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

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderTopology α
β : Type u_3
inst✝² : LinearOrder β
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
f : β → α
x y : α
hf : DenseRange f
hmono : Monotone f
hlt : x < y
⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), f (u n) ∈ Ioo x y) ∧ Tendsto (f ∘ u) atTop (𝓝 x)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
d : α
h : ∀ (b : α), d < b → ∀ᶠ (x : γ) in l, f x < b
hd : IsGLB (Ioi d) d
h✝ : ∃ b, d < b
⊢ (Ioi d).Nonempty

Tactic:
by

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

Goal state before:
pp: α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
d : α
h : ∀ (b : α), d < b → ∀ᶠ (x : γ) in l, f x < b
hd : IsGLB (Ioi d) d
h✝ : ∃ b, d < b
u : ℕ → α
hu_tt : Tendsto u atTop (𝓝 d)
hu_gt : ∀ (n : ℕ), u n ∈ Ioi d
n : ℕ
⊢ d < u n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
a : α
h : ∀ (b : α), a < b → ∀ᶠ (x : γ) in l, f x < b
d : α
hd : IsGLB (Ioi a) d
H0 : Ioi a = Ici d
⊢ a < d

Tactic:
by

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

Goal state before:
pp: α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
a : α
h : ∀ (b : α), a < b → ∀ᶠ (x : γ) in l, f x < b
⊢ ∀ᶠ (x : γ) in l, f x ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
⊢ ‖0‖ₑ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
⊢ ∀ (x y : ℝ≥0∞), ‖x + y‖ₑ ≤ ‖x‖ₑ + ‖y‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
⊢ ∀ (x : ℝ≥0∞), ‖x‖ₑ = 0 ↔ x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
r _r _y : ℝ
⊢ dist _r _y = ‖-_r + _y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: n : ℕ
⊢ ‖↑n‖ₑ = ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: ⊢ ↑‖2‖₊ = ↑2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: q : ℚ≥0
⊢ ‖↑q‖₊ = ↑q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: r : ℝ
hr : 0 ≤ r
⊢ ‖r‖ₑ = ENNReal.ofReal r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: a : ℝ
ha : 0 ≤ a
⊢ ‖ENNReal.ofReal a‖ₑ = ‖a‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: r : ℝ
⊢ ‖|r|‖₊ = ‖r‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: r : ℝ
⊢ ‖|r|‖ₑ = ‖r‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: a : ℝ≥0∞
ha : a ≠ ∞
⊢ ‖a.toReal‖ₑ = a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: r : ℝ
⊢ ‖r‖ₑ = ENNReal.ofReal |r|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: r : ℝ
hr : 0 ≤ r
⊢ r.toNNReal = ‖r‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: r : ℝ
⊢ ENNReal.ofReal r ≤ ‖r‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
x : E
⊢ ‖‖x‖‖₊ = ‖x‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
x : E
⊢ ‖‖x‖‖ₑ = ‖x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Real

Goal state before:
pp: ε : Type u_9
inst✝ : ENorm ε
x : ε
⊢ ‖‖x‖ₑ‖ₑ = ‖x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.Discrete

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ ∀ (a : α), (nhds a).IsCountablyGenerated

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.Discrete

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
hd : DiscreteTopology α
inst✝ : Countable α
i : α
⊢ instTopologicalSpaceSubtype = generateFrom {univ}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.Discrete

Goal state before:
pp: α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
⊢ ⊥ = Preorder.topology α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.Discrete

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
⊢ DiscreteTopology α ↔ OrderTopology α

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
F : Type u_3
G : ι → Type u_4
inst✝¹ : SeminormedGroup E
inst✝ : SeminormedGroup F
x y : E × F
⊢ dist x y = ‖x⁻¹ * y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedGroup (G i)
x : (i : ι) → G i
r : ℝ
hr : 0 ≤ r
⊢ ‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedGroup (G i)
f : (i : ι) → G i
r : ℝ
inst✝ : Nonempty ι
⊢ ‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedGroup (G i)
x : (i : ι) → G i
r : ℝ
hr : 0 < r
⊢ ‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
inst✝² : Fintype ι
inst✝¹ : SeminormedGroup E
inst✝ : Nonempty ι
a : E
⊢ ‖fun _i ↦ a‖ = ‖a‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
F : Type u_3
inst✝² : Fintype ι
inst✝¹ : SeminormedGroup E
inst✝ : Fintype F
g : ι → E
f : F → ι
⊢ 0 ≤ ‖g‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
F : Type u_3
inst✝² : Fintype ι
inst✝¹ : SeminormedGroup E
inst✝ : Fintype F
g : ι → E
f : F → ι
⊢ ‖g ∘ f‖ ≤ ‖g‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
inst✝² : Fintype ι
inst✝¹ : SeminormedGroup E
inst✝ : Nonempty ι
f : ι → E
⊢ Finset.univ.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
inst✝² : Fintype ι
inst✝¹ : SeminormedGroup E
inst✝ : Nonempty ι
f : ι → E
⊢ IsGreatest (Set.range fun x ↦ ‖f x‖) ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
E : Type u_2
F : Type u_3
inst✝² : Fintype ι
inst✝¹ : SeminormedGroup E
inst✝ : Fintype F
f : ι → F
hf : Surjective f
g : F → E
⊢ ‖g ∘ f‖ = ‖g‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → NormedAddCommGroup (G i)
i : ι
y : G i
⊢ ∀ (b : ι), ‖single i y b‖₊ = single i ‖y‖₊ b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → NormedAddCommGroup (G i)
i : ι
y : G i
⊢ ‖single i y‖₊ = ‖y‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Constructions

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → NormedAddCommGroup (G i)
i : ι
y : G i
⊢ ‖single i y‖ₑ = ‖y‖ₑ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s t : Set α
inst✝¹ : Lattice α
inst✝ : Nonempty α
⊢ ∅ ∈ {s | BddBelow s ∧ BddAbove s}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s t : Set α
inst✝¹ : Lattice α
inst✝ : Nonempty α
⊢ ∀ (x : α), {x} ∈ {s | BddBelow s ∧ BddAbove s}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s : Set α
inst✝¹ : Lattice α
inst✝ : Nonempty α
⊢ IsBounded s ↔ BddBelow s ∧ BddAbove s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : Bornology α
inst✝¹ : Lattice α
inst✝ : Nonempty α
h : inst✝² = orderBornology
s : Set α
⊢ IsBounded s ↔ BddBelow s ∧ BddAbove s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : Bornology α
inst✝¹ : Lattice α
inst✝ : Nonempty α
⊢ IsOrderBornology α ↔ inst✝² = orderBornology

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s✝ t : Set α
inst✝² : Bornology α
inst✝¹ : Preorder α
inst✝ : IsOrderBornology α
s : Set αᵒᵈ
⊢ IsBounded s ↔ BddBelow s ∧ BddAbove s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s✝ t : Set α
inst✝⁵ : Bornology α
inst✝⁴ : Preorder α
inst✝³ : IsOrderBornology α
β : Type u_2
inst✝² : Preorder β
inst✝¹ : Bornology β
inst✝ : IsOrderBornology β
s : Set (α × β)
⊢ IsBounded s ↔ BddBelow s ∧ BddAbove s

Tactic:
by

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

Goal state before:
pp: α✝ : Type u_1
s✝ t : Set α✝
inst✝⁵ : Bornology α✝
inst✝⁴ : Preorder α✝
inst✝³ : IsOrderBornology α✝
ι : Type u_2
α : ι → Type u_3
inst✝² : (i : ι) → Preorder (α i)
inst✝¹ : (i : ι) → Bornology (α i)
inst✝ : ∀ (i : ι), IsOrderBornology (α i)
s : Set ((i : ι) → α i)
⊢ IsBounded s ↔ BddBelow s ∧ BddAbove s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s t : Set α
inst✝³ : Bornology α
inst✝² : Preorder α
inst✝¹ : IsOrderBornology α
inst✝ : NoBotOrder α
⊢ (cobounded α).NeBot

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : Bornology α
inst✝² : Preorder α
inst✝¹ : IsOrderBornology α
inst✝ : NoMaxOrder α
⊢ Filter.atTop ≤ cobounded α

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : Bornology α
inst✝¹ : LinearOrder α
inst✝ : IsOrderBornology α
⊢ cobounded α ≤ Filter.atBot ⊔ Filter.atTop

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : Bornology α
inst✝³ : LinearOrder α
inst✝² : IsOrderBornology α
inst✝¹ : NoMaxOrder α
inst✝ : OrderBot α
s : Set α
x✝ : ∃ i, True ∧ Ici i ⊆ s
b : α
left✝ : True
hb : Ici b ⊆ s
x : α
hx : x ∈ sᶜ
⊢ ⊥ ≤ x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : Bornology α
inst✝³ : LinearOrder α
inst✝² : IsOrderBornology α
inst✝¹ : NoMaxOrder α
inst✝ : OrderBot α
⊢ cobounded α = Filter.atTop

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
s : Set α
x : α
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : OrderClosedTopology β
hf : MonotoneOn f s
hx : ClusterPt x (𝓟 s)
h'x : ContinuousWithinAt f s x
⊢ MonotoneOn f (insert x s)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := x '' t
H : (x '' t).Countable
c : β
hc : c ∈ t
d : β
hd : d ∈ t
hcd : x c = x d
⊢ f (x c) = f (x d)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := x '' t
H : (x '' t).Countable
⊢ InjOn x t

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := x '' t
H : (x '' t).Countable
⊢ {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}.Countable

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := x '' t
c : β
hc : c ∈ t
d : β
hd : d ∈ t
hcd : x c ≠ x d
H : c < d
a : α
xca : x c < a
ayc : a < y c
xda : x d < a
ayd : a < y (f (x d))
this✝ : x d ≤ y c
this : d ≤ c
⊢ False

Tactic:
exact not_le.2 H this

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
⊢ {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}.Countable

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
f : α → β
inst✝ : SecondCountableTopology α
hf : Monotone f
⊢ {c | ∃ x y, x < y ∧ f x = c ∧ f y = c}.Countable

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : LinearOrder β
s : Set α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
hf : MonotoneOn f s
x : α
xs : x ∈ s
hx : ¬ContinuousWithinAt f (s ∩ Ioi x) x
⊢ x ∈ {x | x ∈ s ∧ ∃ z, f x < z ∧ ∀ y ∈ s, x < y → z ≤ f y}

Tactic:
dsimp only [mem_ofPred_eq]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : LinearOrder β
s : Set α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
hf : MonotoneOn f s
x : α
xs : x ∈ s
hx : ¬ContinuousWithinAt f (s ∩ Ioi x) x
⊢ x ∈ s ∧ ∃ z, f x < z ∧ ∀ y ∈ s, x < y → z ≤ f y
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : LinearOrder β
s : Set α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
hf : MonotoneOn f s
x : α
xs : x ∈ s
hx : x ∈ s → ∀ (z : β), f x < z → ∃ y ∈ s, x < y ∧ f y < z
u : β
hu : u > f x
v : α
vs : v ∈ s
xv : x < v
fvu : f v < u
⊢ s ∩ Ioo x v ∈ 𝓝[s ∩ Ioi x] x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : LinearOrder β
s : Set α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
hf : MonotoneOn f s
⊢ {x | x ∈ s ∧ ¬ContinuousWithinAt f (s ∩ Ioi x) x}.Countable

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
a b c : FractionalIdeal R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
⊢ ∃ x, a + FractionalIdeal.spanSingleton R⁰ x * b = c

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
a b c : FractionalIdeal R⁰ K
hac : a ≤ c
ha : a ≠ 0
hb : b ≠ 0
⊢ a + spanSingleton R⁰ (c.divMod b a) * b = c

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I J : FractionalIdeal R⁰ K
⊢ I.divMod 0 J = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I J : FractionalIdeal R⁰ K
⊢ I.divMod J 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I J : FractionalIdeal R⁰ K
⊢ divMod 0 I J = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : LinearOrder β
s : Set α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
hf : MonotoneOn f s
⊢ {x | x ∈ s ∧ ¬ContinuousWithinAt f s x}.Countable

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
a b c : FractionalIdeal R⁰ K
hac : ¬a ≤ c
⊢ c.divMod b a = 0

Tactic:
simp [divMod, hac]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : LinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : SecondCountableTopology β
hf : Monotone f
⊢ {x | ¬ContinuousAt f x}.Countable

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
a b c : FractionalIdeal R⁰ K
hac : ¬a ≤ c
⊢ c.divMod b a = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
ι : Sort u_3
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
bdd : BddAbove (range g)
⊢ f (⨆ i, g i) = ⨆ i, f (g i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
ι : Sort u_3
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iInf g)
Mf : Monotone f
bdd : BddBelow (range g)
⊢ f (⨅ i, g i) = ⨅ i, f (g i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
ι : Sort u_3
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iInf g)
Af : Antitone f
bdd : BddBelow (range g)
⊢ f (⨅ i, g i) = ⨆ i, f (g i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : ConditionallyCompleteLinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderClosedTopology β
ι : Sort u_3
inst✝ : Nonempty ι
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Af : Antitone f
bdd : BddAbove (range g)
⊢ f (⨆ i, g i) = ⨅ i, f (g i)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : CompleteLinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderClosedTopology β
f : α → β
s : Set α
Cf : ContinuousWithinAt f s (sSup s)
Mf : MonotoneOn f s
fbot : f ⊥ = ⊥
⊢ f (sSup s) = sSup (f '' s)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : CompleteLinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderClosedTopology β
ι : Sort u_3
f : α → β
g : ι → α
Cf : ContinuousAt f (iSup g)
Mf : Monotone f
fbot : f ⊥ = ⊥
⊢ f (⨆ i, g i) = ⨆ i, f (g i)

Tactic:
by

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

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x y : α
h_nonempty : (Ioo y x).Nonempty
Mf : MonotoneOn f (Ioo y x)
h_bdd : BddAbove (f '' Ioo y x)
l : β
hl : l < sSup (f '' Ioo y x)
⊢ ∃ a ∈ Ioo y x, l < f a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J J' : FractionalIdeal R⁰ K
h : J ≤ I
hJ' : J' ≠ 0
hI : I ≠ 0
H : I * J' = 0 * J
h' : J' ≤ 0
this : (J' ⊓ spanSingleton R⁰ (divMod 0 I J') * I) * 0 = spanSingleton R⁰ (divMod 0 I J') * J * 0
⊢ J' = 0

Tactic:
by

Goal state after: