mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f : α →ᵇ β
l : Filter (α →ᵇ β)
⊢ l ≤ 𝓝 f ↔ Tendsto (⇑UniformFun.ofFun ∘ DFunLike.coe) l (𝓝 ((⇑UniformFun.ofFun ∘ DFunLike.coe) f))

Tactic:
← tendsto_id',

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f : α →ᵇ β
l : Filter (α →ᵇ β)
⊢ Tendsto id l (𝓝 f) ↔ Tendsto (⇑UniformFun.ofFun ∘ DFunLike.coe) l (𝓝 ((⇑UniformFun.ofFun ∘ DFunLike.coe) f))
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
hF : F.FG
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
h : ∀ (x : E), x ∈ F
this : ⊤ = F
⊢ FiniteDimensional 𝕜 E

Tactic:
← this

Goal state after:
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
hF : ⊤.FG
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
h : ∀ (x : E), x ∈ F
this : ⊤ = F
⊢ FiniteDimensional 𝕜 E
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : CompactSpace α
f : α →ᵇ β
M : ℝ
M0 : 0 < M
⊢ ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M

Tactic:
simp_rw [norm_def, ← dist_zero_right]
  exact dist_lt_iff_of_compact M0

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 β
inst✝ : SeminormedAddCommGroup γ
g : C(γ, α)
⊢ ‖compContinuousCLM β 𝕜 g‖ ≤ 1

Tactic:
refine (compContinuousCLM β 𝕜 g).opNorm_le_bound zero_le_one (fun x ↦ ?_)
  simpa using norm_compContinuous_le x g

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
x y : E
hxy : 1 ≤ ‖x - y‖
⊢ 1 ≤ ‖-(y - x)‖

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
this : Std.Symm fun x y ↦ 1 ≤ ‖x - y‖
⊢ ∃ f, (∀ (n : ℕ), ‖f n‖ ≤ R) ∧ Pairwise fun m n ↦ 1 ≤ ‖f m - f n‖

Tactic:
apply
    exists_seq_of_forall_finset_exists' (fun x : E => ‖x‖ ≤ R) fun (x : E) (y : E) => 1 ≤ ‖x - y‖

Goal state after:
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
this : Std.Symm fun x y ↦ 1 ≤ ‖x - y‖
⊢ ∀ (s : Finset E), (∀ x ∈ s, ‖x‖ ≤ R) → ∃ y, ‖y‖ ≤ R ∧ ∀ x ∈ s, 1 ≤ ‖x - y‖
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
this : Std.Symm fun x y ↦ 1 ≤ ‖x - y‖
⊢ ∀ (s : Finset E), (∀ x ∈ s, ‖x‖ ≤ R) → ∃ y, ‖y‖ ≤ R ∧ ∀ x ∈ s, 1 ≤ ‖x - y‖

Tactic:
rintro s -

Goal state after:
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
this : Std.Symm fun x y ↦ 1 ≤ ‖x - y‖
s : Finset E
⊢ ∃ y, ‖y‖ ≤ R ∧ ∀ x ∈ s, 1 ≤ ‖x - y‖
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
⊢ f * -g = 0

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
hx : x ∈ s
hs : s ≠ univ
⊢ ∃ y ∈ frontier s, infDist x sᶜ = dist x y

Tactic:
rcases Metric.exists_mem_closure_infDist_eq_dist (nonempty_compl.2 hs) x with ⟨y, hys, hyd⟩
  rw [closure_compl] at hys
  refine ⟨y, ⟨Metric.closedBall_infDist_compl_subset_closure hx <|
    Metric.mem_closedBall.2 <| ge_of_eq ?_, hys⟩, hyd⟩
  rwa [dist_comm]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : f * g = 0
⊢ f * -g = 0

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0,
    (cthickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
filter_upwards [eventually_notMem_cthickening_of_infEDist_pos x_mem_closure] with δ hδ
    simp only [hδ, not_false_eq_true, mulIndicator_of_notMem, x_mem_closure]

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
this : rightLim f ⊤ = f ⊤
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 (f ⊤))

Tactic:
← this

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
this : rightLim f ⊤ = f ⊤
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 (rightLim f ⊤))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
⊢ f x ≤ leftLim f y

Tactic:
let : TopologicalSpace α := Preorder.topology α
  have : OrderTopology α := ⟨rfl⟩
  rcases eq_or_neBot (𝓝[<] y) with h' | h'
  · rw [leftLim_eq_of_eq_bot _ h']
    exact hf h.le
  rw [leftLim_eq_sSup hf]
  refine le_csSup ⟨f y, ?_⟩ (mem_image_of_mem _ h)
  simp only [upperBounds, mem_image, mem_Iio, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂, mem_ofPred_eq]
  intro z hz
  exact hf hz.le

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, μ (s i) ≠ ∞
⊢ ∀ i ∈ t, IntegrableOn (fun x ↦ 1) (s i) μ

Tactic:
intro i hi

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, μ (s i) ≠ ∞
i : ι
hi : i ∈ t
⊢ IntegrableOn (fun x ↦ 1) (s i) μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
this : ∫ (x : X) in t, AEStronglyMeasurable.mk f hf x ∂μ = 0
⊢ ∫ (x : X) in t, f x ∂μ = 0

Tactic:
← this

Goal state after:
case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
this : ∫ (x : X) in t, AEStronglyMeasurable.mk f hf x ∂μ = 0
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in t, AEStronglyMeasurable.mk f hf x ∂μ
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ μ K ≤ μ.outerMeasure ↑K

Tactic:
Content.outerMeasure,

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ μ K ≤ (inducedOuterMeasure (fun U hU ↦ μ.innerContent { carrier := U, is_open' := hU }) ⋯ ⋯) ↑K
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ofReal ↑δ ≤ ↑δ

Tactic:
by simp only [ENNReal.ofReal_coe_nnreal, le_rfl]

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
h : IsFundamentalDomain G s μ
t : Set α
⊢ μ t = ∑' (g : G), μ (g • t ∩ s)

Tactic:
by
  simpa only [setLIntegral_one] using h.setLIntegral_eq_tsum' (fun _ => 1) t

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

Goal state before:
pp: R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : SecondCountableTopology R
⊢ borel R ≤ f.outer.caratheodory

Tactic:
by
  rw [borel_eq_generateFrom_Ioi]
  refine MeasurableSpace.generateFrom_le ?_
  simp +contextual [f.measurableSet_Ioi]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
C : ℝ
hs : (μ.restrict s) univ < ∞
hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C
this : IsFiniteMeasure (μ.restrict s)
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * μ.real s

Tactic:
simpa using norm_integral_le_of_norm_le_const hC

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

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a b : R
⊢ f.measure (Ico a b) = ofReal (leftLim (↑f) b - leftLim (↑f) a)

Tactic:
rcases le_or_gt b a with (hab | hab)
  · simp only [hab, measure_empty, Ico_eq_empty, not_lt]
    symm
    simp [ENNReal.ofReal_eq_zero, f.mono.leftLim hab]
  · have A : Disjoint {a} (Ioo a b) := by simp
    simp [← Icc_union_Ioo_eq_Ico le_rfl hab, -singleton_union, f.mono.leftLim_le,
      measure_union A measurableSet_Ioo, f.mono.le_leftLim hab, ← ENNReal.ofReal_add]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝¹ : PartialOrder X
x y : X
inst✝ : NullSingletonClass μ
⊢ ∫ (t : X) in Ico x y, f t ∂μ = ∫ (t : X) in Ioc x y, f t ∂μ

Tactic:
by
  rw [integral_Ico_eq_integral_Ioo, integral_Ioc_eq_integral_Ioo]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : (interior V).Nonempty
⊢ index ((fun h ↦ g * h) '' K) V = index K V

Tactic:
by
  refine le_antisymm (mul_left_index_le hK hV g) ?_
  convert! mul_left_index_le (hK.image <| continuous_const_mul g) hV g⁻¹
  rw [image_image]
  simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ 0 < index (↑K₀) U

Tactic:
· exact index_pos K₀ hU

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
h1K : IsCompact K
h2K : (interior K).Nonempty
⊢ 0 < index (↑K₀) U

Tactic:
· exact index_pos K₀ hU

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫ =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + normSq y

Tactic:
← normSq

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + normSq y =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + normSq y
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₄ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
exact nonneg_le_nonneg_of_sq_le_sq (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
cd : Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₄ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
exact nonneg_le_nonneg_of_sq_le_sq (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ * ‖x‖ = ‖y‖ * ‖y‖

Tactic:
simp only [← @inner_self_eq_norm_mul_norm ℝ, inner_add_left, inner_sub_right, real_inner_comm y x,
    sub_eq_zero, re_to_real]

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, x⟫_ℝ + ⟪y, x⟫_ℝ = ⟪y, x⟫_ℝ + ⟪y, y⟫_ℝ ↔ ⟪x, x⟫_ℝ = ⟪y, y⟫_ℝ
Module: Mathlib.Analysis.InnerProductSpace.Continuous

Goal state before:
pp: E : Type u_4
ι : Type u_6
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
f : ι → E
hf : DenseRange f
h : ∀ (i : ι), ⟪x, f i⟫ = ⟪y, f i⟫
⊢ ∀ v ∈ Set.range f, ⟪x, v⟫ = ⟪y, v⟫

Tactic:
simpa

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

Goal state before:
pp: E : Type u_4
ι : Type u_6
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
f : ι → E
hf : DenseRange f
h : ∀ (i : ι), ⟪f i, x⟫ = ⟪f i, y⟫
⊢ ∀ v ∈ Set.range f, ⟪v, x⟫ = ⟪v, y⟫

Tactic:
simpa

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

Goal state before:
pp: E : Type u_4
ι : Type u_6
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
f : ι → E
hf : DenseRange f
h : ∀ (i : ι), ⟪x, f i⟫ = 0
⊢ ∀ v ∈ Set.range f, ⟪x, v⟫ = 0

Tactic:
simpa

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

Goal state before:
pp: E : Type u_4
ι : Type u_6
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
f : ι → E
hf : DenseRange f
h : ∀ (i : ι), ⟪f i, x⟫ = 0
⊢ ∀ v ∈ Set.range f, ⟪v, x⟫ = 0

Tactic:
simpa

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
tfae_1_to_2 : ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x
⊢ [‖⟪x, y⟫‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x, x = 0 ∨ ∃ r, y = r • x, x = 0 ∨ y ∈ 𝕜 ∙ x].TFAE

Tactic:
tfae_have 2 → 3 := fun h => h.imp_right fun h' => ⟨_, h'⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
tfae_1_to_2 : ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x
tfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x → x = 0 ∨ ∃ r, y = r • x
⊢ [‖⟪x, y⟫‖ = ‖x‖ * ‖y‖, x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x, x = 0 ∨ ∃ r, y = r • x, x = 0 ∨ y ∈ 𝕜 ∙ x].TFAE
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
l₁ l₂ : (i : ι) → G i
s : Finset ι
⊢ ∑ j ∈ s, ∑ i ∈ s, ⟪(V i) (l₁ i), (V j) (l₂ j)⟫ = ∑ j ∈ s, ∑ i ∈ s, if i = j then ⟪(V i) (l₁ i), (V j) (l₂ j)⟫ else 0

Tactic:
by
      congr with i
      congr with j
      apply hV.eq_ite

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

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
ha : a ≠ 0
hb : b ≠ 0
h : ((rankOne 𝕜) a) b = ((rankOne 𝕜) c) d
h₂ : ((rankOne 𝕜) b) a = ((rankOne 𝕜) d) c
⊢ ∃ α β, ∃ (_ : α ≠ 0) (_ : 0 < β), a = α • c ∧ b = (α * β) • d

Tactic:
simp only [ContinuousLinearMap.ext_iff, rankOne_apply] at h h₂

Goal state after:
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c
h₂ : ∀ (x : F), ⟪a, x⟫_𝕜 • b = ⟪c, x⟫_𝕜 • d
⊢ ∃ α β, ∃ (_ : α ≠ 0) (_ : 0 < β), a = α • c ∧ b = (α * β) • d
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : X
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : X
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedEBall y r ⊆ f n
⊢ min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)

Tactic:
show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)

Goal state after:
case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : X
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : X
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedEBall y r ⊆ f n
⊢ min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c
h₂ : ∀ (x : F), ⟪a, x⟫_𝕜 • b = ⟪c, x⟫_𝕜 • d
h₃ : a = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜) • c
h₄ : b = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜 * (⟪c, c⟫_𝕜 / ⟪a, a⟫_𝕜)) • d
h₅ : ⟪d, b⟫_𝕜 ≠ 0
h₆ : c ≠ 0
⊢ ⟪b, b⟫_𝕜 ≠ 0

Tactic:
simpa

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

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (∀ ε > 0,
      ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε) →
    ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε

pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε) →
    ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε

Tactic:
· intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine (abs_sub _ _).trans_lt ?_
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i ∈ s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i ∈ s₂ \ s₁, ‖f i‖ ^ 2) < √ε ^ 2 := by
        rw [← hV.norm_sq_sdiff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      have hη := sq_sqrt (le_of_lt hε)
      linarith

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε) →
    ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε