mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : 𝓝[<] a = ⊥
⊢ leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a

Tactic:
simp [leftLim_eq_of_eq_bot _ ha]

Goal state after:
case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : 𝓝[<] a = ⊥
⊢ ↑(c • f) a = ↑c * ↑f a
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 : PreInnerProductSpace.Core 𝕜 F
this : NormedSpace 𝕜 F := Core.toNormedSpace
x : F
⊢ ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜

Tactic:
have h₁ : ‖x‖ ^ 2 = √(re (cd.inner x x)) ^ 2 := rfl
      have h₂ : 0 ≤ re (cd.inner x x) := InnerProductSpace.Core.inner_self_nonneg
      simp [h₁, sq_sqrt, h₂]

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✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
hF : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
cd : Core 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫_𝕜) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫_𝕜 < 1}
this✝ : NormedAddCommGroup F := Core.toNormedAddCommGroupOfTopology h h'
this : NormedSpace 𝕜 F := Core.toNormedSpaceOfTopology h h'
x : F
⊢ ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜

Tactic:
have h₁ : ‖x‖ ^ 2 = √(re (cd.inner x x)) ^ 2 := rfl
      have h₂ : 0 ≤ re (cd.inner x x) := InnerProductSpace.Core.inner_self_nonneg
      simp [h₁, sq_sqrt, h₂]

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

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
p : X
hp : p ∈ s
y : Y
⊢ (p, y) ∈ s ×ˢ univ

Tactic:
simpa only [prodMk_mem_set_prod_eq, mem_univ, and_true] using hp

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

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
p : X
hp : p ∈ s
⊢ ∃ C, ∀ (y : Y), ‖f p y‖ ≤ C

Tactic:
have : ContinuousOn (f p) k := by
        have : ContinuousOn (fun y ↦ (p, y)) k := by fun_prop
        exact hf.comp this (by simp [MapsTo, hp])

Goal state after:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
p : X
hp : p ∈ s
this : ContinuousOn (f p) k
⊢ ∃ C, ∀ (y : Y), ‖f p y‖ ≤ C
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‖ ^ 2 + ‖x - y‖ ^ 2 = 2 * (‖x‖ ^ 2 + ‖y‖ ^ 2)

Tactic:
simp_rw [sq, parallelogram_law_with_norm_mul 𝕜 x y]

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

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_4
v : ι → E
hv : Orthonormal 𝕜 v
l₁ l₂ : ι → 𝕜
s : Finset ι
⊢ ∑ x ∈ s, (starRingEnd 𝕜) (l₁ x) * ⟪v x, ∑ i ∈ s, l₂ i • v i⟫ = ∑ i ∈ s, (starRingEnd 𝕜) (l₁ i) * l₂ i

Tactic:
refine Finset.sum_congr rfl fun i hi => ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l₁ l₂ : ι → 𝕜
s : Finset ι
i : ι
hi : i ∈ s
⊢ (starRingEnd 𝕜) (l₁ i) * ⟪v i, ∑ i ∈ s, l₂ i • v i⟫ = (starRingEnd 𝕜) (l₁ i) * l₂ i
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: case e'_2.e'_5
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✝ : DecidableEq ι
f : (i : ι) → G i
s₁ s₂ : Finset ι
F : (i : ι) → G i := fun i ↦ if i ∈ s₁ then f i else -f i
hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i
hF₂ : ∀ i ∈ s₂ \ s₁, F i = -f i
hF : ∀ (i : ι), ‖F i‖ = ‖f i‖
this :
  ‖∑ i ∈ s₁ \ s₂, (V i) (F i) + ∑ i ∈ s₂ \ s₁, (V i) (F i)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖F i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖F i‖ ^ 2
⊢ ∑ x ∈ s₁ \ s₂, (V x) (f x) = ∑ i ∈ s₁ \ s₂, (V i) (F i)

Tactic:
refine Finset.sum_congr rfl fun i hi => ?_

Goal state after:
case e'_2.e'_5
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✝ : DecidableEq ι
f : (i : ι) → G i
s₁ s₂ : Finset ι
F : (i : ι) → G i := fun i ↦ if i ∈ s₁ then f i else -f i
hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i
hF₂ : ∀ i ∈ s₂ \ s₁, F i = -f i
hF : ∀ (i : ι), ‖F i‖ = ‖f i‖
this :
  ‖∑ i ∈ s₁ \ s₂, (V i) (F i) + ∑ i ∈ s₂ \ s₁, (V i) (F i)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖F i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖F i‖ ^ 2
i : ι
hi : i ∈ s₁ \ s₂
⊢ (V i) (f i) = (V i) (F i)
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: case e'_2.e'_6
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✝ : DecidableEq ι
f : (i : ι) → G i
s₁ s₂ : Finset ι
F : (i : ι) → G i := fun i ↦ if i ∈ s₁ then f i else -f i
hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i
hF₂ : ∀ i ∈ s₂ \ s₁, F i = -f i
hF : ∀ (i : ι), ‖F i‖ = ‖f i‖
this :
  ‖∑ i ∈ s₁ \ s₂, (V i) (F i) + ∑ i ∈ s₂ \ s₁, (V i) (F i)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖F i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖F i‖ ^ 2
⊢ ∑ x ∈ s₂ \ s₁, -(V x) (f x) = ∑ i ∈ s₂ \ s₁, (V i) (F i)

Tactic:
refine Finset.sum_congr rfl fun i hi => ?_

Goal state after:
case e'_2.e'_6
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✝ : DecidableEq ι
f : (i : ι) → G i
s₁ s₂ : Finset ι
F : (i : ι) → G i := fun i ↦ if i ∈ s₁ then f i else -f i
hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i
hF₂ : ∀ i ∈ s₂ \ s₁, F i = -f i
hF : ∀ (i : ι), ‖F i‖ = ‖f i‖
this :
  ‖∑ i ∈ s₁ \ s₂, (V i) (F i) + ∑ i ∈ s₂ \ s₁, (V i) (F i)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖F i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖F i‖ ^ 2
i : ι
hi : i ∈ s₂ \ s₁
⊢ -(V i) (f i) = (V i) (F i)
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
hf : ∀ 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

Tactic:
ε

Goal state after:
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
hf : ∀ 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 : ℝ
⊢ ε > 0 → ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
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
hf : ∀ 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:
ε

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
hf : ∀ 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 : ℝ
⊢ 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
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 S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x y : E
⊢ ⟪x, T y⟫ - ⟪x, S y⟫ = ⟪x, (T - S) y⟫

Tactic:
← 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
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x y : E
⊢ ⟪x, T y - S y⟫ = ⟪x, (T - S) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫
⊢ Filter.Tendsto (fun t ↦ ⟪u t - x, T (y - T x)⟫) Filter.atTop (nhds 0)

Tactic:
rw [← inner_zero_left (T (y - T x))]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫
⊢ Filter.Tendsto (fun t ↦ ⟪u t - x, T (y - T x)⟫) Filter.atTop (nhds ⟪0, T (y - T x)⟫)
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y ↦ y - f (g y)
hle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖
y : F
⊢ ∃ x, f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖

Tactic:
have hnle : ∀ n : ℕ, ‖h^[n] y‖ ≤ (1 / 2) ^ n * ‖y‖ := by
    intro n
    induction n with
    | zero => simp only [one_div, one_mul, iterate_zero_apply, pow_zero, le_rfl]
    | succ n IH =>
      rw [iterate_succ']
      apply le_trans (hle _) _
      rw [pow_succ', mul_assoc]
      gcongr

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y ↦ y - f (g y)
hle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖
y : F
hnle : ∀ (n : ℕ), ‖h^[n] y‖ ≤ (1 / 2) ^ n * ‖y‖
⊢ ∃ x, f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inl
𝕜 : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
hp : Fact (1 ≤ ∞)
f g : WithLp ∞ (α × β)
⊢ edist f g = ENNReal.ofReal (dist f g)

pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f g : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ edist f g = ENNReal.ofReal (dist f g)

Tactic:
· refine ENNReal.eq_of_forall_le_nnreal_iff fun r ↦ ?_
      simp [prod_edist_eq_sup, prod_dist_eq_sup]

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f g : WithLp p (α × β)
h : 1 ≤ p.toReal
⊢ edist f g = ENNReal.ofReal (dist f g)
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜' F
σ' : 𝕜' →+* 𝕜
inst✝⁵ : RingHomInvPair σ σ'
inst✝⁴ : RingHomIsometric σ
inst✝³ : RingHomIsometric σ'
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : RingHomInvPair σ' σ
f : E →SL[σ] F
hinj : Injective ⇑f
hclo : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Set.range ⇑f)
x : E
⊢ (equivRange hinj hclo).symm ⟨f x, ⋯⟩ = x

Tactic:
simp [ContinuousLinearEquiv.symm_apply_eq, Subtype.ext_iff]

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

Goal state before:
pp: a : ℝ
h : a ≠ 0
⊢ Measure.map (fun x ↦ a * x) volume = ofReal |a⁻¹| • volume

Tactic:
by
  conv_rhs =>
    rw [← Real.smul_map_volume_mul_left h, smul_smul, ← ENNReal.ofReal_mul (abs_nonneg _), ←
      abs_mul, inv_mul_cancel₀ h, abs_one, ENNReal.ofReal_one, one_smul]

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

Goal state before:
pp: case mpr
ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v g : ι → E
t : {i : ι} → ℝ
ht : ∀ {i : ι}, t ∈ Icc 0 1
hg : ∀ {i : ι}, (fun a ↦ a • v i) t = g i
⊢ ∃ t, (∀ (i : ι), t i ∈ Icc (0 i) (1 i)) ∧ ∑ i, g i = ∑ i, t i • v i

Tactic:
refine ⟨@t, @ht, ?_⟩

Goal state after:
case mpr
ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v g : ι → E
t : {i : ι} → ℝ
ht : ∀ {i : ι}, t ∈ Icc 0 1
hg : ∀ {i : ι}, (fun a ↦ a • v i) t = g i
⊢ ∑ i, g i = ∑ i, t • v i
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr
p : ℝ≥0∞
ι : Type u_2
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
β : Type u_5
inst✝ : SeminormedAddCommGroup β
hp : p ≠ ∞
b : β
h : 1 ≤ p.toReal
ne_zero : p.toReal ≠ 0
⊢ (∑ x, ‖Function.const ι b x‖₊ ^ p.toReal) ^ (1 / p.toReal) = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊

Tactic:
Function.const_apply,

Goal state after:
case inr
p : ℝ≥0∞
ι : Type u_2
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
β : Type u_5
inst✝ : SeminormedAddCommGroup β
hp : p ≠ ∞
b : β
h : 1 ≤ p.toReal
ne_zero : p.toReal ≠ 0
⊢ (∑ x, ‖b‖₊ ^ p.toReal) ^ (1 / p.toReal) = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr
p : ℝ≥0∞
ι : Type u_2
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
β : Type u_5
inst✝ : SeminormedAddCommGroup β
hp : p ≠ ∞
b : β
h : 1 ≤ p.toReal
ne_zero : p.toReal ≠ 0
⊢ (Fintype.card ι • ‖b‖₊ ^ p.toReal) ^ (1 / p.toReal) = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊

Tactic:
nsmul_eq_mul,

Goal state after:
case inr
p : ℝ≥0∞
ι : Type u_2
hp✝ : Fact (1 ≤ p)
inst✝¹ : Fintype ι
β : Type u_5
inst✝ : SeminormedAddCommGroup β
hp : p ≠ ∞
b : β
h : 1 ≤ p.toReal
ne_zero : p.toReal ≠ 0
⊢ (↑(Fintype.card ι) * ‖b‖₊ ^ p.toReal) ^ (1 / p.toReal) = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 ≤ a_1} ∩ {a_1 | a_1 ≤ g a.1}})

Tactic:
refine
    MeasurableSet.inter ?_
      ((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
        (measurableSet_le measurable_snd (hg.comp measurable_fst)))

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {a | a.1 ∈ s}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n ↦ c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
⊢ x ∈ s

Tactic:
convert! s.smul_mem (c ^ n - c ^ m)⁻¹ A

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n ↦ c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
⊢ x = (c ^ n - c ^ m)⁻¹ • (c ^ n - c ^ m) • x
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
i : ι
⊢ (b.repr (fract b m)) i = Int.fract ((b.repr m) i)

Tactic:
rw [fract, map_sub, Finsupp.coe_sub, Pi.sub_apply, repr_floor_apply, Int.fract]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case intro
ι : Type u_2
inst✝ : Finite ι
val✝ : Fintype ι
x : ↥(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ ↑x ∈ Metric.ball 0 1 ↔ x ∈ {0}

Tactic:
mem_ball_zero_iff,

Goal state after:
case intro
ι : Type u_2
inst✝ : Finite ι
val✝ : Fintype ι
x : ↥(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ ‖↑x‖ < 1 ↔ x ∈ {0}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case neg
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
h : 0 ≤ r
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) = μ (ball x r)

Tactic:
addHaar_ball μ x h

Goal state after:
case neg
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
r : ℝ
h : 0 ≤ r
⊢ ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
⊢ ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : E), μ (closedBall x (2 * ε)) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall x ε)

Tactic:
filter_upwards [self_mem_nhdsWithin] with r hr x

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ μ (closedBall x (2 * r)) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall x r)
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
⊢ ∃ J,
    (∀ (n : ℕ), Box.Icc (J n) ⊆ Box.Ioo I) ∧
      Tendsto (lower ∘ ⇑J) atTop (𝓝 I.lower) ∧ Tendsto (upper ∘ ⇑J) atTop (𝓝 I.upper)

Tactic:
choose a b ha_anti hb_mono ha_mem hb_mem hab ha_tendsto hb_tendsto using
    fun i ↦ exists_seq_strictAnti_strictMono_tendsto (I.lower_lt_upper i)
  exact
    ⟨⟨fun k ↦ ⟨flip a k, flip b k, fun i ↦ hab _ _ _⟩, fun k l hkl ↦
        le_iff_bounds.2 ⟨fun i ↦ (ha_anti i).antitone hkl, fun i ↦ (hb_mono i).monotone hkl⟩⟩,
      fun n x hx i _ ↦ ⟨(ha_mem _ _).1.trans_le (hx.1 _), (hx.2 _).trans_lt (hb_mem _ _).2⟩,
      tendsto_pi_nhds.2 ha_tendsto, tendsto_pi_nhds.2 hb_tendsto⟩

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: n : Type u_7
inst✝ : Fintype n
r : ℝ
hr : 0 ≤ r
x : EuclideanSpace ℝ n
this : 0 ≤ ∑ i, x.ofLp i ^ 2
⊢ x ∈ Metric.ball 0 r ↔ ∑ x_1, x.ofLp x_1 ^ 2 < r ^ 2

Tactic:
mem_ball_zero_iff,

Goal state after:
n : Type u_7
inst✝ : Fintype n
r : ℝ
hr : 0 ≤ r
x : EuclideanSpace ℝ n
this : 0 ≤ ∑ i, x.ofLp i ^ 2
⊢ ‖x‖ < r ↔ ∑ x_1, x.ofLp x_1 ^ 2 < r ^ 2