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

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
hsub : c - a = b - c
g : Y →ᵇ ℝ
hg_mem : ∀ (y : Y), g y ∈ Icc a b
ha' : ¬∃ x, f x = a
hd : Disjoint (range e ∪ ⇑g ⁻¹' Ici c) (⇑g ⁻¹' {a})
dg : Y →ᵇ ℝ
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Ici c)
dga : EqOn (⇑dg) (Function.const Y (c - a)) (⇑g ⁻¹' {a})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (c - a)
hgf : ∀ (x : X), (g + dg) (e x) = f x
y : Y
hay : a < (g + dg) y
x : X
left✝ : a ≤ f x
hxy : f x < (g + dg) y
⊢ (g + dg) y ≤ b

Tactic:
rcases le_total c (g y) with hc | hc

Goal state after:
case inr.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
hsub : c - a = b - c
g : Y →ᵇ ℝ
hg_mem : ∀ (y : Y), g y ∈ Icc a b
ha' : ¬∃ x, f x = a
hd : Disjoint (range e ∪ ⇑g ⁻¹' Ici c) (⇑g ⁻¹' {a})
dg : Y →ᵇ ℝ
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Ici c)
dga : EqOn (⇑dg) (Function.const Y (c - a)) (⇑g ⁻¹' {a})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (c - a)
hgf : ∀ (x : X), (g + dg) (e x) = f x
y : Y
hay : a < (g + dg) y
x : X
left✝ : a ≤ f x
hxy : f x < (g + dg) y
hc : c ≤ g y
⊢ (g + dg) y ≤ b

case inr.inr
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
hsub : c - a = b - c
g : Y →ᵇ ℝ
hg_mem : ∀ (y : Y), g y ∈ Icc a b
ha' : ¬∃ x, f x = a
hd : Disjoint (range e ∪ ⇑g ⁻¹' Ici c) (⇑g ⁻¹' {a})
dg : Y →ᵇ ℝ
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Ici c)
dga : EqOn (⇑dg) (Function.const Y (c - a)) (⇑g ⁻¹' {a})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (c - a)
hgf : ∀ (x : X), (g + dg) (e x) = f x
y : Y
hay : a < (g + dg) y
x : X
left✝ : a ≤ f x
hxy : f x < (g + dg) y
hc : g y ≤ c
⊢ (g + dg) y ≤ b
Module: Mathlib.RingTheory.Norm.Transitivity

Goal state before:
pp: case e_6
R : Type u_1
S : Type u_2
n : Type u_4
m : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Matrix m m S
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
k : m
inst✝¹ : Fintype m
inst✝ : Fintype n
f : S →+* Matrix n n R
i✝ j✝ : m
⊢ (evalRingHom 0).mapMatrix (cornerAddX M k) i✝ j✝ = M i✝ j✝

Tactic:
simp [cornerAddX, diagonal, apply_ite]

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
hsub : c - a = b - c
g : Y →ᵇ ℝ
hg_mem : ∀ (y : Y), g y ∈ Icc a b
ha' : ¬∃ x, f x = a
hd : Disjoint (range e ∪ ⇑g ⁻¹' Ici c) (⇑g ⁻¹' {a})
dg : Y →ᵇ ℝ
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Ici c)
dga : EqOn (⇑dg) (Function.const Y (c - a)) (⇑g ⁻¹' {a})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (c - a)
hgf : ∀ (x : X), (g + dg) (e x) = f x
y : Y
hay : a < (g + dg) y
x : X
left✝ : a ≤ f x
hxy : f x < (g + dg) y
hc : g y ≤ c
⊢ c + (c - a) = b

Tactic:
rw [hsub, add_sub_cancel]

Goal state after:
Module: Mathlib.RingTheory.Norm.Transitivity

Goal state before:
pp: case e_6.e_6
R : Type u_1
S : Type u_2
n : Type u_4
m : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
M : Matrix m m S
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
k : m
inst✝¹ : Fintype m
inst✝ : Fintype n
f : S →+* Matrix n n R
i j : m
⊢ (evalRingHom 0).mapMatrix (cornerAddX M k) i j = M i j

Tactic:
simp [cornerAddX, diagonal, apply_ite]

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
b : ℝ
hb : IsLUB (range ⇑f) b
g : Y →ᵇ ℝ
xl : Y → X
hxl : ∀ (y : Y), f (xl y) ≤ g y
hgb : ∀ (y : Y), g y ≤ b
hb' : ¬∃ x, f x = b
dg : Y →ᵇ ℝ
hgf : ∀ (x : X), (g - dg) (e x) = f x
y : Y
hyb : (g - dg) y < b
xu : X
hyxu : (g - dg) y < f xu
right✝ : f xu ≤ b
x : X
ha : IsGLB (range ⇑f) (f x)
hmem : ∀ (x_1 : X), f x_1 ∈ Icc (f x) b
hle : f x ≤ b
hlt : f x < b
c : ℝ := (f x + b) / 2
hac : f x < c
hcb : c < b
hsub : c - f x = b - c
hd : Disjoint (range e ∪ ⇑g ⁻¹' Iic c) (⇑g ⁻¹' {b})
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Iic c)
dgb : EqOn (⇑dg) (Function.const Y (b - c)) (⇑g ⁻¹' {b})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (b - c)
hc : c < g y
⊢ f x = c - (b - c)

Tactic:
rw [← hsub, sub_sub_cancel]

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f) b
hmem : ∀ (x : X), f x ∈ Icc a b
hle : a ≤ b
hlt : a < b
c : ℝ := (a + b) / 2
hac : a < c
hcb : c < b
hsub : c - a = b - c
g : Y →ᵇ ℝ
xl : Y → X
hxl : ∀ (y : Y), f (xl y) ≤ g y
hgb : ∀ (y : Y), g y ≤ b
hb' : ¬∃ x, f x = b
hd : Disjoint (range e ∪ ⇑g ⁻¹' Iic c) (⇑g ⁻¹' {b})
dg : Y →ᵇ ℝ
dg0 : EqOn (⇑dg) (Function.const Y 0) (range e ∪ ⇑g ⁻¹' Iic c)
dgb : EqOn (⇑dg) (Function.const Y (b - c)) (⇑g ⁻¹' {b})
dgmem : ∀ (x : Y), dg x ∈ Icc 0 (b - c)
hgf : ∀ (x : X), (g - dg) (e x) = f x
y : Y
hyb : (g - dg) y < b
xu : X
hyxu : (g - dg) y < f xu
right✝ : f xu ≤ b
hc : c < g y
h✝ : a ∉ range ⇑f
⊢ a = c - (b - c)

Tactic:
rw [← hsub, sub_sub_cancel]

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : C(X, ℝ)
t : Set ℝ
e : X → Y
hs : t.OrdConnected
hf : ∀ (x : X), f x ∈ t
hne : t.Nonempty
he : IsClosedEmbedding e
h : ℝ ≃o ↑(Ioo (-1) 1)
⊢ Continuous[inst✝², _] (Subtype.val ∘ ⇑h ∘ ⇑f)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.TietzeExtension

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
⊢ Continuous toReal

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z w : ℍ
⊢ (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =
    dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))

Tactic:
congr 1

Goal state after:
case e_a
z w : ℍ
⊢ 2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2 = dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2
Module: Mathlib.Geometry.Euclidean.Inversion.Basic

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c : P
R : ℝ
hR : R ≠ 0
⊢ ContinuousAt (dist c) c

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Geometry.Euclidean.Inversion.Basic

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
c : P
R : ℝ
hR : R ≠ 0
hdist : Tendsto (dist c) (𝓝[≠] c) (𝓝[>] 0)
hratio : Tendsto (fun x ↦ dist c (inversion c R x)) (𝓝[≠] c) atTop
⊢ Tendsto (fun x ↦ dist c (inversion c R x)) (𝓝[≠] c) atTop

Tactic:
simpa using hratio

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
α : Type u_4
inst✝ : TopologicalSpace α
f : α → GL n R
hf : Continuous[inst✝, Units.instTopologicalSpaceUnits] f
i : n
⊢ Continuous[instTopologicalSpaceMatrix, Pi.topologicalSpace] fun A ↦ A i

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
α : Type u_4
inst✝ : TopologicalSpace α
f : α → GL n R
hf : Continuous[inst✝, Units.instTopologicalSpaceUnits] f
i : n
⊢ Continuous[inst✝, instTopologicalSpaceMatrix] fun x ↦ ↑(f x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z w : ℍ
h : z.re = w.re
h₀ : 0 < z.im / w.im
⊢ dist z.im (w.im * ((z.im / w.im + w.im / z.im) / 2)) = w.im * |(z.im / w.im - w.im / z.im) / 2|

Tactic:
skip

Goal state after:
z w : ℍ
h : z.re = w.re
h₀ : 0 < z.im / w.im
⊢ dist z.im (w.im * ((z.im / w.im + w.im / z.im) / 2)) = w.im * |(z.im / w.im - w.im / z.im) / 2|
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z w : ℍ
h : z.re = w.re
h₀ : 0 < z.im / w.im
⊢ |z.im - w.im * ((z.im / w.im + w.im / z.im) / 2)| = |w.im * ((z.im / w.im - w.im / z.im) / 2)|

Tactic:
congr 1

Goal state after:
z w : ℍ
h : z.re = w.re
h₀ : 0 < z.im / w.im
⊢ z.im - w.im * ((z.im / w.im + w.im / z.im) / 2) = w.im * ((z.im / w.im - w.im / z.im) / 2)
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: case refine_2
z✝ w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
⊢ ∃ i, 0 < i ∧ ball z i ⊆ UpperHalfPlane.coe ⁻¹' ball (↑z) R

Tactic:
have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)

Goal state after:
case refine_2
z✝ w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / z.im + 1
⊢ ∃ i, 0 < i ∧ ball z i ⊆ UpperHalfPlane.coe ⁻¹' ball (↑z) R
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
α : Type u_4
inst✝ : TopologicalSpace α
f : α → SL n R
hf : Continuous[inst✝, instTopologicalSpace] f
i : n
⊢ Continuous[instTopologicalSpaceMatrix, Pi.topologicalSpace] fun A ↦ A i

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
α : Type u_4
inst✝ : TopologicalSpace α
f : α → SL n R
hf : Continuous[inst✝, instTopologicalSpace] f
i : n
⊢ Continuous[inst✝, instTopologicalSpaceMatrix] fun x ↦ ↑(f x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : CommRing S
inst✝ : TopologicalSpace S
f : R →+* S
hf : IsInducing ⇑f
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] ⇑(map f)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : CommRing S
inst✝ : TopologicalSpace S
f : R →+* S
hf : IsEmbedding ⇑f
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] ⇑(map f)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Matrix

Goal state before:
pp: n : Type u_1
R : Type u_2
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : CommRing R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalRing R
inst✝ : T0Space R
⊢ Continuous[Units.instTopologicalSpaceUnits, Units.instTopologicalSpaceUnits] ⇑GeneralLinearGroup.det

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Metric

Goal state before:
pp: z✝ w : ℍ
r✝ : ℝ
z : ℍ
r : ℝ
⊢ IsCompact (closedBall (↑(z.center r)) (z.im * Real.sinh r))

Tactic:
apply isCompact_closedBall

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : Group G
inst✝² : MulAction G X
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalSpace X
x✝ :
  ContinuousSMul G X ∧
    ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
      Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)
cont : ContinuousSMul G X
h :
  ∀ (𝒰 : Ultrafilter (G × X)) (x₁ x₂ : X),
    Tendsto (fun gx ↦ (gx.1 • gx.2, gx.2)) (↑𝒰) (𝓝 (x₁, x₂)) → ∃ g, g • x₂ = x₁ ∧ Tendsto fst (↑𝒰) (𝓝 g)
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Category.CompHaus.Basic

Goal state before:
pp: case mp
X Y : CompHaus
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X ↦ ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
⊢ False

Tactic:
let D := ({y} : Set Y)

Goal state after:
case mp
X Y : CompHaus
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X ↦ ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
D : Set ↑Y.toTop := {y}
⊢ False
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
Y✝ : Type u_2
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
Y : Type u_3
inst✝ : TopologicalSpace Y
f : C(X, Y)
y : Y
h : Tendsto (⇑f) (coclosedCompact X) (𝓝 y)
⊢ Continuous[instTopologicalSpace, inst✝] fun x ↦ x.elim y ⇑f

Tactic:
rw [continuous_iff]
    refine ⟨h, f.continuous⟩

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace X
inst✝² : ContinuousSMul G X
inst✝¹ : IsTopologicalGroup G
inst✝ : IsPretransitive G X
x : X
hx : IsProperMap fun g ↦ g • x
f : G × G → G × X := Prod.map id fun g ↦ g • x
hfsurj : Function.Surjective f
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] f

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.Basic

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
inst✝⁴ : TopologicalSpace G
inst✝³ : TopologicalSpace X
inst✝² : ContinuousSMul G X
inst✝¹ : IsTopologicalGroup G
inst✝ : IsPretransitive G X
x : X
hx : IsProperMap fun g ↦ g • x
f : G × G → G × X := Prod.map id fun g ↦ g • x
hfsurj : Function.Surjective f
⊢ Continuous[instTopologicalSpaceProd, instTopologicalSpaceProd] fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.CompactlyGenerated

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : MulAction G X
inst✝² : CompactlyGeneratedSpace (X × X)
inst✝¹ : T2Space X
inst✝ : ContinuousSMul G X
h : ∀ {U V : Set X}, IsCompact U → IsCompact V → IsCompact {g | (g • U ∩ V).Nonempty}
⊢ Continuous fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.CompactlyGenerated

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : MulAction G X
inst✝² : CompactlyGeneratedSpace (X × X)
inst✝¹ : T2Space X
inst✝ : ContinuousSMul G X
h : ∀ {U V : Set X}, IsCompact U → IsCompact V → IsCompact {g | (g • U ∩ V).Nonempty}
K : Set (X × X)
hK : IsCompact K
U : Set X := fst '' K
V : Set X := snd '' K
hU : IsCompact U
hV : IsCompact V
this : IsCompact ((fun gx ↦ (gx.1 • gx.2, gx.2)) ⁻¹' U ×ˢ V)
⊢ Continuous fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Algebra.ProperAction.CompactlyGenerated

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : MulAction G X
inst✝² : CompactlyGeneratedSpace (X × X)
inst✝¹ : T2Space X
inst✝ : ContinuousSMul G X
h : ∀ {U V : Set X}, IsCompact U → IsCompact V → IsCompact {g | (g • U ∩ V).Nonempty}
K : Set (X × X)
hK : IsCompact K
U : Set X := fst '' K
V : Set X := snd '' K
hU : IsCompact U
hV : IsCompact V
⊢ Continuous fun gx ↦ (gx.1 • gx.2, gx.2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.Compactification.OnePoint.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X
⊢ ¬Continuous[_, instTopologicalSpace] ⇑CofiniteTopology.of.symm

Tactic:
simp only [continuous_iff_continuousAt, ContinuousAt, not_forall]

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X
⊢ ∃ x, ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 x) (𝓝 (CofiniteTopology.of.symm x))
Module: Mathlib.Topology.Compactness.CompactlyGeneratedSpace

Goal state before:
pp: X : Type w
Y : Type x
tX : TopologicalSpace X
tY : TopologicalSpace Y
inst✝ : SequentialSpace X
s : Set X
h : ∀ (S : CompHaus) (f : C(↑S.toTop, X)), IsClosed (⇑f ⁻¹' s)
u : ℕ → X
p : X
hu : ∀ (n : ℕ), u n ∈ s
hup : Tendsto u atTop (𝓝 p)
g : ULift.{u, 0} (OnePoint ℕ) → X := ⇑(continuousMapMkNat u p hup) ∘ ULift.down
this : Tendsto OnePoint.some atTop (𝓝 ∞)
⊢ Continuous[_, tX] g

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction

Goal state before:
pp: g : GL (Fin 2) ℝ
hg : 0 < ↑(Matrix.GeneralLinearGroup.det g)
⊢ Continuous fun h ↦ ↑(Matrix.GeneralLinearGroup.det h)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction

Goal state before:
pp: g : GL (Fin 2) ℝ
hg : ↑(Matrix.GeneralLinearGroup.det g) < 0
⊢ Continuous fun h ↦ ↑(Matrix.GeneralLinearGroup.det h)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
α : Type u_2
s : Finset α
f : α → ℂ → E
hf : ∀ a ∈ s, Meromorphic (f a)
r : ℝ
h₂f : ∀ i ∈ s, CircleIntegrable (fun x ↦ log⁺ ‖f i x‖) 0 r
⊢ circleAverage (fun x ↦ ∑ c ∈ s, log⁺ ‖f c x‖ + log ↑s.card) 0 r =
    circleAverage ((∑ i ∈ s, fun x ↦ log⁺ ‖f i x‖) + fun x ↦ log ↑s.card) 0 r

Tactic:
congr 1

Goal state after:
case e_f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
α : Type u_2
s : Finset α
f : α → ℂ → E
hf : ∀ a ∈ s, Meromorphic (f a)
r : ℝ
h₂f : ∀ i ∈ s, CircleIntegrable (fun x ↦ log⁺ ‖f i x‖) 0 r
⊢ (fun x ↦ ∑ c ∈ s, log⁺ ‖f c x‖ + log ↑s.card) = (∑ i ∈ s, fun x ↦ log⁺ ‖f i x‖) + fun x ↦ log ↑s.card
Module: Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction

Goal state before:
pp: ⊢ Continuous fun g ↦ g • I

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction

Goal state before:
pp: K : Set ℍ
hK : IsCompact K
A : ℝ
hA : ∀ (g : SL(2, ℝ)), g • I ∈ K → ↑g 0 0 ^ 2 + ↑g 0 1 ^ 2 ≤ A
A' : ℝ
hA' : ∀ (g : SL(2, ℝ)), g • I ∈ K → ↑g 1 0 ^ 2 + ↑g 1 1 ^ 2 ≤ A'
this✝² : SeminormedAddCommGroup (Matrix (Fin 2) (Fin 2) ℝ) := Matrix.seminormedAddCommGroup
this✝¹ : ProperSpace (Matrix (Fin 2) (Fin 2) ℝ)
this✝ : IsCompact {m | ∀ (i j : Fin 2), |m i j| ≤ max √A √A'}
this : IsCompact (Subtype.val ⁻¹' {m | ∀ (i j : Fin 2), |m i j| ≤ max √A √A'})
⊢ Continuous fun g ↦ g • I

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: E✝ : Type u_1
inst✝² : NormedAddCommGroup E✝
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D₁ D₂ : locallyFinsupp E ℤ
r : ℝ
this : ∀ {A B C D : ℝ}, A + B + (C + D) = A + C + (B + D)
⊢ ∑ᶠ (z : E), (↑(((toClosedBall r) D₁) z) + ↑(((toClosedBall r) D₂) z)) * log (r * ‖z‖⁻¹) +
      (↑(D₁ 0) + ↑(D₂ 0)) * log r =
    ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * log (r * ‖z‖⁻¹) +
        ∑ᶠ (z : E), ↑(((toClosedBall r) D₂) z) * log (r * ‖z‖⁻¹) +
      (↑(D₁ 0) * log r + ↑(D₂ 0) * log r)

Tactic:
congr 1

Goal state after:
case e_a
E✝ : Type u_1
inst✝² : NormedAddCommGroup E✝
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D₁ D₂ : locallyFinsupp E ℤ
r : ℝ
this : ∀ {A B C D : ℝ}, A + B + (C + D) = A + C + (B + D)
⊢ ∑ᶠ (z : E), (↑(((toClosedBall r) D₁) z) + ↑(((toClosedBall r) D₂) z)) * log (r * ‖z‖⁻¹) =
    ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * log (r * ‖z‖⁻¹) + ∑ᶠ (z : E), ↑(((toClosedBall r) D₂) z) * log (r * ‖z‖⁻¹)

case e_a
E✝ : Type u_1
inst✝² : NormedAddCommGroup E✝
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D₁ D₂ : locallyFinsupp E ℤ
r : ℝ
this : ∀ {A B C D : ℝ}, A + B + (C + D) = A + C + (B + D)
⊢ (↑(D₁ 0) + ↑(D₂ 0)) * log r = ↑(D₁ 0) * log r + ↑(D₂ 0) * log r
Module: Mathlib.Analysis.Complex.ValueDistribution.Cartan

Goal state before:
pp: f : ℂ → ℂ
h₁ : MeromorphicAt f 0
h₂ : meromorphicOrderAt f 0 = 0
a : ℂ
ha_sphere : a ∈ sphere 0 |1|
ha_ne : a ∈ {meromorphicTrailingCoeffAt f 0}ᶜ
⊢ MeromorphicAt (fun x ↦ a) 0

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Cartan

Goal state before:
pp: f : ℂ → ℂ
h : ¬MeromorphicAt f 0
a : ℂ
⊢ MeromorphicAt (fun x ↦ a) 0

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Cartan

Goal state before:
pp: f : ℂ → ℂ
h : meromorphicOrderAt f 0 = 0
hf : ¬MeromorphicAt f 0
a : ℂ
⊢ MeromorphicAt (fun x ↦ a) 0

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Cartan

Goal state before:
pp: f : ℂ → ℂ
R : ℝ
h : Meromorphic f
hR : R ≠ 0
a : ℂ
⊢ Meromorphic (f - fun x ↦ a)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation

Goal state before:
pp: f : ℂ → ℂ
R : ℝ
h : Meromorphic f
this : Measurable f
⊢ AEStronglyMeasurable (fun p ↦ cartanKernel f R p.1 p.2)
    ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation

Goal state before:
pp: f : ℂ → ℂ
h : Meromorphic f
R : ℝ
F : ℝ → ℝ → ℝ := Cartan.cartanKernel f R
⊢ (2 * π)⁻¹ * ∫ (β : ℝ) in 0..2 * π, (2 * π)⁻¹ * ∫ (α : ℝ) in 0..2 * π, F α β =
    (2 * π)⁻¹ * ∫ (β : ℝ) in 0..2 * π, log⁺ ‖f (circleMap 0 R β)‖

Tactic:
congr 1

Goal state after:
case e_a
f : ℂ → ℂ
h : Meromorphic f
R : ℝ
F : ℝ → ℝ → ℝ := Cartan.cartanKernel f R
⊢ ∫ (β : ℝ) in 0..2 * π, (2 * π)⁻¹ * ∫ (α : ℝ) in 0..2 * π, F α β = ∫ (β : ℝ) in 0..2 * π, log⁺ ‖f (circleMap 0 R β)‖
Module: Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic

Goal state before:
pp: case h₁.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
hD : 0 ≤ D
a b : ℝ
ha : 0 < a
hb : 0 < b
a✝⁴ : a ≤ b
s : Set E := ((toClosedBall b) D).support
hs : s.Finite
z : E
h₁ : ¬D z = 0
a✝³ : ¬a = 0
a✝² : ¬z = 0
a✝¹ : ¬a * ‖z‖⁻¹ = 1
a✝ : ¬a * ‖z‖⁻¹ = -1
this : z ∈ closedBall 0 |a|
⊢ ¬((toClosedBall b) D) z = 0

Tactic:
rwa [toClosedBall_eval_within]

Goal state after:
case h₁.h.ha
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
D : locallyFinsupp E ℤ
hD : 0 ≤ D
a b : ℝ
ha : 0 < a
hb : 0 < b
a✝⁴ : a ≤ b
s : Set E := ((toClosedBall b) D).support
hs : s.Finite
z : E
h₁ : ¬D z = 0
a✝³ : ¬a = 0
a✝² : ¬z = 0
a✝¹ : ¬a * ‖z‖⁻¹ = 1
a✝ : ¬a * ‖z‖⁻¹ = -1
this : z ∈ closedBall 0 |a|
⊢ z ∈ closedBall 0 |b|