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

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
f1 : α → Set α := fun a ↦ Ioi a
f2 : α → Set α := fun a ↦ Iio a
s : Set α
hs : s ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a}
⊢ s ∈ range f1 ∪ range f2

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
x : α
hx : x ∈ e.source
⊢ x ∈ ↑e ⁻¹' ↑e.symm ⁻¹' s ↔ x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Field

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : 𝕜
hx : 0 < x
p : 𝕜 → Prop
h : ∀ᶠ (ε : 𝕜) in 𝓝[>] 0, p ε
⊢ ∀ᶠ (ε : 𝕜) in 𝓝[>] 0, p (x * ε)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : ∀ (x : α), ↑e x = ↑e' x
hsymm : ∀ (x : β), ↑e.symm x = ↑e'.symm x
hs : e.source = e'.source
⊢ ↑e = ↑e'

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : Countable α
⊢ SecondCountableTopology α

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : ∀ (x : α), ↑e x = ↑e' x
hsymm : ∀ (x : β), ↑e.symm x = ↑e'.symm x
hs : e.source = e'.source
A : ↑e = ↑e'
⊢ ↑e.symm = ↑e'.symm

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝ : Preorder α
t : OrderTopology α
⊢ OrderTopology αᵒᵈ

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : TopologicalSpace β
f : β → α
⊢ Continuous[inst✝, ts] f ↔ ∀ (a : α), IsOpen[inst✝] (f ⁻¹' Ioi a) ∧ IsOpen[inst✝] (f ⁻¹' Iio a)

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
t : OrderTopology α
s : Set α
⊢ IsOpen[ts] s ↔ GenerateOpen {s | ∃ a, s = Ioi a ∨ s = Iio a} s

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : ∀ (x : α), ↑e x = ↑e' x
hsymm : ∀ (x : β), ↑e.symm x = ↑e'.symm x
hs : e.source = e'.source
⊢ e = e'

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
h : e.source ⊆ s
⊢ (e.restr s).source = e.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
s : Set α
⊢ ((PartialEquiv.refl α).restr s).source = s

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
s : Set α
⊢ ((PartialEquiv.refl α).restr s).target = s

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
a : α
b : β
a' : α
ha' : a' ∈ {a}
⊢ const β a (const α b a') = a'

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
a : α
b b' : β
hb' : b' ∈ {b}
⊢ const α b (const β a b') = b'

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
a : α
⊢ 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e'✝ e' : PartialEquiv β γ
h : e.target = e'.source
x : α
hx : x ∈ e.source
⊢ (↑e' ∘ ↑e) x ∈ e'.target

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e'✝ e' : PartialEquiv β γ
h : e.target = e'.source
y : γ
hy : y ∈ e'.target
⊢ (↑e.symm ∘ ↑e'.symm) y ∈ e.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e'✝ e' : PartialEquiv β γ
h : e.target = e'.source
x : α
hx : x ∈ e.source
⊢ (↑e.symm ∘ ↑e'.symm) ((↑e' ∘ ↑e) x) = x

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e'✝ e' : PartialEquiv β γ
h : e.target = e'.source
y : γ
hy : y ∈ e'.target
⊢ (↑e' ∘ ↑e) ((↑e.symm ∘ ↑e'.symm) y) = y

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
ts : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
f : β → α
a : α
x : Filter β
⊢ Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ (b : β) in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ (b : β) in x, f b < a'

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv β γ
⊢ (e.trans e').source = e.source ∩ ↑e ⁻¹' (e.target ∩ e'.source)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv β γ
⊢ (e.trans e').source = ↑e.symm '' (e.target ∩ e'.source)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
e'' : PartialEquiv γ δ
⊢ ((e.trans e').trans e'').source = (e.trans (e'.trans e'')).source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
⊢ (e.trans (PartialEquiv.refl β)).source = e.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
⊢ ((PartialEquiv.refl α).trans e).source = e.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set β
⊢ (e.trans ((PartialEquiv.refl β).restr s)).source = (e.restr (↑e ⁻¹' s)).source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set β
⊢ (e.trans ((PartialEquiv.refl β).restr s)).source = (e.restr (e.source ∩ ↑e ⁻¹' s)).source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv β γ
s : Set α
⊢ ((e.restr s).trans e').source = ((e.trans e').restr s).source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv α γ
x : α
he : x ∈ e.source
he' : x ∈ e'.source
⊢ ↑e x ∈ ↑e.symm.symm.symm ⁻¹' e'.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
⊢ Equivalence EqOnSource

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
⊢ e.target = e'.target

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
ts : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
a : α
⊢ TendstoIxxClass Icc (𝓝 a) (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
h : e ≈ e'
⊢ e.symm ≈ e'.symm

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
hα : IsEmpty α
⊢ ∅.Countable

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: 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 ≈ e'.trans f'

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set α
⊢ e.restr s ≈ e'.restr s

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e e' : PartialEquiv α β
he : e ≈ e'
s : Set β
⊢ e.source ∩ ↑e ⁻¹' s = e'.source ∩ ↑e' ⁻¹' s

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
⊢ (e.trans e.symm).source = e.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
A : (e.trans e.symm).source = e.source
⊢ (e.trans e.symm).source = (ofSet e.source).source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
⊢ e.trans e.symm ≈ ofSet e.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

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

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e✝ : PartialEquiv α β
e'✝ : PartialEquiv β γ
e : PartialEquiv α β
e' : PartialEquiv γ δ
p : α × γ
hp : p ∈ e.source ×ˢ e'.source
⊢ (↑e p.1, ↑e' p.2) ∈ e.target ×ˢ e'.target

Tactic:
by

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

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

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e✝ : PartialEquiv α β
e'✝ : PartialEquiv β γ
e : PartialEquiv α β
e' : PartialEquiv γ δ
p : β × δ
hp : p ∈ e.target ×ˢ e'.target
⊢ (↑e.symm p.1, ↑e'.symm p.2) ∈ e.source ×ˢ e'.source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e✝ : PartialEquiv α β
e'✝ : PartialEquiv β γ
e : PartialEquiv α β
e' : PartialEquiv γ δ
p : α × γ
hp : p ∈ e.source ×ˢ e'.source
⊢ (↑e.symm (↑e p.1, ↑e' p.2).1, ↑e'.symm (↑e p.1, ↑e' p.2).2) = p

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e✝ : PartialEquiv α β
e'✝ : PartialEquiv β γ
e : PartialEquiv α β
e' : PartialEquiv γ δ
p : β × δ
hp : p ∈ e.target ×ˢ e'.target
⊢ (↑e (↑e.symm p.1, ↑e'.symm p.2).1, ↑e' (↑e.symm p.1, ↑e'.symm p.2).2) = p

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv γ δ
⊢ (e.prod e').symm = e.symm.prod e'.symm

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
⊢ (PartialEquiv.refl α).prod (PartialEquiv.refl β) = PartialEquiv.refl (α × β)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
η : Type u_5
ε : Type u_6
e : PartialEquiv α β
f : PartialEquiv β γ
e' : PartialEquiv δ η
f' : PartialEquiv η ε
⊢ (e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f')

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: ι : Type u_5
αi : ι → Type u_6
⊢ (PartialEquiv.pi fun i ↦ PartialEquiv.refl (αi i)) = PartialEquiv.refl ((i : ι) → αi i)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: ι : Type u_5
αi : ι → Type u_6
βi : ι → Type u_7
γi : ι → Type u_8
ei : (i : ι) → PartialEquiv (αi i) (βi i)
ei' : (i : ι) → PartialEquiv (βi i) (γi i)
⊢ (PartialEquiv.pi ei).trans (PartialEquiv.pi ei') = PartialEquiv.pi fun i ↦ (ei i).trans (ei' i)

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
h : e.target = univ
⊢ e.source ⊆ univ

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
h : e.target = univ
⊢ univ ⊆ e.target

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
e : PartialEquiv α β
h : e.source = univ
⊢ Injective ↑e

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : α ≃ β
e' : β ≃ γ
⊢ (e.trans e').toPartialEquiv.source = (e.toPartialEquiv.trans e'.toPartialEquiv).source

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : α ≃ β
f' : PartialEquiv β γ
f'' : PartialEquiv γ δ
⊢ (e.transPartialEquiv f').trans f'' = e.transPartialEquiv (f'.trans f'')

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : α ≃ β
e' : β ≃ γ
f'' : PartialEquiv γ δ
⊢ (e.trans e').transPartialEquiv f'' = e.transPartialEquiv (e'.transPartialEquiv f'')

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
f' : β ≃ γ
f'' : γ ≃ δ
⊢ (e.transEquiv f').transEquiv f'' = e.transEquiv (f'.trans f'')

Tactic:
by

Goal state after:
Module: Mathlib.Logic.Equiv.PartialEquiv

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
f'' : γ ≃ δ
⊢ (e.trans e').transEquiv f'' = e.trans (e'.transEquiv f'')

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
c : Set α
h : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
k : Set (Set α)
k_fin : k.Finite
hk : k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
kl : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Ioi a}
kr : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Iio a}
⊢ k = kl ∪ kr

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
c : Set α
h : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
k : Set (Set α)
k_fin : k.Finite
hk : k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
kl : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Ioi a}
kr : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Iio a}
k_eq : k = kl ∪ kr
⊢ kl ⊆ k

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
c : Set α
h : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
k : Set (Set α)
k_fin : k.Finite
hk : k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
kl : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Ioi a}
kr : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Iio a}
k_eq : k = kl ∪ kr
this : Finite ↑kl
⊢ kr ⊆ k

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
c : Set α
h : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
k : Set (Set α)
k_fin : k.Finite
hk : k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
kl : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Ioi a}
kr : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Iio a}
k_eq : k = kl ∪ kr
this✝ : Finite ↑kl
this : Finite ↑kr
al : ↑kl → α
alc : ∀ (s : ↑kl), al s ∈ c
hal : ∀ (s : ↑kl), ↑s = Ioi (al s)
ar : ↑kr → α
arc : ∀ (s : ↑kr), ar s ∈ c
har : ∀ (s : ↑kr), ↑s = Iio (ar s)
⊢ range al ⊆ c

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
c : Set α
h : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
k : Set (Set α)
k_fin : k.Finite
hk : k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
kl : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Ioi a}
kr : Set (Set α) := {s | s ∈ k ∧ ∃ a ∈ c, s = Iio a}
k_eq : k = kl ∪ kr
this✝ : Finite ↑kl
this : Finite ↑kr
al : ↑kl → α
alc : ∀ (s : ↑kl), al s ∈ c
hal : ∀ (s : ↑kl), ↑s = Ioi (al s)
ar : ↑kr → α
arc : ∀ (s : ↑kr), ar s ∈ c
har : ∀ (s : ↑kr), ↑s = Iio (ar s)
⊢ range ar ⊆ c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Grp.EnoughInjectives

Goal state before:
pp: A_ : AddCommGrpCat
⊢ (fun i ↦ { down := i 0 }) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Grp.EnoughInjectives

Goal state before:
pp: A_ : AddCommGrpCat
⊢ ∀ (x y : ↑A_),
    { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun (x + y) =
      { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun x +
        { toFun := fun a i ↦ { down := i a }, map_zero' := ⋯ }.toFun y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Grp.Images

Goal state before:
pp: G H : AddCommGrpCat
f : G ⟶ H
⊢ factorThruImage f ≫ ι f = f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Grp.Images

Goal state before:
pp: G H : AddCommGrpCat
f : G ⟶ H
F' : MonoFactorisation f
⊢ (ConcreteCategory.hom F'.e) ↑(Classical.indefiniteDescription (fun x ↦ (Hom.hom f) x = ↑0) ⋯) = 0

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝ : Preorder α
c : Set α
h : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
⊢ IsTopologicalBasis {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}

Tactic:
by

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

Goal state before:
pp: α : Type u
ts : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
a : α
hu : ∃ u, a < u
hl : ∃ l, l < a
⊢ 𝓝 a = ⨅ l, ⨅ (_ : l < a), ⨅ u, ⨅ (_ : a < u), 𝓟 (Ioo l u)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Grp.Images

Goal state before:
pp: G H : AddCommGrpCat
f : G ⟶ H
F' : MonoFactorisation f
⊢ ∀ (x y : ↥(Hom.hom f).range),
    (ConcreteCategory.hom F'.e) ↑(Classical.indefiniteDescription (fun x_1 ↦ (Hom.hom f) x_1 = ↑(x + y)) ⋯) =
      (ConcreteCategory.hom F'.e) ↑(Classical.indefiniteDescription (fun x_1 ↦ (Hom.hom f) x_1 = ↑x) ⋯) +
        (ConcreteCategory.hom F'.e) ↑(Classical.indefiniteDescription (fun x ↦ (Hom.hom f) x = ↑y) ⋯)

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
ts : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
f : β → α
a : α
x : Filter β
hu : ∃ u, a < u
hl : ∃ l, l < a
h : ∀ (l u : α), l < a → a < u → ∀ᶠ (b : β) in x, l < f b ∧ f b < u
⊢ Tendsto f x (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Category.Grp.Images

Goal state before:
pp: G H : AddCommGrpCat
f : G ⟶ H
F' : MonoFactorisation f
⊢ lift F' ≫ F'.m = ι f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.EpiMono

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderSubobjects
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hS : S.ShortExact
h₂ : P S.X₂
⊢ P S.X₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.EpiMono

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
P : ObjectProperty C
inst✝¹ : P.IsClosedUnderQuotients
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hS : S.ShortExact
h₂ : P S.X₂
⊢ P S.X₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.EpiMono

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
P : ObjectProperty C
⊢ ∀ {X Y : C} (f : X ⟶ Y) [Mono f], ⊤ Y → ⊤ X

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.EpiMono

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
P : ObjectProperty C
⊢ ∀ {X Y : C} (f : X ⟶ Y) [Epi f], ⊤ X → ⊤ Y

Tactic:
by

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

Goal state before:
pp: α✝ : Type u
β : Type v
γ : Type w
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Preorder (α i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), OrderTopology (α i)
f : (i : ι) → α i
⊢ TendstoIxxClass Icc (𝓝 f) (𝓝 f)

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
⊢ induced f inst✝¹ ≤ Preorder.topology α

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Extensions

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
D : Type u'
inst✝¹ : Category.{v', u'} D
P : ObjectProperty C
inst✝ : HasZeroMorphisms C
⊢ ∀ {S : ShortComplex C}, S.ShortExact → ⊤ S.X₁ → ⊤ S.X₃ → ⊤ S.X₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.Extensions

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u'
inst✝⁶ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : P.IsClosedUnderExtensions
F : D ⥤ C
inst✝³ : HasZeroMorphisms D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : PreservesFiniteLimits F
inst✝ : PreservesFiniteColimits F
S✝ : ShortComplex D
hS : S✝.ShortExact
h₁ : P.inverseImage F S✝.X₁
h₃ : P.inverseImage F S✝.X₃
⊢ P.inverseImage F S✝.X₂

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : ∀ {x y : α}, f x < f y ↔ x < y
H₁ : ∀ {a : α} {b : β} {x : α}, b < f a → ¬b < f x → ∃ y < a, b ≤ f y
H₂ : ∀ {a : α} {b : β} {x : α}, f a < b → ¬f x < b → ∃ y, a < y ∧ f y ≤ b
⊢ induced f inst✝¹ = Preorder.topology α

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : LinearOrder β
t : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : StrictMono f
hc : (range f).OrdConnected
⊢ induced f t = Preorder.topology α

Tactic:
by

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

Goal state before:
pp: α✝ : Type u
β : Type v
γ : Type w
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
t : Set α
ht : t.OrdConnected
⊢ (range Subtype.val).OrdConnected

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
a : α
⊢ 𝓝[≥] a = (⨅ u, ⨅ (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : OrderTopology α
a : α
ha : ∃ u, a < u
⊢ 𝓝[≥] a = ⨅ u, ⨅ (_ : a < u), 𝓟 (Ico a u)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : Preorder α
inst✝¹ : OrderTop α
inst✝ : OrderTopology α
⊢ 𝓝 ⊤ = ⨅ l, ⨅ (_ : l < ⊤), 𝓟 (Ioi l)

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTop α
inst✝¹ : OrderTopology α
inst✝ : Nontrivial α
⊢ (𝓝 ⊤).HasBasis (fun a ↦ a < ⊤) fun a ↦ Ioi a

Tactic:
by

Goal state after: