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

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∃ l, l < f x
z₁ : γ
z₁lt : z₁ < f x
h₁ : Ioc z₁ (f x) ⊆ u
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₁z : f z > z₁
h₂z : g z > z₂
⊢ f z + g z > y

Tactic:
have A1 : min (f z) (f x) ∈ u := by
        by_cases! H : f z ≤ f x
        · simpa [H] using h₁ ⟨h₁z, H⟩
        · simpa [H.le]

Goal state after:
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∃ l, l < f x
z₁ : γ
z₁lt : z₁ < f x
h₁ : Ioc z₁ (f x) ⊆ u
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₁z : f z > z₁
h₂z : g z > z₂
A1 : min (f z) (f x) ∈ u
⊢ f z + g z > y
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: ⊢ ∑' (m : ℕ), 2 ^ (-1 - ↑m) = 1

Tactic:
[

Goal state after:
⊢ ∑' (m : ℕ), 2 ^ (-1 - ↑m) = 1
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: a b : EReal
h : ContinuousAt (fun p ↦ p.1 * p.2) (a, b)
this : (fun p ↦ p.1 * p.2) = (fun x ↦ -x) ∘ (fun p ↦ p.1 * p.2) ∘ fun p ↦ (-p.1, p.2)
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (-a, b)

Tactic:
this

Goal state after:
a b : EReal
h : ContinuousAt (fun p ↦ p.1 * p.2) (a, b)
this : (fun p ↦ p.1 * p.2) = (fun x ↦ -x) ∘ (fun p ↦ p.1 * p.2) ∘ fun p ↦ (-p.1, p.2)
⊢ ContinuousAt ((fun x ↦ -x) ∘ (fun p ↦ p.1 * p.2) ∘ fun p ↦ (-p.1, p.2)) (-a, b)
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∃ l, l < f x
z₁ : γ
z₁lt : z₁ < f x
h₁ : Ioc z₁ (f x) ⊆ u
hx₂ : ∀ (x_1 : γ), g x ≤ x_1
z : α
h₁z : f z > z₁
⊢ f z + g z > y

Tactic:
have A1 : min (f z) (f x) ∈ u := by
        by_cases! H : f z ≤ f x
        · simpa [H] using h₁ ⟨h₁z, H⟩
        · simpa [H.le]

Goal state after:
α : Type u_1
inst✝⁵ : TopologicalSpace α
s : Set α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
y : γ
hy : (fun z ↦ f z + g z) x > y
u v : Set γ
u_open : IsOpen[inst✝¹] u
xu : f x ∈ u
v_open : IsOpen[inst✝¹] v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∃ l, l < f x
z₁ : γ
z₁lt : z₁ < f x
h₁ : Ioc z₁ (f x) ⊆ u
hx₂ : ∀ (x_1 : γ), g x ≤ x_1
z : α
h₁z : f z > z₁
A1 : min (f z) (f x) ∈ u
⊢ f z + g z > y
Module: Mathlib.Topology.Instances.EReal.Lemmas

Goal state before:
pp: case h.refine_1
a : ℝ
h : 0 < a
x : ℝ
p : EReal × EReal
p1_gt : ↑(2 * max (x + 1) 0 / a) < p.1
p2_gt : ↑(a / 2) < p.2
p1_pos : 0 < p.1
a2_pos : 0 < ↑(a / 2)
lock : ↑(2 * max (x + 1) 0 / a) * ↑(a / 2) ≤ p.1 * ↑(a / 2)
key : p.1 * ↑(a / 2) ≤ p.1 * p.2
⊢ ↑x < p.1 * p.2

Tactic:
replace lock := le_trans lock key

Goal state after:
case h.refine_1
a : ℝ
h : 0 < a
x : ℝ
p : EReal × EReal
p1_gt : ↑(2 * max (x + 1) 0 / a) < p.1
p2_gt : ↑(a / 2) < p.2
p1_pos : 0 < p.1
a2_pos : 0 < ↑(a / 2)
key : p.1 * ↑(a / 2) ≤ p.1 * p.2
lock : ↑(2 * max (x + 1) 0 / a) * ↑(a / 2) ≤ p.1 * p.2
⊢ ↑x < p.1 * p.2
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousAt f x
hg : LowerSemicontinuousAt g x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
⊢ LowerSemicontinuousAt (fun z ↦ f z + g z) x

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁵ : TopologicalSpace α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousAt f x
hg : LowerSemicontinuousAt g x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
⊢ LowerSemicontinuousAt (fun z ↦ f z + g z) x
Module: Mathlib.MeasureTheory.OuterMeasure.OfFunction

Goal state before:
pp: α : Type u_1
m : Set (OuterMeasure α)
s : Set α
h : m.Nonempty
⊢ (sInf m) s = ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ μ ∈ m, μ (t n)

Tactic:
[

Goal state after:
α : Type u_1
m : Set (OuterMeasure α)
s : Set α
h : m.Nonempty
⊢ (sInf m) s = ⨅ t, ⨅ (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ μ ∈ m, μ (t n)
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
x : α
ι : Type u_4
γ : Type u_5
inst✝⁵ : AddCommMonoid γ
inst✝⁴ : LinearOrder γ
inst✝³ : IsOrderedAddMonoid γ
inst✝² : TopologicalSpace γ
inst✝¹ : OrderTopology γ
inst✝ : ContinuousAdd γ
f : ι → α → γ
a : Finset ι
ha : ∀ i ∈ a, LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun z ↦ ∑ i ∈ a, f i z) x

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁶ : TopologicalSpace α
x : α
ι : Type u_4
γ : Type u_5
inst✝⁵ : AddCommMonoid γ
inst✝⁴ : LinearOrder γ
inst✝³ : IsOrderedAddMonoid γ
inst✝² : TopologicalSpace γ
inst✝¹ : OrderTopology γ
inst✝ : ContinuousAdd γ
f : ι → α → γ
a : Finset ι
ha : ∀ i ∈ a, LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun z ↦ ∑ i ∈ a, f i z) x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
x : α
ι : Sort u_4
δ' : Type u_6
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝 x, BddAbove (range fun i ↦ f i y)
h : ∀ (i : ι), LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun x' ↦ ⨆ i, f i x') x

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
x : α
ι : Sort u_4
δ' : Type u_6
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝 x, BddAbove (range fun i ↦ f i y)
h : ∀ (i : ι), LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun x' ↦ ⨆ i, f i x') x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
x : α
ι : Type u_4
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
⊢ LowerSemicontinuousWithinAt (fun x' ↦ ∑' (i : ι), f i x') s x

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
x : α
ι : Type u_4
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
⊢ LowerSemicontinuousWithinAt (fun x' ↦ ∑' (i : ι), f i x') s x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x : α
ι : Type u_4
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun x' ↦ ∑' (i : ι), f i x') x

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
x : α
ι : Type u_4
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun x' ↦ ∑' (i : ι), f i x') x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x : α
ι : Type u_4
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), LowerSemicontinuousAt (f i) x
⊢ LowerSemicontinuousAt (fun x' ↦ ∑' (i : ι), f i x') x

Tactic:
simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at *
  exact lowerSemicontinuousWithinAt_tsum h

Goal state after:
Module: Mathlib.Order.SetDissipate

Goal state before:
pp: β : Type u_2
s : ℕ → Set β
n : ℕ
⊢ dissipate s n = ⋂ k, ⋂ (_ : k < n + 1), s k

Tactic:
[

Goal state after:
β : Type u_2
s : ℕ → Set β
n : ℕ
⊢ dissipate s n = ⋂ k, ⋂ (_ : k < n + 1), s k
Module: Mathlib.Order.SetDissipate

Goal state before:
pp: case h₁
α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, dissipate s x ⊆ ⋂ x, s x

Tactic:
[

Goal state after:
case h₁
α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, dissipate s x ⊆ ⋂ x, s x
Module: Mathlib.Order.SetDissipate

Goal state before:
pp: case h₂
α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, s x ⊆ ⋂ x, dissipate s x

Tactic:
[

Goal state after:
case h₂
α : Type u_1
β : Type u_2
s : α → Set β
inst✝ : Preorder α
⊢ ⋂ x, s x ⊆ ⋂ x, dissipate s x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : UpperSemicontinuousAt f x
hg : UpperSemicontinuousAt g x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
⊢ UpperSemicontinuousAt (fun z ↦ f z + g z) x

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁵ : TopologicalSpace α
x : α
γ : Type u_5
inst✝⁴ : AddCommMonoid γ
inst✝³ : LinearOrder γ
inst✝² : IsOrderedAddMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : UpperSemicontinuousAt f x
hg : UpperSemicontinuousAt g x
hcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)
⊢ UpperSemicontinuousAt (fun z ↦ f z + g z) x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
x : α
ι : Type u_4
γ : Type u_5
inst✝⁵ : AddCommMonoid γ
inst✝⁴ : LinearOrder γ
inst✝³ : IsOrderedAddMonoid γ
inst✝² : TopologicalSpace γ
inst✝¹ : OrderTopology γ
inst✝ : ContinuousAdd γ
f : ι → α → γ
a : Finset ι
ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x
⊢ UpperSemicontinuousAt (fun z ↦ ∑ i ∈ a, f i z) x

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁶ : TopologicalSpace α
x : α
ι : Type u_4
γ : Type u_5
inst✝⁵ : AddCommMonoid γ
inst✝⁴ : LinearOrder γ
inst✝³ : IsOrderedAddMonoid γ
inst✝² : TopologicalSpace γ
inst✝¹ : OrderTopology γ
inst✝ : ContinuousAdd γ
f : ι → α → γ
a : Finset ι
ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x
⊢ UpperSemicontinuousAt (fun z ↦ ∑ i ∈ a, f i z) x
Module: Mathlib.Analysis.SpecificLimits.Basic

Goal state before:
pp: case hfh
R : Type u_4
inst✝⁵ : TopologicalSpace R
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
a : R
ha : 0 ≤ a
A : Tendsto (fun x ↦ a + x⁻¹) atTop (𝓝 a)
x : R
hx : 1 ≤ x
⊢ ↑⌈a * x⌉₊ / x ≤ a + x⁻¹

Tactic:
simp [div_le_iff₀ (zero_lt_one.trans_le hx), inv_mul_cancel₀ (zero_lt_one.trans_le hx).ne',
      (Nat.ceil_lt_add_one (mul_nonneg ha (zero_le_one.trans hx))).le, add_mul]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: case neg
α : Type u_1
inst✝³ : TopologicalSpace α
s : Set α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
h₁ : LowerSemicontinuousWithinAt f s x
h₂ : UpperSemicontinuousWithinAt f s x
v : Set γ
hv : v ∈ 𝓝 (f x)
Hl : ∀ (l : γ), f x ≤ l
Hu : ∀ (u : γ), u ≤ f x
a : α
this : f a = f x
⊢ f a ∈ v

Tactic:
this

Goal state after:
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
s : Set α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
h₁ : LowerSemicontinuousWithinAt f s x
h₂ : UpperSemicontinuousWithinAt f s x
v : Set γ
hv : v ∈ 𝓝 (f x)
Hl : ∀ (l : γ), f x ≤ l
Hu : ∀ (u : γ), u ≤ f x
a : α
this : f a = f x
⊢ f x ∈ v
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
⊢ ContinuousAt f x ↔ LowerSemicontinuousAt f x ∧ UpperSemicontinuousAt f x

Tactic:
[

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
x : α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
⊢ ContinuousAt f x ↔ LowerSemicontinuousAt f x ∧ UpperSemicontinuousAt f x
Module: Mathlib.Topology.Semicontinuity.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
⊢ Continuous[inst✝³, inst✝¹] f ↔ LowerSemicontinuous f ∧ UpperSemicontinuous f

Tactic:
[

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
γ : Type u_4
inst✝² : LinearOrder γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
⊢ Continuous[inst✝³, inst✝¹] f ↔ LowerSemicontinuous f ∧ UpperSemicontinuous f
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case h.left
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔
    a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅

Tactic:
[

Goal state after:
case h.left
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔
    a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: case refine_2
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b : b ∈ Finset.image (fun x ↦ ↑x) T
⊢ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) b ∈ g b

Tactic:
[

Goal state after:
case refine_2
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset ↑s
f : ↑s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b : b ∈ Finset.image (fun x ↦ ↑x) T
⊢ Function.extend (fun x ↦ ↑x) f (fun x ↦ ∅) b ∈ g b
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
ι : Type u_4
s : ι → Set α
S : Set ι
s' : Set α
⊢ s' ∈ piiUnionInter (fun i ↦ {s i}) S ↔ s' ∈ {s' | ∃ t, ∃ (_ : ↑t ⊆ S), s' = ⋂ i ∈ t, s i}

Tactic:
[

Goal state after:
α : Type u_3
ι : Type u_4
s : ι → Set α
S : Set ι
s' : Set α
⊢ s' ∈ piiUnionInter (fun i ↦ {s i}) S ↔ s' ∈ {s' | ∃ t, ∃ (_ : ↑t ⊆ S), s' = ⋂ i ∈ t, s i}
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
⊢ t1 ∩ t2 ∈ piiUnionInter π S

Tactic:
[

Goal state after:
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
⊢ t1 ∩ t2 ∈ piiUnionInter π S
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
g : ι → Set α := fun n ↦ (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ
hp_union_ss : ↑(p1 ∪ p2) ⊆ S
⊢ (⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x = ⋂ i ∈ p1 ∪ p2, g i

Tactic:
[

Goal state after:
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
hpi : ∀ (x : ι), IsPiSystem (π x)
S : Set ι
t1 : Set α
p1 : Finset ι
hp1S : ↑p1 ⊆ S
f1 : ι → Set α
hf1m : ∀ x ∈ p1, f1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
t2 : Set α
p2 : Finset ι
hp2S : ↑p2 ⊆ S
f2 : ι → Set α
hf2m : ∀ x ∈ p2, f2 x ∈ π x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
h_nonempty : (t1 ∩ t2).Nonempty
g : ι → Set α := fun n ↦ (if n ∈ p1 then f1 n else univ) ∩ if n ∈ p2 then f2 n else univ
hp_union_ss : ↑(p1 ∪ p2) ⊆ S
⊢ (⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x = ⋂ i ∈ p1 ∪ p2, g i
Module: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s = ⨅ t, μ ↑t

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ s = ⨅ t, μ ↑t
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
α : Type u_3
d : DynkinSystem α
s : Set α
h : d.Has s
t : Set α
hts : d.Has (t ∩ s)
this : tᶜ ∩ s = (t ∩ s)ᶜ \ sᶜ
⊢ d.Has (tᶜ ∩ s)

Tactic:
[

Goal state after:
α✝ : Type u_1
β : Type u_2
α : Type u_3
d : DynkinSystem α
s : Set α
h : d.Has s
t : Set α
hts : d.Has (t ∩ s)
this : tᶜ ∩ s = (t ∩ s)ᶜ \ sᶜ
⊢ d.Has (tᶜ ∩ s)
Module: Mathlib.MeasureTheory.PiSystem

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
α : Type u_3
d : DynkinSystem α
s : Set α
h : d.Has s
t : Set α
hts : d.Has (t ∩ s)
this : tᶜ ∩ s = (t ∩ s)ᶜ \ sᶜ
⊢ d.Has (tᶜ ∩ s)

Tactic:
this

Goal state after:
α✝ : Type u_1
β : Type u_2
α : Type u_3
d : DynkinSystem α
s : Set α
h : d.Has s
t : Set α
hts : d.Has (t ∩ s)
this : tᶜ ∩ s = (t ∩ s)ᶜ \ sᶜ
⊢ d.Has ((t ∩ s)ᶜ \ sᶜ)
Module: Mathlib.MeasureTheory.Function.AEMeasurableSequence

Goal state before:
pp: ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : ι → α → β
μ : Measure α
p : α → (ι → β) → Prop
inst✝ : Countable ι
hf : ∀ (i : ι), AEMeasurable (f i) μ
hp : ∀ᵐ (x : α) ∂μ, p x fun n ↦ f n x
hf_eq : ∀ (i : ι), f i =ᵐ[μ] AEMeasurable.mk (f i) ⋯
⊢ μ {x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) ⋯ x) ∧ p x fun n ↦ f n x}ᶜ = 0

Tactic:
[

Goal state after:
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : ι → α → β
μ : Measure α
p : α → (ι → β) → Prop
inst✝ : Countable ι
hf : ∀ (i : ι), AEMeasurable (f i) μ
hp : ∀ᵐ (x : α) ∂μ, p x fun n ↦ f n x
hf_eq : ∀ (i : ι), f i =ᵐ[μ] AEMeasurable.mk (f i) ⋯
⊢ μ {x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) ⋯ x) ∧ p x fun n ↦ f n x}ᶜ = 0
Module: Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated

Goal state before:
pp: α : Type u_1
s t : Set α
⊢ MeasurableSet t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ

Tactic:
[

Goal state after:
α : Type u_1
s t : Set α
⊢ MeasurableSet t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ
Module: Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated

Goal state before:
pp: α : Type u_1
s t : Set α
⊢ (∃ s', MeasurableSet s' ∧ (fun x ↦ x ∈ s) ⁻¹' s' = t) ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ

Tactic:
[

Goal state after:
α : Type u_1
s t : Set α
⊢ (∃ s', MeasurableSet s' ∧ (fun x ↦ x ∈ s) ⁻¹' s' = t) ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
⊢ m.trim s = ⨅ t, m ↑t

Tactic:
by
  simp [iInf_subtype, iInf_and, trim_eq_iInf]

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.Induced

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
⊢ (⨆ i, μ i).trim = ⨆ i, (μ i).trim

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
⊢ (⨆ i, μ i).trim = ⨆ i, (μ i).trim
Module: Mathlib.MeasureTheory.MeasurableSpace.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A ↦ (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
this : Aᶜ = g '' Bᶜ
⊢ ↑Aᶜ ≃ᵐ ↑Bᶜ

Tactic:
this

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A ↦ (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
this : Aᶜ = g '' Bᶜ
⊢ ↑(g '' Bᶜ) ≃ᵐ ↑Bᶜ
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
⊢ map f μ = 0 ↔ μ = 0

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
⊢ map f μ = 0 ↔ μ = 0
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure α
s : Set β
t : Set β := f '' toMeasurable μ (f ⁻¹' s) ∪ (range f)ᶜ
htm : MeasurableSet t
⊢ s ∩ range f ⊆ f '' toMeasurable μ (f ⁻¹' s)

Tactic:
← image_preimage_eq_inter_range

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
m1 : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ : Measure α
s : Set β
t : Set β := f '' toMeasurable μ (f ⁻¹' s) ∪ (range f)ᶜ
htm : MeasurableSet t
⊢ f '' f ⁻¹' s ⊆ f '' toMeasurable μ (f ⁻¹' s)
Module: Mathlib.MeasureTheory.Measure.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α ≃ᵐ β
μ : Measure α
s : Set β
⊢ s ∈ Filter.map (⇑f) (ae μ) ↔ s ∈ ae (map (⇑f) μ)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α ≃ᵐ β
μ : Measure α
s : Set β
⊢ s ∈ Filter.map (⇑f) (ae μ) ↔ s ∈ ae (map (⇑f) μ)
Module: Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving

Goal state before:
pp: case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
e : α ≃ α
he : QuasiMeasurePreserving (⇑e) μ μ
he' : QuasiMeasurePreserving (⇑e.symm) μ μ
hs : ⇑e '' s =ᵐ[μ] s
k : ℕ
⊢ ⇑(e ^ (-↑k)).symm ⁻¹' s =ᵐ[μ] s

Tactic:
rw [zpow_neg, zpow_natCast]
    replace hs : e ⁻¹' s =ᵐ[μ] s := by
      convert! he.preimage_ae_eq hs.symm
      rw [Equiv.preimage_image]
    replace he : (⇑e)^[k] ⁻¹' s =ᵐ[μ] s := he.preimage_iterate_ae_eq k hs
    rwa [Equiv.Perm.iterate_eq_pow e k] at he

Goal state after: