l3lab/ntp-mathlib huggingface dataset:
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · 

Proof state:
case intro.intro.intro.refine'_2.right
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
⊢ (f x + f (-x)) / 2 ≤ 1

Next tactic:
linarith [(B x).2, (B (-x)).2]
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · 

Proof state:
case intro.intro.intro.refine'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
⊢ (support fun x => (f x + f (-x)) / 2) = ball 0 1

Next tactic:
refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · 

Proof state:
case intro.intro.intro.refine'_3.refine'_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∈ ball 0 1
⊢ (f x + f (-x)) / 2 ≠ 0

Next tactic:
apply ne_of_gt
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      

Proof state:
case intro.intro.intro.refine'_3.refine'_1.h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∈ ball 0 1
⊢ 0 < (f x + f (-x)) / 2

Next tactic:
have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∈ ball 0 1
⊢ 0 < f x

Next tactic:
apply lt_of_le_of_ne (B x).1 (Ne.symm _)
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∈ ball 0 1
⊢ f x ≠ 0

Next tactic:
rwa [← f_support] at hx
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      

Proof state:
case intro.intro.intro.refine'_3.refine'_1.h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∈ ball 0 1
this : 0 < f x
⊢ 0 < (f x + f (-x)) / 2

Next tactic:
linarith [(B (-x)).1]
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · 

Proof state:
case intro.intro.intro.refine'_3.refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
⊢ (f x + f (-x)) / 2 = 0

Next tactic:
have I1 : x ∉ support f := by rwa [f_support]
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by 

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
⊢ x ∉ support f

Next tactic:
rwa [f_support]
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      

Proof state:
case intro.intro.intro.refine'_3.refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : x ∉ support f
⊢ (f x + f (-x)) / 2 = 0

Next tactic:
have I2 : -x ∉ support f := by
        rw [f_support]
        simpa using hx
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      have I2 : -x ∉ support f := by
        

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : x ∉ support f
⊢ -x ∉ support f

Next tactic:
rw [f_support]
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      have I2 : -x ∉ support f := by
        rw [f_support]
        

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : x ∉ support f
⊢ -x ∉ ball 0 1

Next tactic:
simpa using hx
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      have I2 : -x ∉ support f := by
        rw [f_support]
        simpa using hx
      

Proof state:
case intro.intro.intro.refine'_3.refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : x ∉ support f
I2 : -x ∉ support f
⊢ (f x + f (-x)) / 2 = 0

Next tactic:
simp only [mem_support, Classical.not_not] at I1 I2
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      have I2 : -x ∉ support f := by
        rw [f_support]
        simpa using hx
      simp only [mem_support, Classical.not_not] at I1 I2
      

Proof state:
case intro.intro.intro.refine'_3.refine'_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : f x = 0
I2 : f (-x) = 0
⊢ (f x + f (-x)) / 2 = 0

Next tactic:
simp only [I1, I2, add_zero, zero_div]
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      have I2 : -x ∉ support f := by
        rw [f_support]
        simpa using hx
      simp only [mem_support, Classical.not_not] at I1 I2
      simp only [I1, I2, add_zero, zero_div]
  · 

Proof state:
case intro.intro.intro.refine'_4
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
⊢ ∀ (x : E), (fun x => (f x + f (-x)) / 2) (-x) = (fun x => (f x + f (-x)) / 2) x

Next tactic:
intro x
theorem u_exists :
    ∃ u : E → ℝ,
      ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by
  have A : IsOpen (ball (0 : E) 1) := isOpen_ball
  obtain ⟨f, f_support, f_smooth, f_range⟩ :
    ∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1
  exact A.exists_smooth_support_eq
  have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
  refine' ⟨fun x => (f x + f (-x)) / 2, _, _, _, _⟩
  · exact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _
  · intro x
    simp only [mem_Icc]
    constructor
    · linarith [(B x).1, (B (-x)).1]
    · linarith [(B x).2, (B (-x)).2]
  · refine' support_eq_iff.2 ⟨fun x hx => _, fun x hx => _⟩
    · apply ne_of_gt
      have : 0 < f x := by
        apply lt_of_le_of_ne (B x).1 (Ne.symm _)
        rwa [← f_support] at hx
      linarith [(B (-x)).1]
    · have I1 : x ∉ support f := by rwa [f_support]
      have I2 : -x ∉ support f := by
        rw [f_support]
        simpa using hx
      simp only [mem_support, Classical.not_not] at I1 I2
      simp only [I1, I2, add_zero, zero_div]
  · intro x; 

Proof state:
case intro.intro.intro.refine'_4
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ⊤ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
⊢ (fun x => (f x + f (-x)) / 2) (-x) = (fun x => (f x + f (-x)) / 2) x

Next tactic:
simp only [add_comm, neg_neg]
theorem u_compact_support : HasCompactSupport (u : E → ℝ) := by
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ HasCompactSupport u

Next tactic:
rw [hasCompactSupport_def, u_support, closure_ball (0 : E) one_ne_zero]
theorem u_compact_support : HasCompactSupport (u : E → ℝ) := by
  rw [hasCompactSupport_def, u_support, closure_ball (0 : E) one_ne_zero]
  

Proof state:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ IsCompact (closedBall 0 1)

Next tactic:
exact isCompact_closedBall _ _
theorem u_int_pos : 0 < ∫ x : E, u x ∂μ := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < ∫ (x : E), u x ∂μ

Next tactic:
refine' (integral_pos_iff_support_of_nonneg u_nonneg _).mpr _
theorem u_int_pos : 0 < ∫ x : E, u x ∂μ := by
  refine' (integral_pos_iff_support_of_nonneg u_nonneg _).mpr _
  · 

Proof state:
case refine'_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ Integrable fun i => u i

Next tactic:
exact (u_continuous E).integrable_of_hasCompactSupport (u_compact_support E)
theorem u_int_pos : 0 < ∫ x : E, u x ∂μ := by
  refine' (integral_pos_iff_support_of_nonneg u_nonneg _).mpr _
  · exact (u_continuous E).integrable_of_hasCompactSupport (u_compact_support E)
  · 

Proof state:
case refine'_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < ↑↑μ (support fun i => u i)

Next tactic:
rw [u_support]
theorem u_int_pos : 0 < ∫ x : E, u x ∂μ := by
  refine' (integral_pos_iff_support_of_nonneg u_nonneg _).mpr _
  · exact (u_continuous E).integrable_of_hasCompactSupport (u_compact_support E)
  · rw [u_support]; 

Proof state:
case refine'_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < ↑↑μ (ball 0 1)

Next tactic:
exact measure_ball_pos _ _ zero_lt_one
theorem w_def (D : ℝ) :
    (w D : E → ℝ) = fun x => ((∫ x : E, u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x) := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
⊢ w D = fun x => ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x)

Next tactic:
ext1 x
theorem w_def (D : ℝ) :
    (w D : E → ℝ) = fun x => ((∫ x : E, u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x) := by
  ext1 x; 

Proof state:
case h
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ w D x = ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x)

Next tactic:
rfl
theorem w_nonneg (D : ℝ) (x : E) : 0 ≤ w D x := by
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ w D x

Next tactic:
apply mul_nonneg _ (u_nonneg _)
theorem w_nonneg (D : ℝ) (x : E) : 0 ≤ w D x := by
  apply mul_nonneg _ (u_nonneg _)
  

Proof state:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹

Next tactic:
apply inv_nonneg.2