactic:
← e.image_source_eq_target,

Goal state after:
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Opens X
hs : Nonempty ↥s
⊢ (e.subtypeRestr hs).target ⊆ ↑e '' e.source
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U : Opens X
hU : Nonempty ↥U
y : Y
hy : y ∈ (e.subtypeRestr hU).target
⊢ (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y = ↑e.symm y

Tactic:
e.eq_symm_apply _ hy.1

Goal state after:
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U : Opens X
hU : Nonempty ↥U
y : Y
hy : y ∈ (e.subtypeRestr hU).target
⊢ ↑e ((Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y) = y

X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U : Opens X
hU : Nonempty ↥U
y : Y
hy : y ∈ (e.subtypeRestr hU).target
⊢ (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y ∈ e.source
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
⊢ ∃ i_1 i_2,
    IsOpen[instTopologicalSpaceProd] i_2 ∧
      (Z.localTrivAsPartialEquiv i).source =
        (Z.localTrivAsPartialEquiv i_1).source ∩ ↑(Z.localTrivAsPartialEquiv i_1) ⁻¹' i_2

Tactic:
refine ⟨i, Z.baseSet i ×ˢ univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_⟩

Goal state after:
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i : ι
⊢ (Z.localTrivAsPartialEquiv i).source =
    (Z.localTrivAsPartialEquiv i).source ∩ ↑(Z.localTrivAsPartialEquiv i) ⁻¹' Z.baseSet i ×ˢ univ
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
e✝ : Pretrivialization F proj
x : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
z : ↑(proj ⁻¹' s)
hz : z ∈ Subtype.val ⁻¹' e.source
⊢ ⟨e.invFun (proj ↑z, (↑e ↑z).2), ⋯⟩ = z

Tactic:
simp_rw [← e.proj_toFun _ hz]

Goal state after:
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
e✝ : Pretrivialization F proj
x : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
z : ↑(proj ⁻¹' s)
hz : z ∈ Subtype.val ⁻¹' e.source
⊢ ⟨e.invFun ((↑e.toPartialEquiv ↑z).1, (↑e ↑z).2), ⋯⟩ = z
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: case hc
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
e✝ : Pretrivialization F proj
x : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
z : ↑(proj ⁻¹' s)
hz : z ∈ Subtype.val ⁻¹' e.source
⊢ (proj ↑z, (↑e ↑z).2) ∈ e.target

Tactic:
simp_rw [← e.proj_toFun _ hz]

Goal state after:
case hc
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
e✝ : Pretrivialization F proj
x : Z
e' : Pretrivialization F TotalSpace.proj
b : B
y : E b
e : Pretrivialization F proj
s : Set B
inst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))
z : ↑(proj ⁻¹' s)
hz : z ∈ Subtype.val ⁻¹' e.source
⊢ ((↑e.toPartialEquiv ↑z).1, (↑e ↑z).2) ∈ e.target
Module: Mathlib.Topology.SeparatedMap

Goal state before:
pp: X : Type u_1
Y : Type u_2
A : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
f : X → Y
hf : IsLocallyInjective f
g : A → X
cont : Continuous[inst✝, inst✝¹] g
hg : Function.Injective g
⊢ IsLocallyInjective (f ∘ g)

Tactic:
rw [isLocallyInjective_iff_isOpen_diagonal] at hf ⊢
  rw [← hg.preimage_pullbackDiagonal]
  apply hf.preimage (cont.mapPullback cont)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
n : ℕ
a : A
ha : IsStrictlyPositive a
⊢ log (a ^ n) = n • log a

Tactic:
grind [log_pow]

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

Goal state before:
pp: n : ℤ
⊢ exp (2 * π * ↑n) = 1

Tactic:
by
  simpa only [mul_comm] using exp_int_mul_two_pi n

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

Goal state before:
pp: x y : Circle
⊢ exp
      ((if (↑(exp (↑x).arg)).arg ≤ (↑y).arg then (↑y).arg - (↑(exp (↑x).arg)).arg
        else 2 * π + (↑y).arg - (↑(exp (↑x).arg)).arg) +
        (↑x).arg) =
    y

Tactic:
split_ifs with hxy <;> simp

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

Goal state before:
pp: r : ℝ
⊢ IsOpen (centeredArc r)

Tactic:
have hset : {x : ℝ | |x| < r} = Ioo (-r) r := by
    ext x
    simp [abs_lt]
  simpa [centeredArc, hset] using
    isLocalHomeomorph_circleExp.isOpenMap (Ioo (-r) r) isOpen_Ioo

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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : F
⊢ ‖(toDual 𝕜 E).symm ((toSesqForm A) x)‖ ≤ ‖A‖ * ‖x‖

Tactic:
LinearIsometryEquiv.norm_map

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : F
⊢ ‖(toSesqForm A) x‖ ≤ ‖A‖ * ‖x‖
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖A‖ ≤ ‖adjointAux A‖

Tactic:
← adjointAux_adjointAux A

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖adjointAux (adjointAux A)‖ ≤ ‖adjointAux A‖
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : E
⊢ ‖(toDual 𝕜 F).symm ((toSesqForm (adjointAux A)) x)‖ ≤ ‖adjointAux A‖ * ‖x‖

Tactic:
LinearIsometryEquiv.norm_map

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
x : E
⊢ ‖(toSesqForm (adjointAux A)) x‖ ≤ ‖adjointAux A‖ * ‖x‖
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖adjoint A‖ * ‖A‖ = ‖A‖ * ‖A‖

Tactic:
LinearIsometryEquiv.norm_map

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖A‖ * ‖A‖ = ‖A‖ * ‖A‖
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
this : s.centroid ∉ affineSpan k (s.points '' {i}ᶜ)
⊢ AffineIndependent k (Function.update s.points i s.centroid)

Tactic:
exact AffineIndependent.affineIndependent_update_of_notMem_affineSpan s.independent this

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ AffineIndependent k (Function.update s.points i s.centroid)

Tactic:
by
  have : s.centroid ∉ affineSpan k (s.points '' {i}ᶜ) :=
    s.centroid_notMem_affineSpan_of_ne_univ (by simp)
  exact AffineIndependent.affineIndependent_update_of_notMem_affineSpan s.independent this

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : DecidableEq P
p : ι → P
hi : AffineIndependent k p
s : Finset ι
sp : AffineSubspace k P
inst✝ : FiniteDimensional k ↥sp.direction
hle : affineSpan k ↑(Finset.image p s) ≤ sp
hc : #s = finrank k ↥sp.direction + 1
hn : s.Nonempty
⊢ (affineSpan k ↑(Finset.image p s)).direction = sp.direction

Tactic:
have hd := direction_le hle

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : DecidableEq P
p : ι → P
hi : AffineIndependent k p
s : Finset ι
sp : AffineSubspace k P
inst✝ : FiniteDimensional k ↥sp.direction
hle : affineSpan k ↑(Finset.image p s) ≤ sp
hc : #s = finrank k ↥sp.direction + 1
hn : s.Nonempty
hd : (affineSpan k ↑(Finset.image p s)).direction ≤ sp.direction
⊢ (affineSpan k ↑(Finset.image p s)).direction = sp.direction
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₀ : P
h : p₀ ∈ s
v : V
hp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
w : V
hw : w ∈ vectorSpan k s
x : V
hx : x ∈ (fun x ↦ x -ᵥ p₀) '' s
⊢ x ∈ ↑(k ∙ v)

Tactic:
SetLike.mem_coe,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₀ : P
h : p₀ ∈ s
v : V
hp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
w : V
hw : w ∈ vectorSpan k s
x : V
hx : x ∈ (fun x ↦ x -ᵥ p₀) '' s
⊢ x ∈ k ∙ v
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: α : Type u_3
E : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
G : Type u_7
inst✝⁴ : Group G
inst✝³ : MeasurableSpace α
inst✝² : MulAction G α
inst✝¹ : MeasurableConstSMul G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
f : α → E
g : G
h : MeasurableEmbedding fun x ↦ g • x
⊢ ∫ (x : α), f (g • x) ∂μ = ∫ (x : α), f x ∂μ

Tactic:
rw [← h.integral_map, MeasureTheory.map_smul]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hp : 0 < p.toReal
⊢ Memℓp f p ↔ ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ ∑' (i : α), ‖f i‖ ^ p.toReal

Tactic:
refine ⟨fun hf ↦ ?_, memℓp_gen'⟩
  obtain ⟨hp₁, hp₂⟩ := ENNReal.toReal_pos_iff.mp hp
  simp only [Memℓp, hp₁.ne', ↓reduceIte, hp₂.ne] at hf
  simpa [upperBounds] using isLUB_hasSum (by intro; positivity) hf.hasSum |>.1

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Nonempty α
f : ↥(lp E ∞)
⊢ IsLUB (Set.range fun i ↦ ‖↑f i‖) ‖f‖

Tactic:
lp.norm_eq_ciSup

Goal state after:
α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Nonempty α
f : ↥(lp E ∞)
⊢ IsLUB (Set.range fun i ↦ ‖↑f i‖) (⨆ i, ‖↑f i‖)
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inl.inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E ∞)
_i : IsEmpty α
⊢ 0 ≤ ‖f‖

Tactic:
lp.norm_eq_ciSup

Goal state after:
case inr.inl.inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E ∞)
_i : IsEmpty α
⊢ 0 ≤ ⨆ i, ‖↑f i‖
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
⊢ ‖f‖ = 0 ↔ f = 0

Tactic:
by
  refine ⟨fun h => ?_, by rintro rfl; exact norm_zero⟩
  rcases p.trichotomy with (rfl | rfl | hp)
  · ext i
    have : { i : α | ¬f i = 0 } = ∅ := by simpa [lp.norm_eq_card_dsupport f] using! h
    have : ¬¬f i = 0 := Set.eq_empty_iff_forall_notMem.mp this i
    tauto
  · rcases isEmpty_or_nonempty α with _i | _i
    · simp [eq_iff_true_of_subsingleton]
    have H : IsLUB (Set.range fun i => ‖f i‖) 0 := by simpa [h] using! lp.isLUB_norm f
    ext i
    have : ‖f i‖ = 0 := le_antisymm (H.1 ⟨i, rfl⟩) (norm_nonneg _)
    simpa using! this
  · have hf : HasSum (fun i : α => ‖f i‖ ^ p.toReal) 0 := by
      have := lp.hasSum_norm hp f
      rwa [h, Real.zero_rpow hp.ne'] at this
    have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i ↦ by positivity
    rw [hasSum_zero_iff_of_nonneg this] at hf
    ext i
    have : f i = 0 ∧ p.toReal ≠ 0 := by
      simpa [Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg (f i))] using! congr_fun hf i
    exact this.1

Goal state after:
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: case right
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m < N
h'm : ↑(m + 1) ≤ N
A : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x
n : α
x : E
⊢ ‖(continuousMultilinearCurryLeftEquiv 𝕜 (fun x ↦ E) F) (iteratedFDeriv 𝕜 (m + 1) (f n) x)‖ ≤ v (m + 1) n

Tactic:
LinearIsometryEquiv.norm_map

Goal state after:
case right
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m < N
h'm : ↑(m + 1) ≤ N
A : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x
n : α
x : E
⊢ ‖iteratedFDeriv 𝕜 (m + 1) (f n) x‖ ≤ v (m + 1) n
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
hr : 0 < r
⊢ ball ((IsometryEquiv.vaddConst c) 0) ‖r‖ = ball c r

Tactic:
simp [abs_of_pos hr]

Goal state after: