mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.Algebra.SMul

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
H' : Type u_4
inst✝¹¹ : TopologicalSpace H'
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
H'' : Type u_6
inst✝⁸ : TopologicalSpace H''
E'' : Type u_7
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
I'' : ModelWithCorners 𝕜 E'' H''
G : Type u_8
inst✝⁵ : TopologicalSpace G
inst✝⁴ : ChartedSpace H G
M : Type u_9
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H' M
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H'' N
n : ℕ∞ω
⊢ ContMDiff (𝓘(𝕜, E →L[𝕜] E).prod 𝓘(𝕜, E)) (𝓘(𝕜, E →L[𝕜] E).prod 𝓘(𝕜, E)) n id

Tactic:
;

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
H' : Type u_4
inst✝¹¹ : TopologicalSpace H'
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
H'' : Type u_6
inst✝⁸ : TopologicalSpace H''
E'' : Type u_7
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
I'' : ModelWithCorners 𝕜 E'' H''
G : Type u_8
inst✝⁵ : TopologicalSpace G
inst✝⁴ : ChartedSpace H G
M : Type u_9
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H' M
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H'' N
n : ℕ∞ω
⊢ ContMDiff (𝓘(𝕜, E →L[𝕜] E).prod 𝓘(𝕜, E)) (𝓘(𝕜, E →L[𝕜] E).prod 𝓘(𝕜, E)) n id
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut ⊆ (extChartAt I c).source

Tactic:
;

Goal state after:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ (extChartAt I c).source ∩ ↑(extChartAt I c) ⁻¹' ball (↑(extChartAt I c) c) f.rOut ⊆ (extChartAt I c).source
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: case h₁.refine_1
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
x : M
⊢ ↑(extChartAt I c) x ∈ range ↑I

Tactic:
;

Goal state after:
case h₁.refine_1
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
s : Set M
hse : s ⊆ (extChartAt I c).source
hsf : ↑(extChartAt I c) '' s ⊆ ball (↑(extChartAt I c) c) f.rOut
x : M
⊢ ↑(extChartAt I c) x ∈ range ↑I
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ 0 < f.rIn

Tactic:
;

Goal state after:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
⊢ 0 < f.rIn
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
x : M
hx : ↑f x = 1 x
⊢ 1 x ≠ 0

Tactic:
;

Goal state after:
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
inst✝ : FiniteDimensional ℝ E
x : M
hx : ↑f x = 1 x
⊢ 1 x ≠ 0
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace Y
e : X → Y
he : IsClosedEmbedding e
α : Type v
s : α → Set X
ho : ∀ (a : α), IsOpen[inst✝²] (s a)
U : α → Set Y
hUo : ∀ (a : α), IsOpen[inst✝¹] (U a)
hU : ∀ (a : α), e ⁻¹' U a = s a
hu : ⋃ a, e ⁻¹' U a = univ
heU : range e ⊆ ⋃ i, U i
V : α → Set Y
hVo : ∀ (i : α), IsOpen[inst✝¹] (V i)
heV : range e ⊆ ⋃ i, V i
hVf : LocallyFinite V
hVU : ∀ (i : α), V i ⊆ U i
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝²] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ e ⁻¹' U a

Tactic:
refine ⟨α, fun a ↦ e ⁻¹' (V a), fun a ↦ (hVo a).preimage he.continuous, ?_,
      hVf.preimage_continuous he.continuous, fun a ↦ ⟨a, preimage_mono (hVU a)⟩⟩

Goal state after:
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace Y
e : X → Y
he : IsClosedEmbedding e
α : Type v
s : α → Set X
ho : ∀ (a : α), IsOpen[inst✝²] (s a)
U : α → Set Y
hUo : ∀ (a : α), IsOpen[inst✝¹] (U a)
hU : ∀ (a : α), e ⁻¹' U a = s a
hu : ⋃ a, e ⁻¹' U a = univ
heU : range e ⊆ ⋃ i, U i
V : α → Set Y
hVo : ∀ (i : α), IsOpen[inst✝¹] (V i)
heV : range e ⊆ ⋃ i, V i
hVf : LocallyFinite V
hVU : ∀ (i : α), V i ⊆ U i
⊢ ⋃ b, (fun a ↦ e ⁻¹' V a) b = univ
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: case refine_2
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find x + 1)
Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s)
this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 ↑x
r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x
hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x)
hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (K n)ᶜ
hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x
T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s)
hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ ↑(T n)
x : X
hx : x ∈ s
⊢ ∃ i, x ∈ B ((fun a ↦ ↑↑a.snd) i) ((fun a ↦ r a.fst ↑a.snd) i)

Tactic:
rcases mem_iUnion₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩

Goal state after:
case refine_2
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed[inst✝³] s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n ↦ K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find x + 1)
Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s)
this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 ↑x
r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x
hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x)
hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (K n)ᶜ
hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x
T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s)
hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n ↦ ↑(T n)
x : X
hx : x ∈ s
c : X
hc : c ∈ Kdiff (K'.find x + 1) ∩ s
hcT : ⟨c, hc⟩ ∈ T (K'.find x)
hcx : x ∈ B (↑⟨c, hc⟩) (r (K'.find x) ⟨↑⟨c, hc⟩, ⋯⟩)
⊢ ∃ i, x ∈ B ((fun a ↦ ↑↑a.snd) i) ((fun a ↦ r a.fst ↑a.snd) i)
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: case mk
ι : Type u
X : Type v
inst✝⁴ : TopologicalSpace X
E : Type u_1
inst✝³ : AddCommMonoid E
inst✝² : SMulWithZero ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set X
f g : PartitionOfUnity ι X s
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
sum_eq_one'✝ : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun✝ i) x = 1
sum_le_one'✝ : ∀ (x : X), ∑ᶠ (i : ι), (toFun✝ i) x ≤ 1
h :
  { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
        sum_le_one' := sum_le_one'✝ }.toFun =
    g.toFun
⊢ { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
      sum_le_one' := sum_le_one'✝ } =
    g

Tactic:
;

Goal state after:
case mk
ι : Type u
X : Type v
inst✝⁴ : TopologicalSpace X
E : Type u_1
inst✝³ : AddCommMonoid E
inst✝² : SMulWithZero ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set X
f g : PartitionOfUnity ι X s
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
sum_eq_one'✝ : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun✝ i) x = 1
sum_le_one'✝ : ∀ (x : X), ∑ᶠ (i : ι), (toFun✝ i) x ≤ 1
h :
  { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
        sum_le_one' := sum_le_one'✝ }.toFun =
    g.toFun
⊢ { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
      sum_le_one' := sum_le_one'✝ } =
    g
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: case mk.mk
ι : Type u
X : Type v
inst✝⁴ : TopologicalSpace X
E : Type u_1
inst✝³ : AddCommMonoid E
inst✝² : SMulWithZero ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set X
f : PartitionOfUnity ι X s
toFun✝¹ : ι → C(X, ℝ)
locallyFinite'✝¹ : LocallyFinite fun i ↦ support ⇑(toFun✝¹ i)
nonneg'✝¹ : 0 ≤ toFun✝¹
sum_eq_one'✝¹ : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun✝¹ i) x = 1
sum_le_one'✝¹ : ∀ (x : X), ∑ᶠ (i : ι), (toFun✝¹ i) x ≤ 1
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
sum_eq_one'✝ : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun✝ i) x = 1
sum_le_one'✝ : ∀ (x : X), ∑ᶠ (i : ι), (toFun✝ i) x ≤ 1
h :
  { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, sum_eq_one' := sum_eq_one'✝¹,
        sum_le_one' := sum_le_one'✝¹ }.toFun =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
        sum_le_one' := sum_le_one'✝ }.toFun
⊢ { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, sum_eq_one' := sum_eq_one'✝¹,
      sum_le_one' := sum_le_one'✝¹ } =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
      sum_le_one' := sum_le_one'✝ }

Tactic:
;

Goal state after:
case mk.mk
ι : Type u
X : Type v
inst✝⁴ : TopologicalSpace X
E : Type u_1
inst✝³ : AddCommMonoid E
inst✝² : SMulWithZero ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set X
f : PartitionOfUnity ι X s
toFun✝¹ : ι → C(X, ℝ)
locallyFinite'✝¹ : LocallyFinite fun i ↦ support ⇑(toFun✝¹ i)
nonneg'✝¹ : 0 ≤ toFun✝¹
sum_eq_one'✝¹ : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun✝¹ i) x = 1
sum_le_one'✝¹ : ∀ (x : X), ∑ᶠ (i : ι), (toFun✝¹ i) x ≤ 1
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
sum_eq_one'✝ : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun✝ i) x = 1
sum_le_one'✝ : ∀ (x : X), ∑ᶠ (i : ι), (toFun✝ i) x ≤ 1
h :
  { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, sum_eq_one' := sum_eq_one'✝¹,
        sum_le_one' := sum_le_one'✝¹ }.toFun =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
        sum_le_one' := sum_le_one'✝ }.toFun
⊢ { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, sum_eq_one' := sum_eq_one'✝¹,
      sum_le_one' := sum_le_one'✝¹ } =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, sum_eq_one' := sum_eq_one'✝,
      sum_le_one' := sum_le_one'✝ }
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: case mk
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f g : BumpCovering ι X s
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
le_one'✝ : toFun✝ ≤ 1
eventuallyEq_one'✝ : ∀ x ∈ s, ∃ i, ⇑(toFun✝ i) =ᶠ[𝓝 x] 1
h :
  { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
        eventuallyEq_one' := eventuallyEq_one'✝ }.toFun =
    g.toFun
⊢ { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
      eventuallyEq_one' := eventuallyEq_one'✝ } =
    g

Tactic:
;

Goal state after:
case mk
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f g : BumpCovering ι X s
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
le_one'✝ : toFun✝ ≤ 1
eventuallyEq_one'✝ : ∀ x ∈ s, ∃ i, ⇑(toFun✝ i) =ᶠ[𝓝 x] 1
h :
  { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
        eventuallyEq_one' := eventuallyEq_one'✝ }.toFun =
    g.toFun
⊢ { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
      eventuallyEq_one' := eventuallyEq_one'✝ } =
    g
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: case mk.mk
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
toFun✝¹ : ι → C(X, ℝ)
locallyFinite'✝¹ : LocallyFinite fun i ↦ support ⇑(toFun✝¹ i)
nonneg'✝¹ : 0 ≤ toFun✝¹
le_one'✝¹ : toFun✝¹ ≤ 1
eventuallyEq_one'✝¹ : ∀ x ∈ s, ∃ i, ⇑(toFun✝¹ i) =ᶠ[𝓝 x] 1
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
le_one'✝ : toFun✝ ≤ 1
eventuallyEq_one'✝ : ∀ x ∈ s, ∃ i, ⇑(toFun✝ i) =ᶠ[𝓝 x] 1
h :
  { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, le_one' := le_one'✝¹,
        eventuallyEq_one' := eventuallyEq_one'✝¹ }.toFun =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
        eventuallyEq_one' := eventuallyEq_one'✝ }.toFun
⊢ { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, le_one' := le_one'✝¹,
      eventuallyEq_one' := eventuallyEq_one'✝¹ } =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
      eventuallyEq_one' := eventuallyEq_one'✝ }

Tactic:
;

Goal state after:
case mk.mk
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
toFun✝¹ : ι → C(X, ℝ)
locallyFinite'✝¹ : LocallyFinite fun i ↦ support ⇑(toFun✝¹ i)
nonneg'✝¹ : 0 ≤ toFun✝¹
le_one'✝¹ : toFun✝¹ ≤ 1
eventuallyEq_one'✝¹ : ∀ x ∈ s, ∃ i, ⇑(toFun✝¹ i) =ᶠ[𝓝 x] 1
toFun✝ : ι → C(X, ℝ)
locallyFinite'✝ : LocallyFinite fun i ↦ support ⇑(toFun✝ i)
nonneg'✝ : 0 ≤ toFun✝
le_one'✝ : toFun✝ ≤ 1
eventuallyEq_one'✝ : ∀ x ∈ s, ∃ i, ⇑(toFun✝ i) =ᶠ[𝓝 x] 1
h :
  { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, le_one' := le_one'✝¹,
        eventuallyEq_one' := eventuallyEq_one'✝¹ }.toFun =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
        eventuallyEq_one' := eventuallyEq_one'✝ }.toFun
⊢ { toFun := toFun✝¹, locallyFinite' := locallyFinite'✝¹, nonneg' := nonneg'✝¹, le_one' := le_one'✝¹,
      eventuallyEq_one' := eventuallyEq_one'✝¹ } =
    { toFun := toFun✝, locallyFinite' := locallyFinite'✝, nonneg' := nonneg'✝, le_one' := le_one'✝,
      eventuallyEq_one' := eventuallyEq_one'✝ }
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝ : NormalSpace X
hs : IsClosed s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, {i | x ∈ u i}.Finite
us : s ⊆ ⋃ i, u i
this✝ : Nonempty (PartialRefinement u s ⊤)
this : ∀ (c : Set (PartialRefinement u s ⊤)), IsChain (fun x1 x2 ↦ x1 ≤ x2) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub
v : PartialRefinement u s ⊤
hv : IsMax v
i : ι
⊢ i ∈ v.carrier

Tactic:
;

Goal state after:
ι : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝ : NormalSpace X
hs : IsClosed s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, {i | x ∈ u i}.Finite
us : s ⊆ ⋃ i, u i
this✝ : Nonempty (PartialRefinement u s ⊤)
this : ∀ (c : Set (PartialRefinement u s ⊤)), IsChain (fun x1 x2 ↦ x1 ≤ x2) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub
v : PartialRefinement u s ⊤
hv : IsMax v
i : ι
⊢ i ∈ v.carrier
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝² : TopologicalSpace (TotalSpace F V)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
toFun✝¹ : (x : M) → V x
contMDiff_toFun✝¹ : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦ ⟨x, toFun✝¹ x⟩
toFun✝ : (x : M) → V x
contMDiff_toFun✝ : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦ ⟨x, toFun✝ x⟩
h :
  { toFun := toFun✝¹, contMDiff_toFun := contMDiff_toFun✝¹ }.toFun =
    { toFun := toFun✝, contMDiff_toFun := contMDiff_toFun✝ }.toFun
⊢ { toFun := toFun✝¹, contMDiff_toFun := contMDiff_toFun✝¹ } = { toFun := toFun✝, contMDiff_toFun := contMDiff_toFun✝ }

Tactic:
;

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝² : TopologicalSpace (TotalSpace F V)
inst✝¹ : (x : M) → TopologicalSpace (V x)
inst✝ : FiberBundle F V
toFun✝¹ : (x : M) → V x
contMDiff_toFun✝¹ : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦ ⟨x, toFun✝¹ x⟩
toFun✝ : (x : M) → V x
contMDiff_toFun✝ : ContMDiff I (I.prod 𝓘(𝕜, F)) n fun x ↦ ⟨x, toFun✝ x⟩
h :
  { toFun := toFun✝¹, contMDiff_toFun := contMDiff_toFun✝¹ }.toFun =
    { toFun := toFun✝, contMDiff_toFun := contMDiff_toFun✝ }.toFun
⊢ { toFun := toFun✝¹, contMDiff_toFun := contMDiff_toFun✝¹ } = { toFun := toFun✝, contMDiff_toFun := contMDiff_toFun✝ }
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s : Cₛ^n⟮I; F, V⟯
⊢ ⇑(0 • s) = 0

Tactic:
;

Goal state after:
case zero
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s : Cₛ^n⟮I; F, V⟯
⊢ ⇑(0 • s) = 0
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s : Cₛ^n⟮I; F, V⟯
k : ℕ
ih : ⇑(k • s) = k • ⇑s
⊢ ⇑((k + 1) • s) = ⇑(k • s) + ⇑s

Tactic:
;

Goal state after:
case succ
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s : Cₛ^n⟮I; F, V⟯
k : ℕ
ih : ⇑(k • s) = k • ⇑s
⊢ ⇑((k + 1) • s) = ⇑(k • s) + ⇑s
Module: Mathlib.Topology.MetricSpace.PartitionOfUnity

Goal state before:
pp: ι : Type u_1
X : Type u_2
inst✝ : MetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
δ : C(X, ℝ≥0)
hδ0 : ∀ (x : X), 0 < δ x
hδ : ∀ (i : ι), ∀ x ∈ K i, closedEBall x ↑(δ x) ⊆ U i
i : ι
x : X
hx : x ∈ K i
⊢ closedBall x ↑(δ x) ⊆ U i

Tactic:
rw [← Metric.closedEBall_coe]

Goal state after:
ι : Type u_1
X : Type u_2
inst✝ : MetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
δ : C(X, ℝ≥0)
hδ0 : ∀ (x : X), 0 < δ x
hδ : ∀ (i : ι), ∀ x ∈ K i, closedEBall x ↑(δ x) ⊆ U i
i : ι
x : X
hx : x ∈ K i
⊢ closedEBall x ↑(δ x) ⊆ U i