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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : Function.End α
h : LocallyLipschitz f
⊢ LocallyLipschitz (f ^ 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : Function.End α
h : LocallyLipschitz f
n : ℕ
⊢ LocallyLipschitz (f ^ (n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h : a ≤ b + c
⊢ a - c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h : a ≤ b - c
⊢ a + c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
⊢ b' < b

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h : a < b - c
⊢ a + c < b

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h : a < b + c
⊢ a < b - -c

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h : a < b + c
⊢ a < c + b

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
p : ι → Prop
s : ι → Set α
x : α
h : x ∉ {x | {n | p n ∧ x ∈ s n}.Infinite}
⊢ {x_1 | p x_1 ∧ ¬s x_1 ⊆ {x}ᶜ}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
p : ι → Prop
s : ι → Set α
x : α
h : x ∉ {x | {n | p n ∧ x ∈ s n}.Infinite}
⊢ x ∉ {x}ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
p : ι → Prop
s : ι → Set α
⊢ blimsup s cofinite p = {x | {n | p n ∧ x ∈ s n}.Infinite}

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
p : ι → Prop
s : ι → Set α
⊢ bliminf s cofinite p = {x | {n | p n ∧ x ∉ s n}.Finite}

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
s : ι → Set α
⊢ limsup s cofinite = {x | {n | x ∈ s n}.Infinite}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
hb'b : b' < b
⊢ ∀ᶠ (r : α) in atTop, limsup (fun x ↦ u x r) cofinite ≤ b'

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
s : ι → Set α
⊢ liminf s cofinite = {x | {n | x ∉ s n}.Finite}

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : y = ⊤
⊢ 0 ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
hb'b : b' < b
this : ∀ᶠ (r : α) in atTop, limsup (fun x ↦ u x r) cofinite ≤ b'
⊢ ∃ r, ∀ s ≥ r, limsup (fun x ↦ u x s) cofinite ≤ b'

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
l : Filter β
b : ι → Set β
q : ι → Prop
hl : l.HasBasis q b
u : β → Set α
p : β → Prop
x : α
hx : x ∈ blimsup u l p
⊢ ∃ f, ∀ (i : ↑{i | q i}), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b ↑i

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
l : Filter β
b : ι → Set β
hl : l.HasBasis (fun x ↦ True) b
u : β → Set α
p : β → Prop
x : α
hx : x ∈ blimsup u l p
⊢ ∃ f, ∀ (i : ι), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b i

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder α
a : α
hf : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
h : a < f.limsSup
⊢ ∃ᶠ (n : α) in f, a < n

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
⊢ ∃ c, ∃ (_ : c ∈ {c | ∀ᶠ (n : α) in f, c ≤ u n}), b < c

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
⊢ ∀ᶠ (a : α) in f, b < u a

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : AddZeroClass α
inst✝ : AddLeftStrictMono α
x ε : α
u : ℕ → α
hu_bdd : IsBoundedUnder LE.le atTop u
hu : limsup u atTop ≤ x
hε : 0 < ε
⊢ ∃ n, u ↑n < x + ε

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x z t : EReal
h' : z < t
hx_top : x ≠ ⊤
y : ℝ
h : x ≤ ↑y
hy_bot : ↑y ≠ ⊥
hy_top : ¬↑y = ⊤
hxy : x = ↑y
htz_top : ¬t - z = ⊤
⊢ 0 ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : AddZeroClass α
inst✝ : AddLeftStrictMono α
x ε : α
u : ℕ → α
hu_bdd : IsBoundedUnder GE.ge atTop u
hu : x ≤ liminf u atTop
hε : ε < 0
⊢ ∃ n, x + ε < u ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : ¬y = ⊤
hxy : ¬x = y
⊢ x < y

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
⊢ x - t < y - z

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
b : β
hu : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h : b < limsup u f
⊢ ∃ᶠ (x : α) in f, b < u x

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : EReal
⊢ (∀ (z : ℝ), x < ↑z → y ≤ ↑z) ↔ y ≤ x

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : EReal
⊢ (∀ (z : ℝ), ↑z < y → ↑z ≤ x) ↔ y ≤ x

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
hc : c < a + b
⊢ ∃ a' < a, c < a' + b

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
hc : c < a + b
⊢ ∃ b' < b, c < a + b'

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h : ∀ a' < a, ∀ b' < b, a' + b' ≤ c
⊢ a + b ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c : EReal
h₁ : a ≠ ⊥ ∨ b ≠ ⊤
h₂ : a ≠ ⊤ ∨ b ≠ ⊥
h : ∀ a' > a, ∀ b' > b, c ≤ a' + b'
⊢ c ≤ a + b

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : ℝ≥0∞
hy_top : y ≠ ∞
h_le : y ≤ x
⊢ ↑(x - y) = ↑x - ↑y

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x : EReal
h : 0 < x
⊢ ⊤ * x = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x : EReal
h : x < 0
⊢ ⊤ * x = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x : ℝ≥0∞
hx : x ≠ 0
⊢ ↑x * ⊤ = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
x : β
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ limsup u f ≤ x ↔ ∀ y > x, ∀ᶠ (a : α) in f, u a < y

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x : EReal
h : 0 < x
⊢ ⊥ * x = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < ↑δ
hδ : ↑δ * ↑(3 * K) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : edist x x' < ↑δ
A : s ∩ Metric.eball x ↑δ ∈ 𝓝[s] x
B : t ∩ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x ↑δ
hbt : (a, b).2 ∈ t
hby : (a, b).2 ∈ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2}
⊢ edist (f (a, b)) (f (x', b)) + edist (f (x', b)) (f (x', y)) + edist (f (x', y)) (f (x, y)) ≤
    ↑K * (↑δ + ↑δ) + ε / 2 + ↑K * ↑δ

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x : EReal
h : x < 0
⊢ ⊥ * x = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
inst✝ : DenselyOrdered β
x : β
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ limsup u f ≤ x ↔ ∀ y > x, ∀ᶠ (a : α) in f, u a ≤ y

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : EReal
⊢ (x * y).toReal = x.toReal * y.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: case inr
ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
hb'b : b' < b
this : ∀ᶠ (r : α) in atTop, limsup (fun x ↦ u x r) cofinite ≤ b'
r : α
hr : ∀ s ≥ r, limsup (fun x ↦ u x s) cofinite ≤ b'
b'' : β
hb'b'' : b' < b''
hb''b : b'' < b
⊢ ∃ b'' ∈ Set.Ico b' b, ∀ᶠ (n : ι) in cofinite, u n r ≤ b''

Tactic:
· refine ⟨b'', ⟨hb'b''.le, hb''b⟩ , ?_⟩
      have h_lt := eventually_lt_of_limsup_lt ((hr r le_rfl).trans_lt hb'b'')
      filter_upwards [h_lt] with n hn using hn.le

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
hb'b : b' < b
this : ∀ᶠ (r : α) in atTop, limsup (fun x ↦ u x r) cofinite ≤ b'
r : α
hr : ∀ s ≥ r, limsup (fun x ↦ u x s) cofinite ≤ b'
⊢ ∃ b'' ∈ Set.Ico b' b, ∀ᶠ (n : ι) in cofinite, u n r ≤ b''

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < ↑δ
hδ : ↑δ * ↑(3 * K) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : edist x x' < ↑δ
A : s ∩ Metric.eball x ↑δ ∈ 𝓝[s] x
B : t ∩ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x ↑δ
hbt : (a, b).2 ∈ t
hby : (a, b).2 ∈ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2}
⊢ ↑K * (↑δ + ↑δ) + ε / 2 + ↑K * ↑δ = ↑δ * ↑(3 * K) + ε / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < ↑δ
hδ : ↑δ * ↑(3 * K) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : edist x x' < ↑δ
A : s ∩ Metric.eball x ↑δ ∈ 𝓝[s] x
B : t ∩ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x ↑δ
hbt : (a, b).2 ∈ t
hby : (a, b).2 ∈ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2}
⊢ ↑δ * ↑(3 * K) + ε / 2 ≤ ε / 2 + ε / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
hb'b : b' < b
this✝ : ∀ᶠ (r : α) in atTop, limsup (fun x ↦ u x r) cofinite ≤ b'
r : α
hr : ∀ s ≥ r, limsup (fun x ↦ u x s) cofinite ≤ b'
b'' : β
hb''b : b'' ∈ Set.Ico b' b
hb'' : ∀ᶠ (n : ι) in cofinite, u n r ≤ b''
n : ι
this : ∀ᶠ (r : α) in atTop, u n r ≤ b'
⊢ ∃ r, ∀ s ≥ r, u n s ≤ b''

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
⊢ ContinuousOn f (s ×ˢ t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
K : ℝ≥0
s : Set α
hs : Dense s
ha : ∀ a ∈ s, Continuous[inst✝¹, PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦ f (a, y)
hb : ∀ (b : β), LipschitzWith K fun x ↦ f (x, b)
⊢ Continuous[instTopologicalSpaceProd, PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] f

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Encodable.Lattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Encodable β
inst✝ : CompleteLattice α
f : β → α
⊢ ⨆ i, ⨆ b ∈ decode₂ β i, f b = ⨆ b, f b

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Encodable.Lattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
C : Set α → Prop
H0 : C ∅
H1 : ∀ (b : β), C (f b)
n : ℕ
⊢ C (⋃ b ∈ none, f b)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Encodable.Lattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
C : Set α → Prop
H0 : C ∅
H1 : ∀ (b : β), C (f b)
n : ℕ
b : β
⊢ C (⋃ b_1 ∈ some b, f b_1)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Encodable.Lattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
hd : Pairwise (Disjoint on f)
i j : ℕ
ij : i ≠ j
x : α
this : ∀ (a : β), encode a = i → x ∈ f a → ∀ (b : β), encode b = j → x ∉ f b
⊢ x ∈ (fun i ↦ ⋃ b ∈ decode₂ β i, f b) i → x ∉ (fun i ↦ ⋃ b ∈ decode₂ β i, f b) j

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Encodable.Lattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
hd : Pairwise (Disjoint on f)
⊢ Pairwise (Disjoint on fun i ↦ ⋃ b ∈ decode₂ β i, f b)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
⊢ atTop = ⨅ x, 𝓟 (Ici {x})

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝ : Preorder β
f : β → Finset α
h : Monotone f
h' : ∀ (x : α), ∃ n, x ∈ f n
⊢ Tendsto f atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: β : Type u_4
γ : Type u_5
inst✝ : DecidableEq β
i : β → γ
j : γ → β
h : Function.LeftInverse j i
a : β
⊢ a ∈ Finset.image j {i a}

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
s : Set (Finset α)
t : Finset α
H : ∀ (b : Finset α), t ⊆ b → b ∈ s
b : Finset (α ⊕ β)
hb : t.disjSum ∅ ⊆ b
⊢ t ⊆ b.toLeft

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
⊢ Tendsto Finset.toLeft atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: ⊢ ∀ {a b : EReal}, a * b = 0 → a = 0 ∨ b = 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
x : β
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ x ≤ limsup u f ↔ ∀ y < x, ∃ᶠ (a : α) in f, y < u a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
s : Set (Finset β)
t : Finset β
H : ∀ (b : Finset β), t ⊆ b → b ∈ s
b : Finset (α ⊕ β)
hb : ∅.disjSum t ⊆ b
⊢ t ⊆ b.toRight

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
⊢ Tendsto Finset.toRight atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
⊢ Tendsto Finset.powerset atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrderBot α
h✝ : Nonempty α
h : IsDirectedOrder α
s : Finset α
a : α
ha : ∀ i ∈ s, i ≤ a
b : α
hb : a ≤ b
c : α
hc : c ∈ s
⊢ c ∈ Finset.Iic b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrderBot α
⊢ Tendsto Finset.Iic atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
i : α
⊢ ∀ᶠ (s : Finset α) in atTop, i ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrder β
f : Filter α
u : α → β
inst✝ : DenselyOrdered β
x : β
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ x ≤ limsup u f ↔ ∀ y < x, ∃ᶠ (a : α) in f, y ≤ u a

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
inst✝ : Infinite α
⊢ map Finset.card atTop = atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
inst✝ : Fintype α
⊢ map Finset.card atTop = pure (Fintype.card α)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
inst✝ : Infinite α
⊢ Tendsto Finset.card atTop atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
hb'b : b' < b
this : ∀ᶠ (r : α) in atTop, limsup (fun x ↦ u x r) cofinite ≤ b'
r : α
hr : ∀ s ≥ r, limsup (fun x ↦ u x s) cofinite ≤ b'
b'' : β
hb''b : b'' ∈ Set.Ico b' b
hb'' : ∀ᶠ (n : ι) in cofinite, u n r ≤ b''
n : ι
⊢ ∃ r, ∀ s ≥ r, u n s ≤ b''

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
inst✝ : Fintype α
⊢ Tendsto Finset.card atTop (pure (Fintype.card α))

Tactic:
by

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b : EReal
⊢ 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝ : Infinite α
f : ℕ → β
l : Filter β
⊢ Tendsto (fun s ↦ f s.card) atTop l ↔ Tendsto f atTop l

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Finset

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝ : Fintype α
f : ℕ → β
l : Filter β
⊢ Tendsto (fun s ↦ f s.card) atTop l ↔ Tendsto f (pure (Fintype.card α)) l

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u v : α → β
h : ∃ᶠ (x : α) in f, u x ≤ v x
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h✝ : f.NeBot
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.SummationFilter

Goal state before:
pp: β : Type u_2
L : SummationFilter β
⊢ L.NeBot ∨ L.filter = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLinearOrder β
f : Filter α
u v : α → β
h : ∃ᶠ (x : α) in f, u x ≤ v x
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
h✝ : f.NeBot
h₃ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.SummationFilter

Goal state before:
pp: β : Type u_2
L : SummationFilter β
c : Set β
hL : ∀ (t' : Set β), (∀ᶠ (a : Finset β) in L.filter, t' ⊆ ↑a) → t' ⊆ c
x : β
hx : ∀ᶠ (s : Finset β) in L.filter, x ∈ s
⊢ ∀ᶠ (a : Finset β) in L.filter, {x} ⊆ ↑a

Tactic:
by

Goal state after: