mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
p : HeightOneSpectrum R
ι : Type u_4
inst✝¹ : Finite ι
inst✝ : Nonempty ι
I : ι → Ideal R
H : ∀ (i : ι), FiniteMultiplicity p.asIdeal (I i)
hI : ∀ (i : ι), I i = ⊥
⊢ ∃ i, I i = ⊥

Tactic:
]

Goal state after:
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
p : HeightOneSpectrum R
ι : Type u_4
inst✝¹ : Finite ι
inst✝ : Nonempty ι
I : ι → Ideal R
H : ∀ (i : ι), FiniteMultiplicity p.asIdeal (I i)
hI : ∀ (i : ι), I i = ⊥
⊢ ∃ i, I i = ⊥
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝ : CompleteLattice α
f : Filter β
p q : β → Prop
u : β → α
h : ∀ᶠ (x : β) in f, u x ≠ ⊥ → (p x ↔ q x)
a : α
b : β
hb : u b ≠ ⊥ → (p b ↔ q b)
hu : u b ≠ ⊥
⊢ q b → u b ≤ a ↔ q b → u b ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case a
ι : Type u_6
R : Type u_7
F : Filter ι
inst✝ : CompleteLattice R
a : ι → R
⊢ sInf {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1} ≤ sInf ((fun I ↦ sSup (a '' I)) '' F.sets)

Tactic:
]

Goal state after:
case a
ι : Type u_6
R : Type u_7
F : Filter ι
inst✝ : CompleteLattice R
a : ι → R
⊢ sInf {a_1 | ∀ᶠ (n : ι) in F, a n ≤ a_1} ≤ sInf ((fun I ↦ sSup (a '' I)) '' F.sets)
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_6
β : Type u_7
inst✝ : CompleteLattice β
f : Filter α
u : α → β
x : β
h : ∃ᶠ (a : α) in f, u a ≤ x
⊢ sSup {a | ∀ᶠ (n : α) in f, a ≤ u n} ≤ x

Tactic:
]

Goal state after:
α : Type u_6
β : Type u_7
inst✝ : CompleteLattice β
f : Filter α
u : α → β
x : β
h : ∃ᶠ (a : α) in f, u a ≤ x
⊢ sSup {a | ∀ᶠ (n : α) in f, a ≤ u n} ≤ x
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
K : ℝ≥0
f : α → β
x y : α
r : ℝ≥0∞
h : LipschitzWith K f
hK : K ≠ 0
hr : edist x y < r
⊢ ↑K * edist x y < ↑K * r

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
K : ℝ≥0
f : α → β
x y : α
r : ℝ≥0∞
h : LipschitzWith K f
hK : K ≠ 0
hr : edist x y < r
⊢ ↑K * edist x y < ↑K * r
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
⊢ ⨅ i, f (⨆ i_1, (⇑f)^[i_1 + i] a) = ⨅ n, ⨆ i, (⇑f)^[i + n] a

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
⊢ ⨅ i, f (⨆ i_1, (⇑f)^[i_1 + i] a) = ⨅ n, ⨆ i, (⇑f)^[i + n] a
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
K : ℝ≥0
f : α → β
h : LipschitzWith K f
x y : α
⊢ edist (f x) (f y) / ↑K ≤ edist x y

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
K : ℝ≥0
f : α → β
h : LipschitzWith K f
x y : α
⊢ edist (f x) (f y) / ↑K ≤ edist x y
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case inl
x : ℝ≥0
⊢ ↑↑x = ↑↑x

Tactic:
]

Goal state after:
case inl
x : ℝ≥0
⊢ ↑↑x = ↑↑x
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case inr
x : ℝ
h : x ≤ 0
⊢ ↑x = ↑x

pp: case inr.hr
x : ℝ
h : x ≤ 0
⊢ 0 ≤ -x

Tactic:
]

Goal state after:
case inr.hr
x : ℝ
h : x ≤ 0
⊢ 0 ≤ -x
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
Kf Kg : ℝ≥0
f : β → γ
g : α → β
hf : LipschitzWith Kf f
hg : LipschitzWith Kg g
x y : α
⊢ ↑Kf * ↑Kg * edist x y = ↑Kf * ↑Kg * edist x y

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : BoundedLENhdsClass α
inst✝ : BoundedLENhdsClass β
f : Filter ι
u : ι → α
a✝ : α
x : α × β
a : α
ha : ∀ᶠ (x : α) in 𝓝 x.1, (fun x1 x2 ↦ x1 ≤ x2) x a
b : β
hb : ∀ᶠ (x : β) in 𝓝 x.2, (fun x1 x2 ↦ x1 ≤ x2) x b
⊢ IsBounded (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x.1 ×ˢ 𝓝 x.2)

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : BoundedLENhdsClass α
inst✝ : BoundedLENhdsClass β
f : Filter ι
u : ι → α
a✝ : α
x : α × β
a : α
ha : ∀ᶠ (x : α) in 𝓝 x.1, (fun x1 x2 ↦ x1 ≤ x2) x a
b : β
hb : ∀ᶠ (x : β) in 𝓝 x.2, (fun x1 x2 ↦ x1 ≤ x2) x b
⊢ IsBounded (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x.1 ×ˢ 𝓝 x.2)
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
f : α → β
Kf : ℝ≥0
hf : LipschitzWith Kf f
g : α → γ
Kg : ℝ≥0
hg : LipschitzWith Kg g
x y : α
⊢ max (edist ((fun x ↦ (f x, g x)) x).1 ((fun x ↦ (f x, g x)) y).1)
      (edist ((fun x ↦ (f x, g x)) x).2 ((fun x ↦ (f x, g x)) y).2) ≤
    max (↑Kf * edist x y) (↑Kg * edist x y)

Tactic:
]

Goal state after:
α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
f : α → β
Kf : ℝ≥0
hf : LipschitzWith Kf f
g : α → γ
Kg : ℝ≥0
hg : LipschitzWith Kg g
x y : α
⊢ max (edist ((fun x ↦ (f x, g x)) x).1 ((fun x ↦ (f x, g x)) y).1)
      (edist ((fun x ↦ (f x, g x)) x).2 ((fun x ↦ (f x, g x)) y).2) ≤
    max (↑Kf * edist x y) (↑Kg * edist x y)
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteDistribLattice α
f : Filter β
u : β → α
s : Set β
inst✝ : DecidablePred fun x ↦ x ∈ s
v : β → α
⊢ ((blimsup (s.piecewise u v) f fun x ↦ x ∈ s) ⊔ blimsup (s.piecewise u v) f fun x ↦ x ∉ s) =
    (blimsup u f fun x ↦ x ∈ s) ⊔ blimsup v f fun x ↦ x ∉ s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : CompleteDistribLattice α
f : Filter β
u : β → α
s : Set β
inst✝ : DecidablePred fun x ↦ x ∈ s
v : β → α
⊢ ((blimsup (s.piecewise u v) f fun x ↦ x ∈ s) ⊔ blimsup (s.piecewise u v) f fun x ↦ x ∉ s) =
    (blimsup u f fun x ↦ x ∈ s) ⊔ blimsup v f fun x ↦ x ∉ s
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝⁹ : Preorder α
inst✝⁸ : Preorder β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : BoundedLENhdsClass α
inst✝⁴ : BoundedLENhdsClass β
f : Filter ι
u : ι → α
a : α
inst✝³ : Finite ι
inst✝² : (i : ι) → Preorder (π i)
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), BoundedLENhdsClass (π i)
x : (i : ι) → π i
⊢ IsBounded (fun x1 x2 ↦ x1 ≤ x2) (pi fun i ↦ 𝓝 (x i))

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝⁹ : Preorder α
inst✝⁸ : Preorder β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : BoundedLENhdsClass α
inst✝⁴ : BoundedLENhdsClass β
f : Filter ι
u : ι → α
a : α
inst✝³ : Finite ι
inst✝² : (i : ι) → Preorder (π i)
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), BoundedLENhdsClass (π i)
x : (i : ι) → π i
⊢ IsBounded (fun x1 x2 ↦ x1 ≤ x2) (pi fun i ↦ 𝓝 (x i))
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
K : ℝ≥0
f : α → α
hf : LipschitzWith K f
n : ℕ
⊢ LipschitzWith (K ^ n * K) f^[n + 1]

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
K : ℝ≥0
f : α → α
hf : LipschitzWith K f
n : ℕ
⊢ LipschitzWith (K ^ n * K) f^[n + 1]
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
K : ℝ≥0
f : α → α
hf : LipschitzWith K f
x : α
n : ℕ
⊢ edist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * edist x (f x)

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
K : ℝ≥0
f : α → α
hf : LipschitzWith K f
x : α
n : ℕ
⊢ edist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * edist x (f x)
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : Function.End α
K : ℝ≥0
h : LipschitzWith K f
n : ℕ
⊢ LipschitzWith (K ^ n * K) (f ^ n * f)

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : Function.End α
K : ℝ≥0
h : LipschitzWith K f
n : ℕ
⊢ LipschitzWith (K ^ n * K) (f ^ n * f)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case pos
x y : ℝ
hy : 0 ≤ ↑y
hxy : x ≤ y
⊢ 0 = (↑x).toENNReal - (↑y).toENNReal

Tactic:
]

Goal state after:
case pos
x y : ℝ
hy : 0 ≤ ↑y
hxy : x ≤ y
⊢ 0 = (↑x).toENNReal - (↑y).toENNReal
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case neg
x y : ℝ
hy : 0 ≤ ↑y
hxy : ¬x ≤ y
⊢ ENNReal.ofReal x - ENNReal.ofReal y = (↑x).toENNReal - (↑y).toENNReal

Tactic:
]

Goal state after:
case neg
x y : ℝ
hy : 0 ≤ ↑y
hxy : ¬x ≤ y
⊢ ENNReal.ofReal x - ENNReal.ofReal y = (↑x).toENNReal - (↑y).toENNReal
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c d : EReal
h1 : c ≠ ⊥ ∨ d ≠ ⊤
h2 : c ≠ ⊤ ∨ d ≠ ⊥
⊢ a + b + (-c + -d) = a + -c + (b + -d)

Tactic:
]

Goal state after:
a b c d : EReal
h1 : c ≠ ⊥ ∨ d ≠ ⊤
h2 : c ≠ ⊤ ∨ d ≠ ⊥
⊢ a + b + (-c + -d) = a + -c + (b + -d)
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : CompleteDistribLattice α
f : Filter β
u : β → α
a : α
⊢ ⨆ i ∈ f, (⨅ a ∈ i, u a) ⊔ a = ⨆ s ∈ f, ⨅ a_1 ∈ s, a ⊔ u a_1

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : CompleteDistribLattice α
f : Filter β
u : β → α
a : α
⊢ ⨆ i ∈ f, (⨅ a ∈ i, u a) ⊔ a = ⨆ s ∈ f, ⨅ a_1 ∈ s, a ⊔ u a_1
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
s : Set α
f : α → β
g : α → γ
Kf Kg : ℝ≥0
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
x✝ : α
hx : x✝ ∈ s
y✝ : α
hy : y✝ ∈ s
⊢ max (edist ((fun x ↦ (f x, g x)) x✝).1 ((fun x ↦ (f x, g x)) y✝).1)
      (edist ((fun x ↦ (f x, g x)) x✝).2 ((fun x ↦ (f x, g x)) y✝).2) ≤
    max (↑Kf * edist x✝ y✝) (↑Kg * edist x✝ y✝)

Tactic:
]

Goal state after:
α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
s : Set α
f : α → β
g : α → γ
Kf Kg : ℝ≥0
hf : LipschitzOnWith Kf f s
hg : LipschitzOnWith Kg g s
x✝ : α
hx : x✝ ∈ s
y✝ : α
hy : y✝ ∈ s
⊢ max (edist ((fun x ↦ (f x, g x)) x✝).1 ((fun x ↦ (f x, g x)) y✝).1)
      (edist ((fun x ↦ (f x, g x)) x✝).2 ((fun x ↦ (f x, g x)) y✝).2) ≤
    max (↑Kf * edist x✝ y✝) (↑Kg * edist x✝ y✝)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a : EReal
b : ℝ
⊢ a = a

Tactic:
]

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

Goal state before:
pp: a : EReal
b : ℝ
⊢ -a = -a

Tactic:
]

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

Goal state before:
pp: case coe
a c : EReal
b : ℝ
hb : ↑b ≠ ⊥ ∨ c ≠ ⊥
ht : ↑b ≠ ⊤ ∨ c ≠ ⊤
⊢ a + ↑b ≤ c ↔ a + ↑b ≤ c

Tactic:
]

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
f : Filter β
inst✝ : CountableInterFilter f
u : β → α
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ ∀ (b : α), limsup u f < b → ∀ᶠ (x : β) in f, u x < b

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
f : Filter β
inst✝ : CountableInterFilter f
u : β → α
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ ∀ (b : α), limsup u f < b → ∀ᶠ (x : β) in f, u x < b
Module: Mathlib.Topology.Order.LiminfLimsup

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : CompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : FirstCountableTopology α
inst✝¹ : OrderTopology α
f : Filter β
inst✝ : CountableInterFilter f
u : β → α
h : u =ᶠ[f] ⊥
⊢ limsup ⊥ f = ⊥

Tactic:
]

Goal state after:
α : Type u_2
β : Type u_3
inst✝⁴ : CompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : FirstCountableTopology α
inst✝¹ : OrderTopology α
f : Filter β
inst✝ : CountableInterFilter f
u : β → α
h : u =ᶠ[f] ⊥
⊢ limsup ⊥ f = ⊥
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f : α → β
hf : LocallyLipschitz f
⊢ ∀ (x : α), ContinuousAt f x

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f : α → β
hf : LocallyLipschitz f
⊢ ∀ (x : α), ContinuousAt f x
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
a : α
⊢ ⨅ i ∈ f, (⨆ a ∈ i, u a) ⊓ aᶜ = ⨅ s ∈ f, ⨆ a_1 ∈ s, u a_1 ⊓ aᶜ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
a : α
⊢ ⨅ i ∈ f, (⨆ a ∈ i, u a) ⊓ aᶜ = ⨅ s ∈ f, ⨆ a_1 ∈ s, u a_1 ⊓ aᶜ
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
inst✝ : f.NeBot
a : α
⊢ (a \ limsup u f)ᶜ = (liminf (fun b ↦ a \ u b) f)ᶜ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
inst✝ : f.NeBot
a : α
⊢ (a \ limsup u f)ᶜ = (liminf (fun b ↦ a \ u b) f)ᶜ
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
a : α
⊢ (a \ liminf u f)ᶜ = (limsup (fun b ↦ a \ u b) f)ᶜ

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
a : α
⊢ (a \ liminf u f)ᶜ = (limsup (fun b ↦ a \ u b) f)ᶜ
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : α → α
hf : LocallyLipschitz f
n : ℕ
⊢ LocallyLipschitz (f^[n] ∘ f)

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : α → α
hf : LocallyLipschitz f
n : ℕ
⊢ LocallyLipschitz (f^[n] ∘ f)
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 * f)

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : Function.End α
h : LocallyLipschitz f
n : ℕ
⊢ LocallyLipschitz (f ^ n * f)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case bot
a b : EReal
h : a = ⊥
⊢ a - ⊥ ≤ b

Tactic:
]

Goal state after:
case bot
a b : EReal
h : a = ⊥
⊢ a - ⊥ ≤ b
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

Goal state after:
a b c : EReal
h : a ≤ b - c
⊢ a + - -c ≤ b
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
⊢ ∀ᶠ (x_1 : α × β) in 𝓝[s ×ˢ t] (x, y), f x_1 ∈ Metric.closedEBall (f (x, y)) ε

Tactic:
]

Goal state after:
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
⊢ ∀ᶠ (x_1 : α × β) in 𝓝[s ×ˢ t] (x, y), f x_1 ∈ Metric.closedEBall (f (x, y)) ε
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

Goal state after:
a b c : EReal
h : a < b + c
⊢ a < b + c
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 < y + (t - z)

Tactic:
]

Goal state after:
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
⊢ x < y + (t - z)
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)ᶜ = {x | {n | p n ∧ x ∉ s n}.Finite}ᶜ

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_4
p : ι → Prop
s : ι → Set α
⊢ (bliminf s cofinite p)ᶜ = {x | {n | p n ∧ x ∉ s n}.Finite}ᶜ
Module: Mathlib.Data.EReal.Operations

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

pp: case pos
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : y = ⊤
⊢ t - z ≠ ⊥

Tactic:
]

Goal state after:
case pos
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : y = ⊤
⊢ x < ⊤

case pos
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : y = ⊤
⊢ t - z ≠ ⊥
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 ∈ ⨅ s ∈ l, ⨆ b, ⨆ (_ : p b ∧ b ∈ s), u b
⊢ ∃ f, ∀ (i : ↑{i | q i}), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b ↑i

Tactic:
]

Goal state after:
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 ∈ ⨅ s ∈ l, ⨆ b, ⨆ (_ : p b ∧ b ∈ s), u b
⊢ ∃ f, ∀ (i : ↑{i | q i}), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b ↑i
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

Goal state after:
case pos
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : ¬y = ⊤
hxy : x = y
⊢ y < y + (t - z)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case neg
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 = ⊤
⊢ ↑y < ↑y + ↑(t - z).toReal

Tactic:
]

Goal state after:
case neg
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 = ⊤
⊢ ↑y < ↑y + ↑(t - z).toReal
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case neg
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : ¬y = ⊤
hxy : ¬x = y
⊢ x + 0 < y + (t - z)

Tactic:
]

Goal state after:
case neg
x y z t : EReal
h : x ≤ y
h' : z < t
hx_top : x ≠ ⊤
hy_bot : y ≠ ⊥
hy_top : ¬y = ⊤
hxy : ¬x = y
⊢ x + 0 < y + (t - z)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : EReal
h : ∀ (z : ℝ), x < ↑z → y ≤ ↑z
⊢ (x < ↑⊤ → y ≤ ↑⊤) ∧ ∀ (x_1 : ℝ), x < ↑↑x_1 → y ≤ ↑↑x_1

Tactic:
]

Goal state after:
x y : EReal
h : ∀ (z : ℝ), x < ↑z → y ≤ ↑z
⊢ (x < ↑⊤ → y ≤ ↑⊤) ∧ ∀ (x_1 : ℝ), x < ↑↑x_1 → y ≤ ↑↑x_1
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : EReal
h : ∀ (z : ℝ), ↑z < y → ↑z ≤ x
⊢ (↑⊤ < y → ↑⊤ ≤ x) ∧ ∀ (x_1 : ℝ), ↑↑x_1 < y → ↑↑x_1 ≤ x

Tactic:
]

Goal state after:
x y : EReal
h : ∀ (z : ℝ), ↑z < y → ↑z ≤ x
⊢ (↑⊤ < y → ↑⊤ ≤ x) ∧ ∀ (x_1 : ℝ), ↑↑x_1 < y → ↑↑x_1 ≤ x
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case pos
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
h : ∀ y > x, ∀ᶠ (a : α) in f, u a < y
h' : ∀ y > x, ∃ z, x < z ∧ z < y
⊢ ∀ ⦃c : β⦄, x < c → limsup u f < c

Tactic:
]

Goal state after:
case pos
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
h : ∀ y > x, ∀ᶠ (a : α) in f, u a < y
h' : ∀ y > x, ∃ z, x < z ∧ z < y
⊢ ∀ ⦃c : β⦄, x < c → limsup u f < c
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'
⊢ -a - b ≤ -c

Tactic:
]

Goal state after:
a b c : EReal
h₁ : a ≠ ⊥ ∨ b ≠ ⊤
h₂ : a ≠ ⊤ ∨ b ≠ ⊥
h : ∀ a' > a, ∀ b' > b, c ≤ a' + b'
⊢ -a - b ≤ -c
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:
rw [← neg_le_neg_iff, neg_add h₁ h₂]

Goal state after:
a b c : EReal
h₁ : a ≠ ⊥ ∨ b ≠ ⊤
h₂ : a ≠ ⊤ ∨ b ≠ ⊥
h : ∀ a' > a, ∀ b' > b, c ≤ a' + b'
⊢ -a - b ≤ -c
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'
a' : EReal
ha' : a' < -a
b' : EReal
hb' : b' < -b
⊢ c ≤ -a' - b'

Tactic:
]

Goal state after:
a b c : EReal
h₁ : a ≠ ⊥ ∨ b ≠ ⊤
h₂ : a ≠ ⊤ ∨ b ≠ ⊥
h : ∀ a' > a, ∀ b' > b, c ≤ a' + b'
a' : EReal
ha' : a' < -a
b' : EReal
hb' : b' < -b
⊢ c ≤ -a' - b'
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

Goal state after:
x : EReal
h : 0 < x
⊢ x * ⊤ = ⊤
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

Goal state after:
x : EReal
h : x < 0
⊢ x * ⊤ = ⊥
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

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

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

Tactic:
]

Goal state after:
x : EReal
h : 0 < x
⊢ x * ⊥ = ⊥
Module: Mathlib.Data.EReal.Operations

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

Tactic:
]

Goal state after:
x : EReal
h : x < 0
⊢ x * ⊥ = ⊤
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
⊢ (∀ y > x, ∀ᶠ (a : α) in f, u a ≤ y) → ∀ ⦃c : β⦄, x < c → limsup u f < c

Tactic:
]

Goal state after:
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
⊢ (∀ y > x, ∀ᶠ (a : α) in f, u a ≤ y) → ∀ ⦃c : β⦄, x < c → limsup u f < c
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case symm
x✝ y✝ : EReal
h : (x✝ * y✝).toReal = x✝.toReal * y✝.toReal
⊢ (x✝ * y✝).toReal = x✝.toReal * y✝.toReal

Tactic:
]

Goal state after:
case symm
x✝ y✝ : EReal
h : (x✝ * y✝).toReal = x✝.toReal * y✝.toReal
⊢ (x✝ * y✝).toReal = x✝.toReal * y✝.toReal
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case pos
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
h : ∀ y < x, ∃ᶠ (a : α) in f, y < u a
h' : ∀ y < x, ∃ z, y < z ∧ z < x
⊢ ∀ ⦃c : β⦄, c < x → c < limsup u f

Tactic:
]

Goal state after:
case pos
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
h : ∀ y < x, ∃ᶠ (a : α) in f, y < u a
h' : ∀ y < x, ∃ z, y < z ∧ z < x
⊢ ∀ ⦃c : β⦄, c < x → c < limsup u f
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
⊢ ⨆ j, ⨆ i, ⨆ (_ : j ∈ decode₂ β i), f j = ⨆ b, f b

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Encodable β
inst✝ : CompleteLattice α
f : β → α
⊢ ⨆ j, ⨆ i, ⨆ (_ : j ∈ decode₂ β i), f j = ⨆ b, f b