l3lab/ntp-mathlib huggingface dataset:
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    

Proof state:
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
i : M
⊢ ∃ g,
    support g = (chartAt H i).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H i)) ⁻¹' s ∧
      Smooth I 𝓘(ℝ, ℝ) g ∧ range g ⊆ Icc 0 1

Next tactic:
apply IsOpen.exists_msmooth_support_eq_aux
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    

Proof state:
case hs
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
i : M
⊢ IsOpen ((chartAt H i).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H i)) ⁻¹' s)

Next tactic:
exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  

Proof state:
case intro
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
A :
  ∀ (c : M),
    ∃ g,
      support g = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s ∧
        Smooth I 𝓘(ℝ, ℝ) g ∧ range g ⊆ Icc 0 1
⊢ ∃ f, support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ ∀ (x : M), 0 ≤ f x

Next tactic:
choose g g_supp g_diff hg using A
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  

Proof state:
case intro
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
⊢ ∃ f, support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ ∀ (x : M), 0 ≤ f x

Next tactic:
have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  

Proof state:
case intro
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
⊢ ∃ f, support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ ∀ (x : M), 0 ≤ f x

Next tactic:
have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  

Proof state:
case intro
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
⊢ ∃ f, support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ ∀ (x : M), 0 ≤ f x

Next tactic:
refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
  · 

Proof state:
case intro.refine_1
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
⊢ (support fun x => ∑ᶠ (c : M), (f c) x * g c (↑(chartAt H c) x)) = s

Next tactic:
refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
  · refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
    · 

Proof state:
case intro.refine_1.refine_1
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∈ s
⊢ ∑ᶠ (c : M), (f c) x * g c (↑(chartAt H c) x) ≠ 0

Next tactic:
apply ne_of_gt
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
  · refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
    · apply ne_of_gt
      

Proof state:
case intro.refine_1.refine_1.h
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∈ s
⊢ 0 < ∑ᶠ (c : M), (f c) x * g c (↑(chartAt H c) x)

Next tactic:
have B : ∃ c, 0 < f c x * g c (chartAt H c x) := by
        obtain ⟨c, hc⟩ : ∃ c, 0 < f c x := f.exists_pos_of_mem (mem_univ x)
        refine ⟨c, mul_pos hc ?_⟩
        apply lt_of_le_of_ne (h'g _ _) (Ne.symm _)
        rw [← mem_support, g_supp, ← mem_preimage, preimage_inter]
        have Hx : x ∈ tsupport (f c) := subset_tsupport _ (ne_of_gt hc)
        simp [(chartAt H c).left_inv (hf c Hx), hx, (chartAt H c).map_source (hf c Hx)]
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
  · refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
    · apply ne_of_gt
      have B : ∃ c, 0 < f c x * g c (chartAt H c x) := by
        

Proof state:
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∈ s
⊢ ∃ c, 0 < (f c) x * g c (↑(chartAt H c) x)

Next tactic:
obtain ⟨c, hc⟩ : ∃ c, 0 < f c x := f.exists_pos_of_mem (mem_univ x)
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
  · refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
    · apply ne_of_gt
      have B : ∃ c, 0 < f c x * g c (chartAt H c x) := by
        obtain ⟨c, hc⟩ : ∃ c, 0 < f c x := f.exists_pos_of_mem (mem_univ x)
        

Proof state:
case intro
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∈ s
c : M
hc : 0 < (f c) x
⊢ ∃ c, 0 < (f c) x * g c (↑(chartAt H c) x)

Next tactic:
refine ⟨c, mul_pos hc ?_⟩
/-- Given an open set in a finite-dimensional real manifold, there exists a nonnegative smooth
function with support equal to `s`. -/
theorem IsOpen.exists_msmooth_support_eq (hs : IsOpen s) :
    ∃ f : M → ℝ, f.support = s ∧ Smooth I 𝓘(ℝ) f ∧ ∀ x, 0 ≤ f x := by
  rcases SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source I M with ⟨f, hf⟩
  have A : ∀ (c : M), ∃ g : H → ℝ,
      g.support = (chartAt H c).target ∩ (chartAt H c).symm ⁻¹' s ∧
      Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1 := by
    intro i
    apply IsOpen.exists_msmooth_support_eq_aux
    exact PartialHomeomorph.isOpen_inter_preimage_symm _ hs
  choose g g_supp g_diff hg using A
  have h'g : ∀ c x, 0 ≤ g c x := fun c x ↦ (hg c (mem_range_self (f := g c) x)).1
  have h''g : ∀ c x, 0 ≤ f c x * g c (chartAt H c x) :=
    fun c x ↦ mul_nonneg (f.nonneg c x) (h'g c _)
  refine ⟨fun x ↦ ∑ᶠ c, f c x * g c (chartAt H c x), ?_, ?_, ?_⟩
  · refine support_eq_iff.2 ⟨fun x hx ↦ ?_, fun x hx ↦ ?_⟩
    · apply ne_of_gt
      have B : ∃ c, 0 < f c x * g c (chartAt H c x) := by
        obtain ⟨c, hc⟩ : ∃ c, 0 < f c x := f.exists_pos_of_mem (mem_univ x)
        refine ⟨c, mul_pos hc ?_⟩
        

Proof state:
case intro
ι : Type uι
E : Type uE
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : FiniteDimensional ℝ E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
t : M → Set F
n : ℕ∞
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M univ
hf : SmoothPartitionOfUnity.IsSubordinate f fun x => (chartAt H x).toPartialEquiv.source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).toPartialEquiv.target ∩ ↑(PartialHomeomorph.symm (chartAt H c)) ⁻¹' s
g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∈ s
c : M
hc : 0 < (f c) x
⊢ 0 < g c (↑(chartAt H c) x)

Next tactic:
apply lt_of_le_of_ne (h'g _ _) (Ne.symm _)