mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ integral μ (piecewise s hs f 0) = integral (μ.restrict s) f

Tactic:
classical
  refine (integral_eq_sum_of_subset ?_).trans
      ((sum_congr rfl fun y hy => ?_).trans (integral_eq_sum_filter _ _).symm)
  · intro y hy
    simp only [mem_filter, mem_range, coe_piecewise, coe_zero, piecewise_eq_indicator,
      mem_range_indicator] at *
    rcases hy with ⟨⟨rfl, -⟩ | ⟨x, -, rfl⟩, h₀⟩
    exacts [(h₀ rfl).elim, ⟨Set.mem_range_self _, h₀⟩]
  · dsimp
    rw [Set.piecewise_eq_indicator, indicator_preimage_of_notMem,
      measureReal_restrict_apply (f.measurableSet_preimage _)]
    exact fun h₀ => (mem_filter.1 hy).2 (Eq.symm h₀)

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

Goal state before:
pp: case neg.hSp
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : ¬g (f a) = 0
h_left_eq : (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = (T (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)}))) (g (f a))
h_left_eq' :
  (T (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)}))) (g (f a)) =
    (T (⋃ y ∈ {b ∈ f.range | g b = g (f a)}, ⇑f ⁻¹' {y})) (g (f a))
⊢ ∀ i ∈ {b ∈ f.range | g b = g (f a)}, μ (⇑f ⁻¹' {i}) ≠ ∞

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case refine_2
α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ (x n).toLp ⋯
hx_tendsto : Tendsto (fun n ↦ eLpNorm (⇑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)
n : ℕ
⊢ ↑↑↑g =ᵐ[μ] ↑↑↑g

Tactic:
rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ↥(Lp E 1 μ)
⊢ integralCLM f = (setToL1 ⋯) f

Tactic:
rfl

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
f : α → E
hE : CompleteSpace E
⊢ (if hf : Integrable (fun a ↦ f a) μ then L1.integralCLM (Integrable.toL1 (fun a ↦ f a) hf) else 0) =
    if hf : Integrable f μ then (L1.setToL1 ⋯) (Integrable.toL1 f hf) else 0

Tactic:
rfl

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
f : β → α → G
s : Set β
b : β
⊢ ∫ (y : α), s.indicator (fun x ↦ f x y) b ∂μ = s.indicator (fun x ↦ ∫ (y : α), f x y ∂μ) b

Tactic:
by
  by_cases hb : b ∈ s <;> simp [hb]

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

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * μ.real s
f : α →ₛ F
⊢ ∑ x ∈ f.range, ‖T (⇑f ⁻¹' {x})‖ * ‖x‖ ≤ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
by
      gcongr
      exact hT_norm _ <| SimpleFunc.measurableSet_fiber _ _

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

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * μ.real s
f : α →ₛ F
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
rfl

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F' : Type u_4
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_norm : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * μ.real s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u v : α → β
l : Filter α
c : β
huv : u ~[l] v
hu : Tendsto u l (𝓝 c)
h : c = 0
⊢ Tendsto v l (𝓝 c)

Tactic:
subst c

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u v : α → β
l : Filter α
huv : u ~[l] v
hu : Tendsto u l (𝓝 0)
⊢ Tendsto v l (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace E
f : α →ₛ E
hfi : Integrable (⇑f) μ
⊢ setToSimpleFunc (weightedSMul μ) f = integral μ f

Tactic:
rfl

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ ∀ (x : α), |f x| = 2 * max (f x) 0 - f x

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ ∀ (x : α), |f x| = 2 * max (-f x) 0 + f x

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case e'_3
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hp_pos : p ≠ 0
H : ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑f') p μ → ∃ g, eLpNorm (⇑f' - g) p μ ≤ δ ∧ P g
η : ℝ≥0∞
ηpos : 0 < η
hη :
  ∀ (f g : α → E),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ → eLpNorm f p μ ≤ η → eLpNorm g p μ ≤ η → eLpNorm (f + g) p μ < ε
f' : α →ₛ E
hf' : eLpNorm (f - ⇑f') p μ < η
f'_mem : MemLp (⇑f') p μ
g : α → E
hg : eLpNorm (⇑f' - g) p μ ≤ η
Pg : P g
⊢ f - g = f - ⇑f' + (⇑f' - g)

Tactic:
simp only [sub_add_sub_cancel]

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

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module R W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f : P →ᴬ[R] Q
h : Continuous (↑f).toFun
x✝ : P
⊢ { toAffineMap := ↑f, cont := h } x✝ = f x✝

Tactic:
rfl

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

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : f.contLinear = 0
x✝ : V
⊢ ↑0 x✝ = 0 x✝

Tactic:
rfl

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

Goal state before:
pp: case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : (↑f).linear = 0
x✝ : V
⊢ 0 x✝ = 0 x✝

Tactic:
rfl

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

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h₁ : f.contLinear = 0 ↔ (↑f).linear = 0
q : Q
h : f = const R P q
p✝ : P
⊢ ↑(const R P q) p✝ = (AffineMap.const R P q) p✝

Tactic:
rfl

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)
i : ℕ
a : α
⊢ (Monotone fun i ↦ f i a) → 0 a ≤ f i a - f 0 a

Tactic:
h_mono

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)
i : ℕ
a : α
h_mono : Monotone fun i ↦ f i a
⊢ 0 a ≤ f i a - f 0 a
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] G)
h_eq : ∀ (a : α), (SimpleFunc.map (fun x ↦ ‖x‖ₑ) (toSimpleFunc f)) a = ‖(toSimpleFunc f) a‖ₑ
x : G
a✝ : x ∈ (toSimpleFunc f).range
hx0 : ¬x = 0
⊢ ‖x‖ₑ * μ (⇑(toSimpleFunc f) ⁻¹' {x}) ≠ ∞

Tactic:
finiteness [SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0]

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

Goal state before:
pp: case H
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
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
A : ↥(Lp E 1 μ) →L[ℝ] F
hA : ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = A ↑f
f✝ : ↥(Lp E 1 μ)
f : ↥(α →₁ₛ[μ] E)
⊢ (A ∘SL coeToLp α E ℝ) f = A ↑f

Tactic:
rfl

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

Goal state before:
pp: case refine_1
𝕜 : Type u_4
inst✝² : RCLike 𝕜
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
r : ℝ
hr : r < 1
x₀ : E
hx₀ : x₀ ∉ F
h : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
hx₀' : x₀ ≠ 0
⊢ (↑‖x₀‖)⁻¹ • x₀ ∉ F

Tactic:
rwa [Submodule.smul_mem_iff]

Goal state after:
case refine_1.s0
𝕜 : Type u_4
inst✝² : RCLike 𝕜
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
r : ℝ
hr : r < 1
x₀ : E
hx₀ : x₀ ∉ F
h : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
hx₀' : x₀ ≠ 0
⊢ (↑‖x₀‖)⁻¹ ≠ 0
Module: Mathlib.Topology.Instances.Matrix

Goal state before:
pp: α : Type u_2
m : Type u_4
n : Type u_5
R : Type u_8
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalSpace α
f : α → m → n → R
⊢ (Continuous fun x ↦ of (f x)) ↔ Continuous f

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃⁱ[𝕜] P₂
x✝ : V
⊢ f.linear x✝ = f.linearIsometry.toLinearMap x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃⁱ[𝕜] P₂
⊢ ⇑f.toAffineMap = ⇑f

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃ[𝕜] P₂
norm_map✝¹ : ∀ (x : V), ‖f.linear x‖ = ‖x‖
norm_map✝ : ∀ (x : V), ‖{ toAffineMap := f, norm_map := norm_map✝¹ }.linear x‖ = ‖x‖
⊢ { toAffineMap := f, norm_map := norm_map✝¹ } =
    { toAffineMap := { toAffineMap := f, norm_map := norm_map✝¹ }.toAffineMap, norm_map := norm_map✝ }

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
f : V →ₗᵢ[𝕜] V₂
x✝ : V
⊢ f.toAffineIsometry.linearIsometry x✝ = f x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
e : P ≃ᵃⁱ[𝕜] P₂
x✝ : V
⊢ e.linear x✝ = e.linearIsometryEquiv.toLinearEquiv x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V₁ : Type u_3
V₂ : Type u_5
P₁ : Type u_8
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V₁
inst✝⁶ : NormedSpace 𝕜 V₁
inst✝⁵ : PseudoMetricSpace P₁
inst✝⁴ : NormedAddTorsor V₁ P₁
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
e : P₁ → P₂
e' : V₁ ≃ₗᵢ[𝕜] V₂
p : P₁
h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p
x✝ : V₁
⊢ (mk' e e' p h).linearIsometryEquiv x✝ = e' x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
e : V ≃ₗᵢ[𝕜] V₂
⊢ ⇑e.toAffineIsometryEquiv = ⇑e

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
e : V ≃ₗᵢ[𝕜] V₂
x✝ : V
⊢ e.toAffineIsometryEquiv.linearIsometryEquiv x✝ = e x✝

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p : P
x✝ : V
⊢ (constVSub 𝕜 p).symm x✝ = ((LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (vaddConst 𝕜 p)) x✝

Tactic:
rfl

Goal state after:
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
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x = SimpleFunc.setToSimpleFunc T f

Tactic:
rfl

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

Goal state before:
pp: case pos
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
g : β → G
hgm : AEStronglyMeasurable g (Measure.map f μ)
⊢ ∫ (y : β), g y ∂Measure.map f μ = ∫ (x : α), g (f x) ∂μ

pp: case neg
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
g : β → G
hgm : ¬AEStronglyMeasurable g (Measure.map f μ)
⊢ ∫ (y : β), g y ∂Measure.map f μ = ∫ (x : α), g (f x) ∂μ

Tactic:
· exact MeasureTheory.integral_map hf.measurable.aemeasurable hgm

Goal state after:
case neg
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
g : β → G
hgm : ¬AEStronglyMeasurable g (Measure.map f μ)
⊢ ∫ (y : β), g y ∂Measure.map f μ = ∫ (x : α), g (f x) ∂μ
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝¹ : Nonempty α
inst✝ : CompactSpace α
w : ∀ (x : α), dist (f x) (g x) < C
c : Continuous[inst✝³, _] fun x ↦ dist (f x) (g x)
⊢ dist f g < C

Tactic:
obtain ⟨x, -, le⟩ :=
    IsCompact.exists_isMaxOn isCompact_univ Set.univ_nonempty (Continuous.continuousOn c)

Goal state after:
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝¹ : Nonempty α
inst✝ : CompactSpace α
w : ∀ (x : α), dist (f x) (g x) < C
c : Continuous[inst✝³, _] fun x ↦ dist (f x) (g x)
x : α
le : IsMaxOn (fun x ↦ dist (f x) (g x)) univ x
⊢ dist f g < C
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Fintype ι
v : Basis ι 𝕜 E
u : E →L[𝕜] F
M : ℝ≥0
hu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M
e : E
φ : E →L[𝕜] ι → 𝕜 := ↑v.equivFunL
⊢ ∑ i, ‖v.equivFun e i • u (v i)‖₊ = ∑ i, ‖v.equivFun e i‖₊ * ‖u (v i)‖₊

Tactic:
by simp only [nnnorm_smul]

Goal state after:
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
ι : Type u_2
f : ι → α →ᵇ R
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
grind

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

Goal state before:
pp: case refine_1
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
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ) ^ (1 / q) ≠ ∞

pp: case refine_2
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
⊢ ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ ≤
    (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ) ^ (1 / q)

Tactic:
· refine ENNReal.mul_ne_top ?_ ?_
    · convert! hf.eLpNorm_ne_top
      rw [eLpNorm_eq_lintegral_rpow_enorm_toReal]
      · rw [ENNReal.toReal_ofReal hpq.nonneg]
      · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
        exact hpq.pos
      · finiteness
    · convert! hg.eLpNorm_ne_top
      rw [eLpNorm_eq_lintegral_rpow_enorm_toReal]
      · rw [ENNReal.toReal_ofReal hpq.symm.nonneg]
      · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
        exact hpq.symm.pos
      · finiteness

Goal state after:
case refine_2
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
⊢ ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ ≤
    (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ) ^ (1 / q)
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
f : α →ᵇ ℝ
⊢ abs ∘ ⇑f = toReal ∘ ⇑f.nnrealPart + toReal ∘ ⇑(-f).nnrealPart

Tactic:
by
  funext x
  dsimp
  simp only [max_zero_add_max_neg_zero_eq_abs_self]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
f : C(α, β)
a✝ : α
⊢ (mkOfCompact f).toContinuousMap a✝ = f a✝

Tactic:
rfl

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

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
f : α →ᵇ β
x✝ : α
⊢ (mkOfCompact f.toContinuousMap) x✝ = f x✝

Tactic:
rfl

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
ι : Type u_5
f : ι → C(α, R)
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
grind

Goal state after: