mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
n : ℕ
h : a ∈ closedBall b r
⊢ a ^ n ∈ closedBall (b ^ n) (n • r)

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
n : ℕ
hn : 0 < n
h : ‖a⁻¹ * b‖ < r
⊢ 0 < ↑n

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
n : ℕ
hn : 0 < n
h : a ∈ ball b r
⊢ a ^ n ∈ ball (b ^ n) (n • r)

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
c : E
⊢ a * c ∈ closedBall (b * c) r ↔ a ∈ closedBall b r

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
c : E
⊢ a * c ∈ ball (b * c) r ↔ a ∈ ball b r

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
⊢ a • closedBall b r = closedBall (a • b) r

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
⊢ a • ball b r = ball (a • b) r

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
m : Multiset E
⊢ ↑‖m.prod‖₊ ≤ ↑(Multiset.map (fun x ↦ ‖x‖₊) m).sum

Tactic:
by

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

Goal state before:
pp: ι : Type u_3
E : Type u_5
inst✝ : SeminormedCommGroup E
s : Finset ι
f : ι → E
⊢ ↑‖∏ a ∈ s, f a‖₊ ≤ ↑(∑ a ∈ s, ‖f a‖₊)

Tactic:
by

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

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
f : E → F
x : E
y : F
⊢ Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
x : E
⊢ (𝓝 x).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {y | ‖y / x‖ < ε}

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 I' J' : FractionalIdeal R⁰ K
H✝ : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
H : Submodule.map ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I = ↑(spanSingleton R⁰ (I'.divMod I J')) * ↑I
⊢ ↑(spanSingleton R⁰ (I'.divMod I J')) * ↑I ≤ ↑(J' + spanSingleton R⁰ (I'.divMod I J') * I)

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
⊢ (𝓤 E).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {p | ‖p.1 / p.2‖ < ε}

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a : E
⊢ ‖a‖ ≤ 0 ↔ a = 1

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a : E
⊢ 0 < ‖a‖ ↔ a ≠ 1

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a b : E
⊢ ‖a / b‖ = 0 ↔ a = b

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a b : E
⊢ 0 < ‖a / b‖ ↔ a ≠ b

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a b : E
h : ‖a / b‖ ≤ 0
⊢ a = b

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a : E
⊢ a = 1 ∨ 0 < ‖a‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : NormedGroup E
a : E
⊢ ‖a‖₊ = 0 ↔ a = 1

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
⊢ t ∈ A

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
⊢ Icc a b ⊆ s

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 I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
⊢ (↥↑I ⧸ Submodule.comap (↑I).subtype ↑J) ≃ₗ[R] ↥↑I' ⧸ Submodule.comap (↑I').subtype ↑J'

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst✝³] s
ht : IsClosed[inst✝³] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬(Icc a b ∩ (s ∩ t)).Nonempty
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
⊢ tᶜ ∩ Ioc z y ∈ 𝓝[>] z

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst✝³] s
ht : IsClosed[inst✝³] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
⊢ (Icc a b ∩ (s ∩ t)).Nonempty

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ ∀ (t t' : Set α),
    IsClosed[inst✝³] t →
      IsClosed[inst✝³] t' →
        Icc a b ⊆ t ∪ t' → (Icc a b ∩ t).Nonempty → (Icc a b ∩ t').Nonempty → (Icc a b ∩ (t ∩ t')).Nonempty

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
⊢ {s | IsPreconnected s} =
    range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
s : Set α
⊢ IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : IsDedekindDomain R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : Algebra S R
inst✝³ : Algebra.IsIntegral S R
inst✝² : IsDomain S
inst✝¹ : Module.IsTorsionFree S R
p : Ideal S
inst✝ : p.IsMaximal
hp : p ≠ 0
h : map (algebraMap S R) p ≠ 0
hF : Fintype ↑{v | v.asIdeal ∣ map (algebraMap S R) p} := ⋯.fintype
x✝¹ : Fintype { v // v.asIdeal ∣ map (algebraMap S R) p } := hF
x✝ : { v // v.asIdeal ∣ map (algebraMap S R) p }
v : HeightOneSpectrum R
property✝ : v.asIdeal ∣ map (algebraMap S R) p
⊢ v.asIdeal.LiesOver p

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hf : ContinuousOn f [[a, b]]
⊢ [[f a, f b]] ⊆ f '' [[a, b]]

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
f : α → α
hf : ContinuousOn f (Icc a b)
hle : a ≤ b
h_surj : SurjOn f (Icc a b) (Icc a b)
x₀ : α
hx₀ : x₀ ∈ Icc a b ∧ f x₀ = a
x₁ : α
hx₁ : x₁ ∈ Icc a b ∧ f x₁ = b
⊢ f x₀ ≤ x₀

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
f : α → α
hf : ContinuousOn f (Icc a b)
hle : a ≤ b
h_surj : SurjOn f (Icc a b) (Icc a b)
x₀ : α
hx₀ : x₀ ∈ Icc a b ∧ f x₀ = a
x₁ : α
hx₁ : x₁ ∈ Icc a b ∧ f x₁ = b
⊢ x₁ ≤ f x₁

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a : α
f : α → δ
hf : ContinuousOn f (Ici a)
htop : Tendsto f atTop atTop
⊢ Ioi (f a) ⊆ f '' Ioi a

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a : α
f : α → δ
hf : ContinuousOn f (Ici a)
htop : Tendsto f atTop atBot
⊢ Iio (f a) ⊆ f '' Ioi a

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a : α
f : α → δ
hf : ContinuousOn f (Iic a)
hbot : Tendsto f atBot atBot
⊢ Iio (f a) ⊆ f '' Iio a

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : IsDedekindDomain R
S : Type u_3
inst✝⁵ : CommRing S
inst✝⁴ : Algebra S R
inst✝³ : Algebra.IsIntegral S R
inst✝² : IsDomain S
inst✝¹ : Module.IsTorsionFree S R
p : Ideal S
inst✝ : p.IsMaximal
hp : p ≠ 0
⊢ map (algebraMap S R) p = ∏ P ∈ (p.primesOver R).toFinset, P ^ P.ramificationIdx S

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a : α
f : α → δ
hf : ContinuousOn f (Iic a)
hbot : Tendsto f atBot atTop
⊢ Ioi (f a) ⊆ f '' Iio a

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
s : Set α
hs : s.OrdConnected
f : α → δ
hf : ContinuousOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ SurjOn f s [[f a, f b]]

Tactic:
by

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

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
p : HeightOneSpectrum R
⊢ Multiset.count p.asIdeal (normalizedFactors I) = multiplicity p.asIdeal I

Tactic:
by

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

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
p : HeightOneSpectrum R
⊢ p.maxPowDividing I = p.asIdeal ^ multiplicity p.asIdeal I

Tactic:
by

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

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
p : HeightOneSpectrum R
⊢ (factorization I) p.asIdeal = multiplicity p.asIdeal I

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅
⊢ ⋃ i, (fun i ↦ ∅) i = ∅ ∧
    ∀ (i : ℕ),
      Bornology.IsBounded ((fun i ↦ ∅) i) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun i ↦ ∅) i)

Tactic:
by

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

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ ⊥
⊢ ∏ᶠ (p : HeightOneSpectrum R), p.asIdeal ^ multiplicity p.asIdeal I = I

Tactic:
by

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

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
p : HeightOneSpectrum R
I J : Ideal R
h : J ≤ I
hJ : J ≠ ⊥
⊢ multiplicity p.asIdeal I ≤ multiplicity p.asIdeal J

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
U : Set α
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
⊢ ∃ f,
    Monotone f ∧
      ⋃ i, f i = U ∧
        ∀ (i : ℕ), Bornology.IsBounded (f i) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
s : Set α
hs : x ∈ s ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ x ∈ s ∩ ball x 1 ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ ball x 1)

Tactic:
by

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

Goal state before:
pp: β : Type u_3
f : ℕ → β
l : Filter β
⊢ Tendsto (fun x ↦ f ↑x) atTop l ↔ Tendsto f atTop l

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
s : Set α
hs : x ∈ s ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ Bornology.IsBounded (s ∩ ball x 1) ∧ s ∩ ball x 1 ⊆ s

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
r : R
n : ℕ
hn : -r ≤ ↑n
⊢ ↑(-↑n) ≤ r

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
⊢ (𝓝 x).HasBasis (fun a ↦ x ∈ a ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a ∧ Bornology.IsBounded a)
    id

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
c : α
x✝ : Set α
⊢ (∃ r, x✝ ⊆ closedBall c r) ↔ ∃ i, True ∧ (closedBall c i)ᶜ ⊆ x✝ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
c : α
x✝² x✝¹ : ℝ
hr : x✝² ≤ x✝¹
x✝ : α
⊢ x✝ ∈ (closedBall c x✝¹)ᶜ → x✝ ∈ (closedBall c x✝²)ᶜ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
f : α → ℤ
l : Filter α
⊢ Tendsto (fun n ↦ ↑(f n)) l atTop ↔ Tendsto f l atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
c : α
x✝ : Set α
⊢ (∃ r, x✝ ⊆ ball c r) ↔ ∃ i, True ∧ (ball c i)ᶜ ⊆ x✝ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
c : α
x✝² x✝¹ : ℝ
hr : x✝² ≤ x✝¹
x✝ : α
⊢ x✝ ∈ (ball c x✝¹)ᶜ → x✝ ∈ (ball c x✝²)ᶜ

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁷ : TopologicalSpace α
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : OrderTopology α
inst✝⁴ : DenselyOrdered α
δ : Type u_1
inst✝³ : LinearOrder δ
inst✝² : TopologicalSpace δ
inst✝¹ : OrderClosedTopology δ
inst✝ : BoundedOrder α
f : α → δ
hf_c : Continuous[inst✝⁷, inst✝²] f
hf : f ⊥ ≤ f ⊤
hf_i : Injective f
⊢ StrictMono f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
c : α
⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i

Tactic:
by

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

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
f : α → ℤ
l : Filter α
⊢ Tendsto (fun n ↦ ↑(f n)) l atBot ↔ Tendsto f l atBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
c : α
⊢ comap (dist c) atTop = cobounded α

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atTop, p x
⊢ ∀ᶠ (n : ℤ) in atTop, p ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
c : α
f : β → α
l : Filter β
⊢ Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
c : α
f : β → α
l : Filter β
⊢ Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
⊢ ∀ᶠ (n : ℤ) in atBot, p ↑n

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
⊢ ∀ᶠ (n : ℤ) in atBot, p ↑n

Tactic:
by
  rw [← Int.comap_cast_atBot (R := R)]; exact h.comap _

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

Goal state before:
pp: R : Type u_2
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
r : R
n : ℕ
hn : r ≤ ↑n
⊢ r ≤ ↑↑n

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
r : R
n : ℕ
hn : -r ≤ ↑n
⊢ ↑(-↑n) ≤ r

Tactic:
by

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

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
f : α → ℚ
l : Filter α
⊢ Tendsto (fun n ↦ ↑(f n)) l atTop ↔ Tendsto f l atTop

Tactic:
by

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

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
f : α → ℚ
l : Filter α
⊢ Tendsto (fun n ↦ ↑(f n)) l atBot ↔ Tendsto f l atBot

Tactic:
by

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

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
p : HeightOneSpectrum R
I J : Ideal R
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ count p.asIdeal (normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod) =
    min (count p.asIdeal (normalizedFactors I)) (count p.asIdeal (normalizedFactors J))

Tactic:
rw [normalizedFactors_prod_inter_eq_inter]

Goal state after:
R : Type u_3
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
p : HeightOneSpectrum R
I J : Ideal R
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ count p.asIdeal (normalizedFactors I ∩ normalizedFactors J) =
    min (count p.asIdeal (normalizedFactors I)) (count p.asIdeal (normalizedFactors J))
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
c : α
⊢ Bornology.IsCobounded s ↔ ∃ r, (closedBall c r)ᶜ ⊆ s

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atTop, p x
⊢ ∀ᶠ (n : ℚ) in atTop, p ↑n

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous[inst✝⁶, inst✝¹] f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := min a x
t : α := max b y
hsa : s ≤ a
hbt : b ≤ t
⊢ StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
s : Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
c : α
h : ∃ r, (closedBall c r)ᶜ ⊆ s
⊢ s ∈ cocompact α

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
⊢ ∀ᶠ (n : ℚ) in atBot, p ↑n

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : PartialOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : IsDirectedOrder α
k : α
⊢ ∀ (a c : α), 0 ≤ c → (a + k ≤ c ↔ a ≤ c - k)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
f : β → α
⊢ (∃ C, ∀ ⦃x : α⦄, x ∈ range f → ∀ ⦃y : α⦄, y ∈ range f → dist x y ≤ C) ↔ ∃ C, ∀ (x y : β), dist (f x) (f y) ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝³ : AddCommGroup α
inst✝² : PartialOrder α
inst✝¹ : IsOrderedAddMonoid α
inst✝ : IsDirectedOrder α
k : α
⊢ ∀ (c : α), 0 ≤ c → c ≤ c - k + k

Tactic:
by

Goal state after: