mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f n x) atTop (𝓝 (-F x))

Tactic:
filter_upwards [h_tendsto] with x hx using hx.neg

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
have h_bound : ∀ᵐ a ∂μ, ∀ i, f' i a ≤ F' a := by
    filter_upwards [hf_bound] with a ha_bound i
    refine ENNReal.ofReal_le_ofReal ?_
    simp only [tsub_le_iff_right, sub_add_cancel, ha_bound i]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f' i a ≤ F' a
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
⊢ ∃ x₀, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖

Tactic:
have x0 : x ≠ 0 := fun H => by simp [H] at xF

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
⊢ ∃ x₀, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ y' ∈ F

Tactic:
simp [y', Submodule.smul_mem _ _ hy]

Goal state after:
Module: Mathlib.Topology.Algebra.Indicator

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
s : Set α
inst✝ : One β
hs : ∀ a ∈ frontier s, f a = 1
hf : Continuous[inst✝², inst✝¹] f
⊢ Continuous[inst✝², inst✝¹] (s.mulIndicator f)

Tactic:
by
  classical exact hf.piecewise hs continuous_const

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝² : PseudoMetricSpace R
inst✝¹ : Mul R
inst✝ : BoundedMul R
f g : X → R
f_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C
g_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C
⊢ ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C

Tactic:
obtain ⟨C, hC⟩ := Metric.isBounded_iff.mp <|
    isBounded_mul (Metric.isBounded_range_iff.mpr f_bdd) (Metric.isBounded_range_iff.mpr g_bdd)
  use C
  intro x y
  exact hC (Set.mul_mem_mul (Set.mem_range_self (f := f) x) (Set.mem_range_self (f := g) x))
           (Set.mul_mem_mul (Set.mem_range_self (f := f) y) (Set.mem_range_self (f := g) y))

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Tactic:
rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
    rwa [← integrable_neg_iff] at hf

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C

Tactic:
rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
  refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
    <;> [left; right]
    <;> apply mem_range_self

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

Goal state before:
pp: case neg
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
h : ¬Nonempty α
⊢ dist f g < C

Tactic:
convert! C0

Goal state after:
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
h : ¬Nonempty α
⊢ dist f g = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case hp_ne_zero
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ ENNReal.ofReal p ≠ 0

Tactic:
rw [Ne, ENNReal.ofReal_eq_zero, not_le]
        exact hpq.pos

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

Goal state before:
pp: case pos
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))

Tactic:
simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
    rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
        simp only [x_mem_closure, indicator_of_mem]]
    exact tendsto_const_nhds

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ₁ δ₂ : ℝ
δ₁_pos : 0 < δ₁
δ₂_pos : 0 < δ₂
hle : δ₁ ≤ δ₂
E : Set α
x : α
⊢ (thickenedIndicator δ₁_pos E) x ≤ (thickenedIndicator δ₂_pos E) x

Tactic:
apply (toNNReal_le_toNNReal (by finiteness) (by finiteness)).mpr

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ₁ δ₂ : ℝ
δ₁_pos : 0 < δ₁
δ₂_pos : 0 < δ₂
hle : δ₁ ≤ δ₂
E : Set α
x : α
⊢ thickenedIndicatorAux δ₁ E x ≤ thickenedIndicatorAux δ₂ E x
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
⊢ Tendsto ((fun n ↦ setToFun μ T hT (fs n)) ∘ x) atTop (𝓝 (setToFun μ T hT f))

Tactic:
have h :
    { x : ι | (fun n => AEStronglyMeasurable (fs n) μ) x } ∩
        { x : ι | (fun n => ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) x } ∈ l :=
    inter_mem hfs_meas h_bound

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
⊢ Tendsto ((fun n ↦ setToFun μ T hT (fs n)) ∘ x) atTop (𝓝 (setToFun μ T hT f))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: f : StieltjesFunction ℝ
x : ℝ
⊢ ⨅ r, ↑f ↑↑r = ⨅ r, ↑f ↑r

Tactic:
refine (Real.iInf_Ioi_eq_iInf_rat_gt _ ?_ f.mono).symm

Goal state after:
f : StieltjesFunction ℝ
x : ℝ
⊢ BddBelow (↑f '' Ioi x)
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 : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
B : ↑f a - ↑f a < ↑δ
⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'

Tactic:
have : (𝓝[>] a).NeBot := nhdsGT_neBot_of_exists_gt ⟨b, hab⟩

Goal state after:
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 : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
B : ↑f a - ↑f a < ↑δ
this : (𝓝[>] a).NeBot
⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
l u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
⊢ f.measure univ = ofReal (u - l)

Tactic:
refine tendsto_nhds_unique (tendsto_measure_Iic_atTop _) ?_

Goal state after:
R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
l u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
⊢ Tendsto (fun x ↦ f.measure (Iic x)) atTop (𝓝 (ofReal (u - l)))
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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x

Tactic:
simpa using hf

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
⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫

Tactic:
simp only [inner_sub_left, inner_sub_right]

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
⊢ ⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫) = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : x ≤ y
hf : ↑g y - ↑f x = ↑g y - ↑g x
⊢ ↑f x = ↑g x

Tactic:
simpa using hf

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

Goal state before:
pp: case inr
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
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ↑f x - ↑g y = ↑g x - ↑g y
⊢ ↑f x = ↑g x

Tactic:
simpa using hf

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

Goal state before:
pp: case neg
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y

Tactic:
convert! cauchy_schwarz_aux' (𝕜 := 𝕜) (⟪x, y⟫ • x) y (t / ‖⟪x, y⟫‖) using 3

Goal state after:
case e'_4.e'_5
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ normSq x * (t * t) = normSq (⟪x, y⟫ • x) * (t / ‖⟪x, y⟫‖) * (t / ‖⟪x, y⟫‖)

case e'_4.e'_6
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ 2 * ‖⟪x, y⟫‖ * t = 2 * re ⟪⟪x, y⟫ • x, y⟫ * (t / ‖⟪x, y⟫‖)
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
𝕝 : Type u_4
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : Module 𝕝 E
inst✝¹ : IsScalarTower 𝕝 𝕜 E
inst✝ : StarModule 𝕝 𝕜
x y : E
r : 𝕝
⊢ r • (starRingEnd 𝕜) ⟪y, x⟫ = r • ⟪x, y⟫

Tactic:
inner_conj_symm

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
𝕝 : Type u_4
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : Module 𝕝 E
inst✝¹ : IsScalarTower 𝕝 𝕜 E
inst✝ : StarModule 𝕝 𝕜
x y : E
r : 𝕝
⊢ r • ⟪x, y⟫ = r • ⟪x, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫

Tactic:
simp only [inner_sub_left, inner_sub_right]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫) = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ⟪x, y⟫ = 0
⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
by
  rw [@norm_add_mul_self 𝕜, add_right_cancel_iff, add_eq_left, mul_eq_zero]
  apply Or.inr
  simp only [h, zero_re]

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
ι₁ : Type u_4
s₁ : Finset ι₁
w₁ : ι₁ → ℝ
v₁ : ι₁ → F
h₁ : ∑ i ∈ s₁, w₁ i = 0
ι₂ : Type u_5
s₂ : Finset ι₂
w₂ : ι₂ → ℝ
v₂ : ι₂ → F
h₂ : ∑ i ∈ s₂, w₂ i = 0
⊢ ∑ x ∈ s₁, w₁ x * 0 + 0 - ∑ x ∈ s₁, w₁ x * ∑ i ∈ s₂, w₂ i * (‖v₁ x - v₂ i‖ * ‖v₁ x - v₂ i‖ / 2) =
    (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (‖v₁ i₁ - v₂ i₂‖ * ‖v₁ i₁ - v₂ i₂‖)) / 2

Tactic:
mul_zero,

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
ι₁ : Type u_4
s₁ : Finset ι₁
w₁ : ι₁ → ℝ
v₁ : ι₁ → F
h₁ : ∑ i ∈ s₁, w₁ i = 0
ι₂ : Type u_5
s₂ : Finset ι₂
w₂ : ι₂ → ℝ
v₂ : ι₂ → F
h₂ : ∑ i ∈ s₂, w₂ i = 0
⊢ ∑ x ∈ s₁, 0 + 0 - ∑ x ∈ s₁, w₁ x * ∑ i ∈ s₂, w₂ i * (‖v₁ x - v₂ i‖ * ‖v₁ x - v₂ i‖ / 2) =
    (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (‖v₁ i₁ - v₂ i₂‖ * ‖v₁ i₁ - v₂ i₂‖)) / 2
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: case mp
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
h : ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1
hx₀ : x ≠ 0
hy₀ : y ≠ 0
⊢ y = (‖y‖ / ‖x‖) • x

Tactic:
exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm

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
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
H₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2
H₂ : re ⟪y, x⟫ = ‖y‖ ^ 2
⊢ re ⟪x - y, x - y⟫ ≤ 0

Tactic:
simp only [inner_sub_left, inner_sub_right]

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
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
H₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2
H₂ : re ⟪y, x⟫ = ‖y‖ ^ 2
⊢ re (⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫)) ≤ 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
x : 𝕜
⊢ ‖x‖ ^ 2 = re (x * star x)

Tactic:
by rw [star_def, mul_conj, ← ofReal_pow, ofReal_re]

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

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
h : l i ≠ 0
⊢ ⟪(V i) v, (V i) (l i)⟫ = ⟪v, l i⟫

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
h : ¬l i ≠ 0
⊢ 0 = ⟪v, l i⟫

Tactic:
· simp only [LinearIsometry.inner_map_map]

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
h : ¬l i ≠ 0
⊢ 0 = ⟪v, l i⟫
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
S T : V →ₗ[ℂ] V
⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S - T = 0

Tactic:
← inner_map_self_eq_zero

Goal state after:
V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
S T : V →ₗ[ℂ] V
⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ ∀ (x : V), ⟪(S - T) x, x⟫_ℂ = 0
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
ι : Type u_4
ι' : Type u_5
ι'' : Type u_6
E' : Type u_7
inst✝³ : SeminormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
E'' : Type u_8
inst✝¹ : SeminormedAddCommGroup E''
inst✝ : InnerProductSpace 𝕜 E''
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
v' : Basis ι' 𝕜 E'
hv' : Orthonormal 𝕜 ⇑v'
e : ι ≃ ι'
⊢ Orthonormal 𝕜 (⇑(v.equiv v' e) ∘ ⇑v)

Tactic:
by
      have h : v.equiv v' e ∘ v = v' ∘ e := by
        ext i
        simp
      rw [h]
      exact hv'.comp _ e.injective

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
⊢ (starRingEnd 𝕜) ⟪x, T y⟫ = ⟪T y, x⟫

Tactic:
inner_conj_symm

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
⊢ ⟪T y, x⟫ = ⟪T y, x⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
T : ι → E →ₗ[𝕜] E
s : Finset ι
hT : ∀ i ∈ s, (T i).IsSymmetric
x✝¹ x✝ : E
⊢ ⟪(∑ i ∈ s, T i) x✝¹, x✝⟫ = ⟪x✝¹, (∑ i ∈ s, T i) x✝⟫

Tactic:
by
  simpa [sum_inner, inner_sum] using Finset.sum_congr rfl fun _ hi ↦ hT _ hi _ _

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

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
h : I = 0
⊢ ↑(re (↑(re ⟪T y, x⟫) + ↑(im ⟪T y, x⟫) * 0)) = ↑(re ⟪T y, x⟫) + ↑(im ⟪T y, x⟫) * 0

Tactic:
mul_zero,

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
h : I = 0
⊢ ↑(re (↑(re ⟪T y, x⟫) + 0)) = ↑(re ⟪T y, x⟫) + 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
u v : E
⊢ (𝕜 ∙ u).reflection v = 2 • (⟪u, v⟫ / ↑‖u‖ ^ 2) • u - v

Tactic:
reflection_apply,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
u v : E
⊢ 2 • (𝕜 ∙ u).starProjection v - v = 2 • (⟪u, v⟫ / ↑‖u‖ ^ 2) • u - v