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

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
H : t.Countable
x : α
hx : x ∈ {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥}
h'x : IsTop x
⊢ x ∈ t ∪ {x | IsTop x}

pp: case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
H : t.Countable
x : α
hx : x ∈ {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥}
h'x : ¬IsTop x
⊢ x ∈ t ∪ {x | IsTop x}

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
H : t.Countable
x : α
hx : x ∈ {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥}
h'x : IsTop x
⊢ x ∈ t ∪ {x | IsTop x}
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
H : t.Countable
x : α
hx : x ∈ {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥}
h'x : ¬IsTop x
⊢ x ∈ t ∪ {x | IsTop x}

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
H : t.Countable
x : α
hx : x ∈ {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥}
h'x : ¬IsTop x
⊢ x ∈ t ∪ {x | IsTop x}
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: case inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
h : ∀ ⦃a : α⦄, a ∈ t → ∀ ⦃b : α⦄, b ∈ t → a ≠ b → a < b → Disjoint (Ioo a (y a)) (Ioo b (y b))
H : b ≤ a
⊢ Disjoint (Ioo a (y a)) (Ioo b (y b))

pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
H : a < b
⊢ Disjoint (Ioo a (y a)) (Ioo b (y b))

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
t : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}
y : α → α
hy : ∀ x ∈ t, y x > x
h'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a ≠ b
h : ∀ ⦃a : α⦄, a ∈ t → ∀ ⦃b : α⦄, b ∈ t → a ≠ b → a < b → Disjoint (Ioo a (y a)) (Ioo b (y b))
H : b ≤ a
⊢ Disjoint (Ioo a (y a)) (Ioo b (y b))
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝⁵ : Semifield 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
inst✝ : ContinuousMul 𝕜
x : 𝕜
hx : x < 1
⊢ ∀ᶠ (b : 𝕜) in 𝓝 1, x < b⁻¹

pp: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝⁵ : Semifield 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
inst✝ : ContinuousMul 𝕜
x : 𝕜
hx : x > 1
⊢ ∀ᶠ (b : 𝕜) in 𝓝 1, b⁻¹ < x

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
α : Type u_2
inst✝⁵ : Semifield 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
inst✝ : ContinuousMul 𝕜
x : 𝕜
hx : x < 1
⊢ ∀ᶠ (b : 𝕜) in 𝓝 1, x < b⁻¹
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: case mp
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioc a u ⊆ s

pp: case mpr
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioc a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioo a u ⊆ s) → ∃ u ∈ Ioi a, Ioc a u ⊆ s
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: case mpr
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioc a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : NoMaxOrder α
inst✝ : DenselyOrdered α
a : α
s : Set α
⊢ (∃ u ∈ Ioi a, Ioc a u ⊆ s) → ∃ u ∈ Ioi a, Ioo a u ⊆ s
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝⁵ : Semifield 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
inst✝ : ContinuousMul 𝕜
x : 𝕜
hx : x > 1
⊢ ∀ᶠ (b : 𝕜) in 𝓝 1, b⁻¹ < x

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
α : Type u_2
inst✝⁵ : Semifield 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : TopologicalSpace 𝕜
inst✝¹ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
inst✝ : ContinuousMul 𝕜
x : 𝕜
hx : x > 1
⊢ ∀ᶠ (b : 𝕜) in 𝓝 1, b⁻¹ < x
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a b : α
hab : a < b
s : Set α
tfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a
tfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a
tfae_1_iff_5 : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_4_to_5 : (∃ u ∈ Ioc a b, Ico a u ⊆ s) → ∃ u ∈ Ioi a, Ico a u ⊆ s
tfae_5_to_4 : (∃ u ∈ Ioi a, Ico a u ⊆ s) → ∃ u ∈ Ioc a b, Ico a u ⊆ s
⊢ [s ∈ 𝓝[≥] a, s ∈ 𝓝[Icc a b] a, s ∈ 𝓝[Ico a b] a, ∃ u ∈ Ioc a b, Ico a u ⊆ s, ∃ u ∈ Ioi a, Ico a u ⊆ s].TFAE

Tactic:
tfae_finish

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

Goal state before:
pp: case refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : CommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedMonoid α
inst✝ : OrderTopology α
a : α
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) = ⨅ x, ⨅ (_ : x > 1), 𝓟 {a_1 | a / a_1 < x}

pp: case refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : CommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedMonoid α
inst✝ : OrderTopology α
a : α
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) = ⨅ x, ⨅ (_ : x > 1), 𝓟 {a_1 | x⁻¹ < a / a_1}

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : CommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedMonoid α
inst✝ : OrderTopology α
a : α
⊢ ⨅ b ∈ Iio a, 𝓟 (Ioi b) = ⨅ x, ⨅ (_ : x > 1), 𝓟 {a_1 | a / a_1 < x}
Module: Mathlib.Topology.Order.LeftRightNhds

Goal state before:
pp: case refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : CommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedMonoid α
inst✝ : OrderTopology α
a : α
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) = ⨅ x, ⨅ (_ : x > 1), 𝓟 {a_1 | x⁻¹ < a / a_1}

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : CommGroup α
inst✝² : LinearOrder α
inst✝¹ : IsOrderedMonoid α
inst✝ : OrderTopology α
a : α
⊢ ⨅ b ∈ Ioi a, 𝓟 (Iio b) = ⨅ x, ⨅ (_ : x > 1), 𝓟 {a_1 | x⁻¹ < a / a_1}
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case hgf
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0)
⊢ ∀ᶠ (b : α) in l, 0 ≤ (abs ∘ fun x ↦ f x * g x) b

pp: case hfh
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0)
⊢ ∀ᶠ (b : α) in l, (abs ∘ fun x ↦ f x * g x) b ≤ |C * g b|

Tactic:
·

Goal state after:
case hgf
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0)
⊢ ∀ᶠ (b : α) in l, 0 ≤ (abs ∘ fun x ↦ f x * g x) b
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case hfh
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0)
⊢ ∀ᶠ (b : α) in l, (abs ∘ fun x ↦ f x * g x) b ≤ |C * g b|

Tactic:
·

Goal state after:
case hfh
𝕜 : Type u_1
α : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x ↦ |C * g x|) l (𝓝 0)
⊢ ∀ᶠ (b : α) in l, (abs ∘ fun x ↦ f x * g x) b ≤ |C * g b|
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
t : Set β
e' : PartialEquiv α β
inst✝¹ : (i : α) → Decidable (i ∈ s)
inst✝ : (i : β) → Decidable (i ∈ t)
h : e.IsImage s t
h' : e'.IsImage s t
x : α
he : x ∈ e.source
hs : x ∈ s
⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewise (↑e) (↑e') x) = x

pp: case inr
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
t : Set β
e' : PartialEquiv α β
inst✝¹ : (i : α) → Decidable (i ∈ s)
inst✝ : (i : β) → Decidable (i ∈ t)
h : e.IsImage s t
h' : e'.IsImage s t
x : α
he : x ∈ e'.source
hs : ¬x ∈ s
⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewise (↑e) (↑e') x) = x

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
t : Set β
e' : PartialEquiv α β
inst✝¹ : (i : α) → Decidable (i ∈ s)
inst✝ : (i : β) → Decidable (i ∈ t)
h : e.IsImage s t
h' : e'.IsImage s t
x : α
he : x ∈ e.source
hs : x ∈ s
⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewise (↑e) (↑e') x) = x
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
t : Set β
e' : PartialEquiv α β
inst✝¹ : (i : α) → Decidable (i ∈ s)
inst✝ : (i : β) → Decidable (i ∈ t)
h : e.IsImage s t
h' : e'.IsImage s t
x : α
he : x ∈ e'.source
hs : ¬x ∈ s
⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewise (↑e) (↑e') x) = x

Tactic:
·

Goal state after:
case inr
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
t : Set β
e' : PartialEquiv α β
inst✝¹ : (i : α) → Decidable (i ∈ s)
inst✝ : (i : β) → Decidable (i ∈ t)
h : e.IsImage s t
h' : e'.IsImage s t
x : α
he : x ∈ e'.source
hs : ¬x ∈ s
⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewise (↑e) (↑e') x) = x
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
c d : 𝕜
⊢ Tendsto (fun x ↦ c * x ^ 0) atTop (𝓝 d) ↔ (c = 0 ∨ 0 = 0) ∧ c = d

pp: case inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d

Tactic:
·

Goal state after:
case inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
c d : 𝕜
⊢ Tendsto (fun x ↦ c * x ^ 0) atTop (𝓝 d) ↔ (c = 0 ∨ 0 = 0) ∧ c = d
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : c < 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d

pp: case inr.inr.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x ↦ 0 * x ^ n) atTop (𝓝 d) ↔ (0 = 0 ∨ n = 0) ∧ 0 = d

pp: case inr.inr.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d

Tactic:
·

Goal state after:
case inr.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : c < 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case inr.inr.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x ↦ 0 * x ^ n) atTop (𝓝 d) ↔ (0 = 0 ∨ n = 0) ∧ 0 = d

pp: case inr.inr.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d

Tactic:
·

Goal state after:
case inr.inr.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
d : 𝕜
hn : n ≠ 0
⊢ Tendsto (fun x ↦ 0 * x ^ n) atTop (𝓝 d) ↔ (0 = 0 ∨ n = 0) ∧ 0 = d
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case inr.inr.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d

Tactic:
·

Goal state after:
case inr.inr.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℕ
c d : 𝕜
hn : n ≠ 0
hc : 0 < c
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)
⊢ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0

pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)
⊢ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)

Tactic:
·

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_2.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)

pp: case refine_2.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)

Tactic:
·

Goal state after:
case refine_2.inl
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: case refine_2.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)

Tactic:
·

Goal state after:
case refine_2.inr
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n < 0 ∧ d = 0
⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d)
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case refine_1
α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
⊢ {s | ∃ a, s = Ioi a ∨ s = Iio a}.Countable

pp: case refine_2
α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
⊢ ts = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a}

Tactic:
·

Goal state after:
case refine_1
α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
⊢ {s | ∃ a, s = Ioi a ∨ s = Iio a}.Countable
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case refine_2
α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
⊢ ts = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a}

Tactic:
·

Goal state after:
case refine_2
α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
⊢ ts = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a}
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case left
α : Type u_1
β : Type u_2
γ : Type u_3
e e' : PartialEquiv α β
f f' : PartialEquiv β γ
he : e ≈ e'
hf : f ≈ f'
⊢ (e.trans f).source = (e'.trans f').source

pp: case right
α : Type u_1
β : Type u_2
γ : Type u_3
e e' : PartialEquiv α β
f f' : PartialEquiv β γ
he : e ≈ e'
hf : f ≈ f'
⊢ EqOn (↑(e.trans f)) (↑(e'.trans f')) (e.trans f).source

Tactic:
·

Goal state after:
case left
α : Type u_1
β : Type u_2
γ : Type u_3
e e' : PartialEquiv α β
f f' : PartialEquiv β γ
he : e ≈ e'
hf : f ≈ f'
⊢ (e.trans f).source = (e'.trans f').source
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case inl
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : IsEmpty α
⊢ ∃ c, c.Countable ∧ ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}

pp: case inr
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : Nonempty α
⊢ ∃ c, c.Countable ∧ ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}

Tactic:
·

Goal state after:
case inl
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : IsEmpty α
⊢ ∃ c, c.Countable ∧ ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case right
α : Type u_1
β : Type u_2
γ : Type u_3
e e' : PartialEquiv α β
f f' : PartialEquiv β γ
he : e ≈ e'
hf : f ≈ f'
⊢ EqOn (↑(e.trans f)) (↑(e'.trans f')) (e.trans f).source

Tactic:
·

Goal state after:
case right
α : Type u_1
β : Type u_2
γ : Type u_3
e e' : PartialEquiv α β
f f' : PartialEquiv β γ
he : e ≈ e'
hf : f ≈ f'
⊢ EqOn (↑(e.trans f)) (↑(e'.trans f')) (e.trans f).source
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case left
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ (e.restr s).source = (e'.restr s).source

pp: case right
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ EqOn (↑(e.restr s)) (↑(e'.restr s)) (e.restr s).source

Tactic:
·

Goal state after:
case left
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ (e.restr s).source = (e'.restr s).source
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case right
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ EqOn (↑(e.restr s)) (↑(e'.restr s)) (e.restr s).source

Tactic:
·

Goal state after:
case right
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ EqOn (↑(e.restr s)) (↑(e'.restr s)) (e.restr s).source
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case inr.a
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : Nonempty α
t : Set (Set α)
t_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}
t_count : t.Countable
ht : ts = generateFrom t
a : Set α → α
ha : ∀ s ∈ t, s = Ioi (a s) ∨ s = Iio (a s)
⊢ ts ≤ generateFrom {s | ∃ a_1 ∈ a '' t, s = Ioi a_1 ∨ s = Iio a_1}

pp: case a
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : Nonempty α
t : Set (Set α)
t_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}
t_count : t.Countable
ht : ts = generateFrom t
a : Set α → α
ha : ∀ s ∈ t, s = Ioi (a s) ∨ s = Iio (a s)
⊢ generateFrom {s | ∃ a_1 ∈ a '' t, s = Ioi a_1 ∨ s = Iio a_1} ≤ ts

Tactic:
·

Goal state after:
case inr.a
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : Nonempty α
t : Set (Set α)
t_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}
t_count : t.Countable
ht : ts = generateFrom t
a : Set α → α
ha : ∀ s ∈ t, s = Ioi (a s) ∨ s = Iio (a s)
⊢ ts ≤ generateFrom {s | ∃ a_1 ∈ a '' t, s = Ioi a_1 ∨ s = Iio a_1}
Module: Mathlib.Topology.Order.Basic

Goal state before:
pp: case a
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : Nonempty α
t : Set (Set α)
t_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}
t_count : t.Countable
ht : ts = generateFrom t
a : Set α → α
ha : ∀ s ∈ t, s = Ioi (a s) ∨ s = Iio (a s)
⊢ generateFrom {s | ∃ a_1 ∈ a '' t, s = Ioi a_1 ∨ s = Iio a_1} ≤ ts

Tactic:
·

Goal state after:
case a
α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : Nonempty α
t : Set (Set α)
t_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}
t_count : t.Countable
ht : ts = generateFrom t
a : Set α → α
ha : ∀ s ∈ t, s = Ioi (a s) ∨ s = Iio (a s)
⊢ generateFrom {s | ∃ a_1 ∈ a '' t, s = Ioi a_1 ∨ s = Iio a_1} ≤ ts
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
s : e.source = univ
t : e.target = univ
x : α
⊢ ↑e x = ↑e' x

pp: case refine_2
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
s : e.source = univ
t : e.target = univ
x : β
⊢ ↑e.symm x = ↑e'.symm x

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
s : e.source = univ
t : e.target = univ
x : α
⊢ ↑e x = ↑e' x
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
s : e.source = univ
t : e.target = univ
x : β
⊢ ↑e.symm x = ↑e'.symm x

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
s : e.source = univ
t : e.target = univ
x : β
⊢ ↑e.symm x = ↑e'.symm x