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

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹¹ : NonUnitalNormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedSpace ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : ContinuousStar A
inst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, quasispectrum ℝ≥0 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, 0 ≤ a x
ha₀ : quasispectrum ℝ≥0 a₀ ⊆ s
ha₀' : 0 ≤ a₀
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ Tendsto (fun x ↦ cfcₙ f (a x)) l (𝓝 (cfcₙ f a₀))

Tactic:
by
  apply continuousOn_cfcₙ_nnreal A hs f |>.continuousWithinAt ⟨ha₀', ha₀⟩ |>.tendsto.comp
  rw [tendsto_nhdsWithin_iff]
  exact ⟨ha_tendsto, ha'.and ha⟩

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : (fun x ↦ (ofMatrix.symm x.1, ofMatrix.symm x.2)) = id
⊢ 𝓤 (CStarMatrix m n A) = 𝓤 (CStarMatrix m n A)

Tactic:
rw [← uniformInducing_toMatrixAux.comap_uniformity, this, Filter.comap_id]

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : (fun x ↦ (ofMatrix.symm x.1, ofMatrix.symm x.2)) = id
⊢ 𝓤 (Matrix m n A) = 𝓤 (CStarMatrix m n A)
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case refine_1
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ cobounded (CStarMatrix m n A) ≤ Filter.comap (⇑ofMatrix.symm) (cobounded (Matrix m n A))

pp: case refine_2
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ Filter.comap (⇑ofMatrix.symm) (cobounded (Matrix m n A)) ≤ cobounded (CStarMatrix m n A)

Tactic:
· exact antilipschitzWith_toMatrixAux.tendsto_cobounded.le_comap

Goal state after:
case refine_2
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ Filter.comap (⇑ofMatrix.symm) (cobounded (Matrix m n A)) ≤ cobounded (CStarMatrix m n A)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range

Goal state before:
pp: case mp
A : Type u_1
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 : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
x✝ : A
⊢ (x✝ ∈ Set.range fun x ↦ cfcₙ x a) → x✝ ∈ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x}

pp: case mpr
A : Type u_1
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 : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
x✝ : A
⊢ x✝ ∈ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x} → x✝ ∈ Set.range fun x ↦ cfcₙ x a

Tactic:
· rintro ⟨f, rfl⟩
    simp only [cfcₙ_nnreal_eq_real f a]
    exact ⟨_, fun _ _ ↦ by positivity, rfl⟩

Goal state after:
case mpr
A : Type u_1
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 : IsTopologicalRing A
inst✝⁴ : T2Space A
inst✝³ : PartialOrder A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : StarOrderedRing A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
x✝ : A
⊢ x✝ ∈ (fun x ↦ cfcₙ x a) '' {f | ∀ x ∈ quasispectrum ℝ a, 0 ≤ f x} → x✝ ∈ Set.range fun x ↦ cfcₙ x a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary

Goal state before:
pp: A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal
u : A
⊢ u ∈ unitary A ↔ IsStarNormal u ∧ spectrum ℂ u ⊆ ↑(unitary ℂ)

Tactic:
rw [← and_iff_right_of_imp isStarNormal_of_mem_unitary]
  refine and_congr_right fun hu ↦ ?_
  nth_rw 1 [← cfc_id ℂ u]
  rw [cfc_unitary_iff id u, Set.subset_def]
  simp only [id_eq, RCLike.star_def, SetLike.mem_coe, Unitary.mem_iff_star_mul_self]

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

Goal state before:
pp: case refine_1
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
he : IsStarProjection e
a : A
ha : 0 ≤ a
ha1 : ‖a‖ ≤ 1
b : A
hb : 0 ≤ b
hb1 : ‖b‖ ≤ 1
x✝ : e ∈ openSegment ℝ a b
t s : ℝ
h0t : 0 < t
h0s : 0 < s
hts : t + s = 1
hlin : t • a + s • b = e
⊢ a = e

Tactic:
have ⟨ha01, hb01⟩ : (a : A⁺¹) ∈ Icc 0 1 ∧ (b : A⁺¹) ∈ Icc 0 1 := by
      grind [inr_mem_Icc_iff_norm_le]

Goal state after:
case refine_1
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
he : IsStarProjection e
a : A
ha : 0 ≤ a
ha1 : ‖a‖ ≤ 1
b : A
hb : 0 ≤ b
hb1 : ‖b‖ ≤ 1
x✝ : e ∈ openSegment ℝ a b
t s : ℝ
h0t : 0 < t
h0s : 0 < s
hts : t + s = 1
hlin : t • a + s • b = e
ha01 : ↑a ∈ Icc 0 1
hb01 : ↑b ∈ Icc 0 1
⊢ a = e
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e : Pretrivialization F proj
x : Z
ex : x ∈ e.source
⊢ ↑e.symm (proj x, (↑e x).2) = x

Tactic:
by
  rw [← e.coe_fst ex, ← e.coe_coe, e.left_inv ex]

Goal state after:
Module: Mathlib.Topology.IsLocalHomeomorph

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
h : ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.domRestrict f)
x : X
hx : x ∈ s
U : Set X
hU : U ∈ 𝓝 x
emb : IsOpenEmbedding (U.domRestrict f)
this : IsOpenEmbedding ((interior U).domRestrict f)
⊢ ∃ e, x ∈ e.source ∧ f = ↑e

Tactic:
obtain ⟨cont, inj, openMap⟩ := isOpenEmbedding_iff_continuous_injective_isOpenMap.mp this

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
h : ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.domRestrict f)
x : X
hx : x ∈ s
U : Set X
hU : U ∈ 𝓝 x
emb : IsOpenEmbedding (U.domRestrict f)
this : IsOpenEmbedding ((interior U).domRestrict f)
cont : Continuous[instTopologicalSpaceSubtype, inst✝] ((interior U).domRestrict f)
inj : Function.Injective ((interior U).domRestrict f)
openMap : IsOpenMap ((interior U).domRestrict f)
⊢ ∃ e, x ∈ e.source ∧ f = ↑e
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case refine_2.refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
y : A
hy : y ∈ Metric.closedBall 0 1
⊢ (fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) y ≤ ‖a‖₊ * ‖a‖₊

Tactic:
exact key (star x) y ((nnnorm_star x).trans_le hx') (mem_closedBall_zero_iff.1 hy)

Goal state after:
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_1.refine_2
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
preim_im : f ⁻¹' f '' U = ⋃ g, (fun x ↦ g • x) ⁻¹' U
this : Nonempty (X → E)
g : G
W : Set X
hWU : W ⊆ f '' U
isOpen : IsOpen[inst✝⁶] (f ⁻¹' W ∩ (fun x ↦ g • x) ⁻¹' U)
e : E
hW : e ∈ f ⁻¹' W
g' : G
he' : (fun m ↦ m • e) g' ∈ U
hfe : f ((fun m ↦ m • e) g') = f e
⊢ (fun x ↦ (g⁻¹ * g') • x) e ∈ (fun x ↦ g • x) ⁻¹' U

Tactic:
apply Set.mem_of_eq_of_mem _ he'; simp_rw [mul_smul, smul_inv_smul]

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: case refine_3
E : Type u_1
X : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace X
f : E → X
s : Set X
inst✝³ : Nonempty (X → E)
ι : Type ?u.20
inst✝² : Nonempty ι
inst✝¹ : TopologicalSpace ι
inst✝ : DiscreteTopology ι
U : ι → Set E
V : Set X
open_V : IsOpen[inst✝⁴] V
open_iff : ∀ (i : ι) {W : Set X}, W ⊆ V → (IsOpen[inst✝⁴] W ↔ IsOpen[inst✝⁵] (f ⁻¹' W ∩ U i))
inj : ∀ (i : ι), Set.InjOn f (U i)
disjoint : Pairwise (Disjoint on U)
exhaustive' : f ⁻¹' V ⊆ ⋃ i, U i
idx : (x : E) → x ∈ f ⁻¹' V → ι
idx_U : ∀ (x : E) (a : x ∈ f ⁻¹' V), x ∈ U (idx x a)
inv : ι → ⦃a : X⦄ → a ∈ V → E
inv_U : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), inv i a_1 ∈ U i
f_inv : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), f (inv i a_1) = a
F : PartialEquiv E (X × ι) :=
  { toFun := fun e ↦ (f e, if he : f e ∈ V then idx e he else Classical.arbitrary ι),
    invFun := fun x ↦ if hx : x.1 ∈ V then inv x.2 hx else Classical.arbitrary (X → E) x.1, source := f ⁻¹' V,
    target := V ×ˢ Set.univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }
open_preim : ∀ {W : Set X}, W ⊆ V → IsOpen[inst✝⁴] W → IsOpen[inst✝⁵] (f ⁻¹' W)
open_source : IsOpen[inst✝⁵] F.source
cont_f : ContinuousOn f F.source
i : ι
W : Set E
open_W : IsOpen[inst✝⁵] W
⊢ IsOpen[inst✝⁴] (V ∩ (fun x ↦ if h : x ∈ V then inv i ⋯ else Classical.arbitrary (X → E) x) ⁻¹' W)

Tactic:
open_iff i Set.inter_subset_left

Goal state after:
case refine_3
E : Type u_1
X : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace X
f : E → X
s : Set X
inst✝³ : Nonempty (X → E)
ι : Type ?u.20
inst✝² : Nonempty ι
inst✝¹ : TopologicalSpace ι
inst✝ : DiscreteTopology ι
U : ι → Set E
V : Set X
open_V : IsOpen[inst✝⁴] V
open_iff : ∀ (i : ι) {W : Set X}, W ⊆ V → (IsOpen[inst✝⁴] W ↔ IsOpen[inst✝⁵] (f ⁻¹' W ∩ U i))
inj : ∀ (i : ι), Set.InjOn f (U i)
disjoint : Pairwise (Disjoint on U)
exhaustive' : f ⁻¹' V ⊆ ⋃ i, U i
idx : (x : E) → x ∈ f ⁻¹' V → ι
idx_U : ∀ (x : E) (a : x ∈ f ⁻¹' V), x ∈ U (idx x a)
inv : ι → ⦃a : X⦄ → a ∈ V → E
inv_U : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), inv i a_1 ∈ U i
f_inv : ∀ (i : ι) ⦃a : X⦄ (a_1 : a ∈ V), f (inv i a_1) = a
F : PartialEquiv E (X × ι) :=
  { toFun := fun e ↦ (f e, if he : f e ∈ V then idx e he else Classical.arbitrary ι),
    invFun := fun x ↦ if hx : x.1 ∈ V then inv x.2 hx else Classical.arbitrary (X → E) x.1, source := f ⁻¹' V,
    target := V ×ˢ Set.univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }
open_preim : ∀ {W : Set X}, W ⊆ V → IsOpen[inst✝⁴] W → IsOpen[inst✝⁵] (f ⁻¹' W)
open_source : IsOpen[inst✝⁵] F.source
cont_f : ContinuousOn f F.source
i : ι
W : Set E
open_W : IsOpen[inst✝⁵] W
⊢ IsOpen[inst✝⁵] (f ⁻¹' (V ∩ (fun x ↦ if h : x ∈ V then inv i ⋯ else Classical.arbitrary (X → E) x) ⁻¹' W) ∩ U i)
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
h : x ≠ y
hdisj : Disjoint (Ioc (↑x).arg (x.angleDiff y + (↑x).arg)) (Ioc (↑y).arg (y.angleDiff x + (↑y).arg))
⊢ Disjoint (⇑(x.path y) '' Ioc 0 1) (⇑(y.path x) '' Ioc 0 1)

Tactic:
rw [path_image_Ioc_of_ne h, path_image_Ioc_of_ne h.symm]

Goal state after:
x y : Circle
h : x ≠ y
hdisj : Disjoint (Ioc (↑x).arg (x.angleDiff y + (↑x).arg)) (Ioc (↑y).arg (y.angleDiff x + (↑y).arg))
⊢ Disjoint (⇑exp '' Ioc (↑x).arg (x.angleDiff y + (↑x).arg)) (⇑exp '' Ioc (↑y).arg (y.angleDiff x + (↑y).arg))
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: case inl
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝ : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝ : IsEmpty ↑(f ⁻¹' {x})
⊢ IsEvenlyCovered f x ↑(f ⁻¹' {x})

Tactic:
exact .of_preimage_eq_empty _ hWx (by simpa using hWV)

Goal state after:
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 : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪(adjoint ↑(toContinuousLinearMap A)) y, x⟫_𝕜 = ⟪y, ↑(toContinuousLinearMap A) x⟫_𝕜

Tactic:
adjoint_eq_toCLM_adjoint

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 : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪↑(ContinuousLinearMap.adjoint (toContinuousLinearMap ↑(toContinuousLinearMap A))) y, x⟫_𝕜 =
    ⟪y, ↑(toContinuousLinearMap A) x⟫_𝕜
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 : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪x, (adjoint ↑(toContinuousLinearMap A)) y⟫_𝕜 = ⟪↑(toContinuousLinearMap A) x, y⟫_𝕜

Tactic:
adjoint_eq_toCLM_adjoint

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 : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
A : E →ₗ[𝕜] F
x : E
y : F
this✝ : CompleteSpace E
this : CompleteSpace F
⊢ ⟪x, ↑(ContinuousLinearMap.adjoint (toContinuousLinearMap ↑(toContinuousLinearMap A))) y⟫_𝕜 =
    ⟪↑(toContinuousLinearMap A) x, y⟫_𝕜
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
inst✝ : CharZero k
n : ℕ
s : Simplex k P n
fs₁ fs₂ : Finset (Fin (n + 1))
m₁ m₂ : ℕ
h₁ : #fs₁ = m₁ + 1
h₂ : #fs₂ = m₂ + 1
h : (affineCombination k univ s.points) (centroidWeightsIndicator k fs₁) = Finset.centroid k fs₂ s.points
⊢ fs₁ = fs₂

Tactic:
Finset.centroid_eq_affineCombination_fintype

Goal state after:
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
inst✝ : CharZero k
n : ℕ
s : Simplex k P n
fs₁ fs₂ : Finset (Fin (n + 1))
m₁ m₂ : ℕ
h₁ : #fs₁ = m₁ + 1
h₂ : #fs₂ = m₂ + 1
h :
  (affineCombination k univ s.points) (centroidWeightsIndicator k fs₁) =
    (affineCombination k univ s.points) (centroidWeightsIndicator k fs₂)
⊢ fs₁ = fs₂
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✝ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
⊢ #{i}ᶜ = n - 1 + 1

Tactic:
by simp [card_compl, NeZero.one_le]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
x✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f
y : 𝕜
h : y • 1 = ↑(↑g).symm ∘SL ↑↑f
hy : y ≠ 0
hfg : ↑↑f = y • ↑↑g
hgf : ↑↑g = star y • ↑↑f
this : ↑↑g = 0
x : H
⊢ f x = (1 • g) x

Tactic:
by simp [by simpa using congr($this x)]

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

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : DivisionRing k
inst✝² : Module k V
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional k V
h : Fintype.card ι = Module.finrank k V + 1
⊢ Nonempty (AffineBasis ι k P)

Tactic:
by
  obtain ⟨s, b, hb⟩ := AffineBasis.exists_affineBasis k V P
  lift s to Finset P using b.finite_set
  refine ⟨b.reindex <| Fintype.equivOfCardEq ?_⟩
  rw [h, ← b.card_eq_finrank_add_one]

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Tactic:
rcases p.dichotomy with (rfl | hp')
        · cases isEmpty_or_nonempty α
          · simp only [lp.eq_zero' f, zero_add, norm_zero, le_refl]
          refine (lp.isLUB_norm (f + g)).2 ?_
          rintro x ⟨i, rfl⟩
          refine le_trans ?_ (add_mem_upperBounds_add
            (lp.isLUB_norm f).1 (lp.isLUB_norm g).1 ⟨_, ⟨i, rfl⟩, _, ⟨i, rfl⟩, rfl⟩)
          exact norm_add_le (f i) (g i)
        · have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
          have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
          have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
          have hf₂ := lp.hasSum_norm hp'' f
          have hg₂ := lp.hasSum_norm hp'' g
          -- apply Minkowski's inequality
          obtain ⟨C, hC₁, hC₂, hCfg⟩ :=
            Real.Lp_add_le_hasSum_of_nonneg hp' hf₁ hg₁ (norm_nonneg' _) (norm_nonneg' _) hf₂ hg₂
          refine le_trans ?_ hC₂
          rw [← Real.rpow_le_rpow_iff (norm_nonneg' (f + g)) hC₁ hp'']
          refine hasSum_le ?_ (lp.hasSum_norm hp'' (f + g)) hCfg
          intro i
          gcongr
          apply norm_add_le

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

Goal state before:
pp: 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
k : ℕ
IH : ↑k ≤ N → (iteratedFDeriv 𝕜 k fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
hk : ↑(k + 1) ≤ N
h'k : ↑k < N
A : Summable fun n ↦ iteratedFDeriv 𝕜 k (f n) 0
⊢ ∀ (n : α) (x : E), ‖fderiv 𝕜 (fun x ↦ iteratedFDeriv 𝕜 k (f n) x) x‖ ≤ v (k + 1) n

Tactic:
intro n x

Goal state after:
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
k : ℕ
IH : ↑k ≤ N → (iteratedFDeriv 𝕜 k fun y ↦ ∑' (n : α), f n y) = fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
hk : ↑(k + 1) ≤ N
h'k : ↑k < N
A : Summable fun n ↦ iteratedFDeriv 𝕜 k (f n) 0
n : α
x : E
⊢ ‖fderiv 𝕜 (fun x ↦ iteratedFDeriv 𝕜 k (f n) x) x‖ ≤ v (k + 1) n
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)
F : α → Type u_5
inst✝ : (i : α) → NormedAddCommGroup (F i)
p : ℝ≥0∞
hp✝ : p ≠ 0
x : ↥(lp E p)
y : ↥(lp F p)
h : ∀ (i : α), ‖↑x i‖ ≤ ‖↑y i‖
hp : 0 < p.toReal
s : Finset α
⊢ ∑ i ∈ s, ‖↑x i‖ ^ p.toReal ≤ ∑ i ∈ s, ‖↑y i‖ ^ p.toReal

Tactic:
gcongr with i _; exact h i

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

Goal state before:
pp: case inr.inr
α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
F : α → Type u_5
inst✝ : (i : α) → NormedAddCommGroup (F i)
p : ℝ≥0∞
hp✝ : p ≠ 0
x : ↥(lp E p)
y : ↥(lp F p)
h : ∀ (i : α), ‖↑x i‖ ≤ ‖↑y i‖
hp : 0 < p.toReal
⊢ ‖x‖ ≤ ‖y‖

Tactic:
exact norm_le_of_forall_sum_le hp (norm_nonneg' _) fun s ↦ calc
      ∑ i ∈ s, ‖x i‖ ^ p.toReal
      _ ≤ ∑ i ∈ s, ‖y i‖ ^ p.toReal := by gcongr with i _; exact h i
      _ ≤ ‖y‖ ^ p.toReal := sum_rpow_le_norm_rpow hp y s

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

Goal state before:
pp: case left.refine_2
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
⊢ ∀ (n : α) (x : E), ‖iteratedFDeriv 𝕜 m (f n) x‖ ≤ v m n

Tactic:
intro n x

Goal state after:
case left.refine_2
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
n : α
x : E
⊢ ‖iteratedFDeriv 𝕜 m (f n) x‖ ≤ v m n
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝¹ : l.NeBot
_i : Fact (1 ≤ p)
inst✝ : ∀ (a : α), CompleteSpace (E a)
⊢ ∀ (u : ℕ → ↥(lp E p)), CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)

Tactic:
intro F hF

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝¹ : l.NeBot
_i : Fact (1 ≤ p)
inst✝ : ∀ (a : α), CompleteSpace (E a)
F : ℕ → ↥(lp E p)
hF : CauchySeq F
⊢ ∃ a, Tendsto F atTop (𝓝 a)