mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: case inr.refine_1
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h✝¹ : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝² : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝¹ : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this : Nonempty E
e : ↑(f ⁻¹' {x})
s : Set X
hs : s ⊆ U'
h : IsOpen[inst✝¹] s
⊢ IsOpen[inst✝²] (f ⁻¹' s ∩ V' e)

Tactic:
convert! ((φ e).isOpen_inter_preimage h).inter (hV e).2 using 1

Goal state after:
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h✝¹ : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝² : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝¹ : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this : Nonempty E
e : ↑(f ⁻¹' {x})
s : Set X
hs : s ⊆ U'
h : IsOpen[inst✝¹] s
⊢ f ⁻¹' s ∩ V' e = (φ e).source ∩ ↑(φ e) ⁻¹' s ∩ V ↑e
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case refine_3
x y : Circle
hxy : x ≠ y
⊢ {x, y}ᶜ ∩ (range ⇑(x.path y) ∩ range ⇑(y.path x)) = ∅

Tactic:
rw [range_path_inter_range_path hxy]

Goal state after:
case refine_3
x y : Circle
hxy : x ≠ y
⊢ {x, y}ᶜ ∩ {x, y} = ∅
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: case h
x✝ : ℝ
⊢ ↑(cexp (↑x✝ * I)).arg = ↑x✝

Tactic:
exp_mul_I,

Goal state after:
case h
x✝ : ℝ
⊢ ↑(Complex.cos ↑x✝ + Complex.sin ↑x✝ * I).arg = ↑x✝
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h✝¹ : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝³ : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝² : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝¹ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this✝ : Nonempty E
e : ↑(f ⁻¹' {x})
s : Set X
hs : s ⊆ U'
h : IsOpen[inst✝²] (f ⁻¹' s ∩ V' e)
this : s ⊆ (φ e).target
⊢ f ⁻¹' s ∩ (φ e).source = f ⁻¹' s ∩ V' e

Tactic:
refine inter_eq_inter_iff_left.mpr ⟨fun e' h ↦ h.2.2, fun e' h ↦ ⟨?_ , h.2⟩⟩

Goal state after:
E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
inst✝ : T2Space E
x : X
hf : IsClosedMap f
fin : (f ⁻¹' {x}).Finite
h✝² : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f
this✝³ : DiscreteTopology ↑(f ⁻¹' {x})
φ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X
hφ : ∀ (e : ↑(f ⁻¹' {x})), ↑e ∈ (φ e).source ∧ ↑(φ e) = f
V : E → Set E
hV : ∀ (x : E), x ∈ V x ∧ IsOpen[inst✝²] (V x)
disj : (f ⁻¹' {x}).PairwiseDisjoint V
V' : ↑(f ⁻¹' {x}) → Set E := fun e ↦ V ↑e ∩ (φ e).source
hV' : ∀ (e : ↑(f ⁻¹' {x})), IsOpen[inst✝²] (V' e)
this✝² : ⋃ e, V' e ∈ 𝓝ˢ (f ⁻¹' {x})
W : Set X
hWx : W ∈ 𝓝 x
hWV : f ⁻¹' W ⊆ ⋃ e, V' e
h✝¹ : Nonempty ↑(f ⁻¹' {x})
U : Set X
hUW : U ⊆ W
hU : IsOpen[inst✝¹] U
hxU : x ∈ U
U' : Set X := U ∩ ⋂ e, f '' V' e
this✝¹ : Finite ↑(f ⁻¹' {x})
hU' : IsOpen[inst✝¹] U'
hUV : ∀ (e : ↑(f ⁻¹' {x})), U' ⊆ f '' V' e
this✝ : Nonempty E
e : ↑(f ⁻¹' {x})
s : Set X
hs : s ⊆ U'
h✝ : IsOpen[inst✝²] (f ⁻¹' s ∩ V' e)
this : s ⊆ (φ e).target
e' : E
h : e' ∈ f ⁻¹' s ∩ (φ e).source
⊢ e' ∈ V ↑e
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : ‖x‖ < π
a✝ : Nontrivial A
this : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane
y : ℝ
hy : -π < y ∧ y < π
⊢ (cexp (↑y * I)).arg = y

Tactic:
rw [← Circle.coe_exp, Circle.arg_exp hy.1 hy.2.le]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
v : ↥(unitary A)
hv : v ∈ closedBall 1 √ε
z : ℂ
hz : z ∈ spectrum ℂ ↑v
⊢ ‖↑v - 1‖ ^ 2 ≤ ε

Tactic:
· refine Real.le_sqrt (by positivity) (by positivity) |>.mp ?_
      simpa [Subtype.dist_eq, dist_eq_norm] using hv

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

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✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
x : E
⊢ 0 ≤ re ⟪x, (adjoint T * T) x⟫_𝕜

Tactic:
simp only [Module.End.mul_apply, adjoint_inner_right, inner_self_eq_norm_sq_to_K]
  norm_cast
  exact sq_nonneg _

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
u : H →L[𝕜] K
⊢ (∀ (x : H), ‖u x‖ = ‖x‖) ↔ adjoint u ∘SL u = 1

Tactic:
by
  rw [LinearMap.norm_map_iff_inner_map_map u, u.inner_map_map_iff_adjoint_comp_self]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
e : H ≃ₗᵢ[𝕜] K
⊢ adjoint ↑↑e ∘SL ↑↑e ∘SL ↑↑e.symm = ↑↑e.symm

Tactic:
← comp_assoc,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
e : H ≃ₗᵢ[𝕜] K
⊢ (adjoint ↑↑e ∘SL ↑↑e) ∘SL ↑↑e.symm = ↑↑e.symm
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
⊢ (∀ (a : H →L[𝕜] H), a ∘SL ↑↑g.symm = ↑(↑g).symm ∘SL ↑↑f ∘SL a ∘SL ↑↑f.symm) ↔ ∃ α, ∀ (x : H), f x = (α • g) x

Tactic:
← comp_assoc,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
⊢ (∀ (a : H →L[𝕜] H), a ∘SL ↑↑g.symm = ((↑(↑g).symm ∘SL ↑↑f) ∘SL a) ∘SL ↑↑f.symm) ↔ ∃ α, ∀ (x : H), f x = (α • g) x
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g g' : α → 𝕜 → F
t : Set 𝕜
y₀ y : 𝕜
hu : Summable u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
h't : IsPreconnected t
hg : ∀ (n : α), ∀ y ∈ t, HasDerivAt (g n) (g' n y) y
hg' : ∀ (n : α), ∀ y ∈ t, ‖g' n y‖ ≤ u n
hy₀ : y₀ ∈ t
hg0 : Summable fun n ↦ g n y₀
hy : y ∈ t
⊢ HasDerivAt (fun z ↦ ∑' (n : α), g n z) (∑' (n : α), g' n y) y

Tactic:
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg ⊢
  convert! hasFDerivAt_tsum_of_isPreconnected hu ht h't hg ?_ hy₀ hg0 hy
  · exact (ContinuousLinearMap.smulRightL 𝕜 𝕜 F 1).map_tsum <|
      .of_norm_bounded hu fun n ↦ hg' n y hy
  · simpa

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inl
α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
f : (a : α) → E a
C : ℝ
_i : Fact (1 ≤ ∞)
F : ι → ↥(lp E ∞)
hF : Bornology.IsBounded (Set.range F)
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
hCF : ∀ (k : ι), ‖F k‖ ≤ C
⊢ Memℓp f ∞

Tactic:
apply memℓp_infty
    use C
    rintro _ ⟨a, rfl⟩
    exact norm_apply_le_of_tendsto (Eventually.of_forall hCF) hf a

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case inr.left
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : ι × ι) in l ×ˢ l, ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
q : ℝ
hqpos : 0 < q
hq : q * r < ε
n : ι × ι
y : E
hy : y ∈ Metric.ball x r
hn : ∀ x_1 ∈ Metric.ball x r, ‖0 - (-f' n.1 x_1 + f' n.2 x_1)‖ < q
⊢ ‖0 - (f n.1 y - f n.2 y - (f n.1 x - f n.2 x))‖ < ε

Tactic:
zero_sub,

Goal state after:
case inr.left
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : ι × ι) in l ×ˢ l, ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
q : ℝ
hqpos : 0 < q
hq : q * r < ε
n : ι × ι
y : E
hy : y ∈ Metric.ball x r
hn : ∀ x_1 ∈ Metric.ball x r, ‖-(-f' n.1 x_1 + f' n.2 x_1)‖ < q
⊢ ‖-(f n.1 y - f n.2 y - (f n.1 x - f n.2 x))‖ < ε
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case inr.right
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
t : Set G
ht : t ∈ map (fun n ↦ f n x) l
ht' : ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
n : ι
hn : f n x ∈ t
n' : ι
hn' : f n' x ∈ t
z : E
a✝ : z ∈ Metric.ball x r
⊢ ‖0 - (f (n, n').1 x - f (n, n').2 x)‖ < ε

Tactic:
zero_sub,

Goal state after:
case inr.right
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
t : Set G
ht : t ∈ map (fun n ↦ f n x) l
ht' : ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
n : ι
hn : f n x ∈ t
n' : ι
hn' : f n' x ∈ t
z : E
a✝ : z ∈ Metric.ball x r
⊢ ‖-(f (n, n').1 x - f (n, n').2 x)‖ < ε
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
⊢ x.smoothTransition = 1 ↔ 1 ≤ x

Tactic:
rcases le_or_gt 1 x with hx | hx

Goal state after:
case inl
x : ℝ
hx : 1 ≤ x
⊢ x.smoothTransition = 1 ↔ 1 ≤ x

case inr
x : ℝ
hx : x < 1
⊢ x.smoothTransition = 1 ↔ 1 ≤ x
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: case pos.inl.inr
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
hF : CompleteSpace F
hE : finrank ℝ E > 0
⊢ μ.real univ • f 0 = |(0 ^ finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ

Tactic:
· have : Nontrivial E := finrank_pos_iff.1 hE
      simp [zero_pow hE.ne', measure_univ_of_isAddLeftInvariant, measureReal_def]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ ν : Measure α
s : Set α
h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u
⊢ ν.IsEverywherePos s ↔ μ.IsEverywherePos s

Tactic:
by
  refine ⟨fun H ↦ H.of_forall_exists_nhds_eq ?_, fun H ↦ H.of_forall_exists_nhds_eq h⟩
  intro x hx
  rcases h x hx with ⟨t, ht, h't⟩
  exact ⟨t, ht, fun u hu ↦ (h't u hu).symm⟩

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedAddCommGroup E'
inst✝¹⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : NormedSpace 𝕜 E
inst✝¹⁴ : NormedSpace 𝕜 E'
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
inst✝¹¹ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁰ : CompleteSpace F
inst✝⁹ : AddGroup G
inst✝⁸ : SFinite μ
inst✝⁷ : SFinite ν
inst✝⁶ : μ.IsAddRightInvariant
inst✝⁵ : MeasurableAdd₂ G
inst✝⁴ : MeasurableNeg G
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
hf : Integrable f ν
hg : Integrable g μ
⊢ Integrable (uncurry fun x t ↦ (L (f t)) (g (x - t))) (μ.prod ν)

Tactic:
by apply hf.convolution_integrand L hg

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ∞
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
this : ⋃ f, support ↑f = s
f : E → ℝ
hf : support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (support ↑⟨f, hf⟩)

Tactic:
exact hf.2.2.1.continuous.isOpen_support

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : HasContDiffBump E
inst✝³ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
⊢ ∫ (x : E) in closedBall c f.rOut, ↑f x ∂μ ≤ ∫ (x : E) in closedBall c f.rOut, 1 ∂μ

Tactic:
by
    apply setIntegral_mono f.integrable.integrableOn _ (fun x ↦ f.le_one)
    simp [measure_closedBall_lt_top]

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
x : E
K_pos : 0 < K
this : ↑f x / ∫ (y : E), ↑f y ∂μ ≤ 1 / ∫ (y : E), ↑f y ∂μ
⊢ 1 * μ.real (closedBall c f.rOut) ≤ K ^ finrank ℝ E * ∫ (x : E), ↑f x ∂μ

pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
x : E
K_pos : 0 < K
this : ↑f x / ∫ (y : E), ↑f y ∂μ ≤ 1 / ∫ (y : E), ↑f y ∂μ
⊢ 0 < μ.real (closedBall c f.rOut)

Tactic:
one_mul,

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
x : E
K_pos : 0 < K
this : ↑f x / ∫ (y : E), ↑f y ∂μ ≤ 1 / ∫ (y : E), ↑f y ∂μ
⊢ μ.real (closedBall c f.rOut) ≤ K ^ finrank ℝ E * ∫ (x : E), ↑f x ∂μ

E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
x : E
K_pos : 0 < K
this : ↑f x / ∫ (y : E), ↑f y ∂μ ≤ 1 / ∫ (y : E), ↑f y ∂μ
⊢ 0 < μ.real (closedBall c f.rOut)
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
⊢ ContDiffOn ℝ ∞ (uncurry fun R x ↦ if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)⁻¹ • x) else 0) (Ioi 1 ×ˢ univ)

Tactic:
suffices
          ContDiffOn ℝ ∞
            (uncurry y ∘ fun p : ℝ × E => ((p.1 - 1) / (p.1 + 1), ((p.1 + 1) / 2)⁻¹ • p.2))
            (Ioi 1 ×ˢ univ) by
          apply this.congr
          rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
          simp only [hR, uncurry_apply_pair, if_true, Function.comp_apply]
        apply (y_smooth E).comp
        · apply ContDiffOn.prodMk
          · refine
              (contDiffOn_fst.sub contDiffOn_const).div (contDiffOn_fst.add contDiffOn_const) ?_
            rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
            apply ne_of_gt
            dsimp only
            linarith
          · apply ContDiffOn.smul _ contDiffOn_snd
            refine ((contDiffOn_fst.add contDiffOn_const).div_const _).inv ?_
            rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
            apply ne_of_gt
            dsimp only
            linarith
        · rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
          have A : 0 < (R - 1) / (R + 1) := by apply div_pos <;> linarith
          have B : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith
          simp only [prodMk_mem_set_prod_eq, mem_Ioo, mem_univ, and_true, A, B]

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.SmoothApprox

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : C(E, F)
⊢ f ∈ closure {f | ContDiff ℝ ∞ ⇑f}

Tactic:
mem_closure_iff_nhds_basis
    (nhds_basis_uniformity uniformity_basis_dist.compactConvergenceUniformity)

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : C(E, F)
⊢ ∀ (i : Set E × ℝ),
    IsCompact i.1 ∧ 0 < i.2 →
      ∃ y ∈ {f | ContDiff ℝ ∞ ⇑f}, y ∈ {y | (y, f) ∈ {fg | ∀ x ∈ i.1, (fg.1 x, fg.2 x) ∈ {p | dist p.1 p.2 < i.2}}}
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
⊢ μ' s = ν s

Tactic:
have sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹) := by
    intro y hy
    by_cases h'y : m ∪ {y} ∈ A
    · have ym : y ∈ m := m_max.mem_of_prop_insert (by simpa using h'y)
      have : y ∈ y • (k * k⁻¹) := by
        simpa using mem_leftCoset y (Set.mul_mem_mul one_k (Set.inv_mem_inv.mpr one_k))
      exact mem_biUnion ym this
    · obtain ⟨x, xm, -, z, zy, zx⟩ : ∃ x ∈ m, y ≠ x ∧ ∃ z, z ∈ y • k ∧ z ∈ x • k := by
        simpa [A, hms, hy, insert_subset_iff, pairwiseDisjoint_insert, hdj, not_disjoint_iff]
          using h'y
      have : y ∈ x • (k * k⁻¹) := by
        rw [show y = x * ((x⁻¹ * z) * (y⁻¹ * z)⁻¹) by group]
        have : (x⁻¹ * z) * (y⁻¹ * z)⁻¹ ∈ k * k⁻¹ := Set.mul_mem_mul ((mem_leftCoset_iff x).mp zx)
          (Set.inv_mem_inv.mpr ((mem_leftCoset_iff y).mp zy))
        exact mem_leftCoset x this
      exact mem_biUnion xm this

Goal state after:
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
⊢ μ' s = ν s