mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J

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)
⊢ Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Ioo y x)))

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 x y).Nonempty
Mf : MonotoneOn f (Ioo x y)
h_bdd : BddBelow (f '' Ioo x y)
m : β
hm : m > sInf (f '' Ioo x y)
⊢ ∃ a ∈ Ioo x y, f a < m

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 x y).Nonempty
Mf : MonotoneOn f (Ioo x y)
h_bdd : BddBelow (f '' Ioo x y)
⊢ Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioo x y)))

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 : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
l : β
hl : l < sSup (f '' Iio x)
⊢ ∃ a < x, l < f a

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 : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
⊢ Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x)))

Tactic:
by

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

Goal state before:
pp: X : Type u
α : Type v
inst✝⁴ : TopologicalSpace X
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderClosedTopology α
inst✝ : PreconnectedSpace X
a b : X
f g : X → α
hf : Continuous[inst✝⁴, inst✝²] f
hg : Continuous[inst✝⁴, inst✝²] g
ha : f a ≤ g a
hb : g b ≤ f b
⊢ ∃ x, f x = g x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Preorder α
h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)
a b : α
hab : ¬a ≤ b
⊢ IsCompact (Icc a b)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PartialOrder α
h : ∀ {a b : α}, a < b → IsCompact (Icc a b)
a✝ b✝ : α
hab : a✝ ≤ b✝
⊢ a✝ = b✝ → IsCompact (Icc a✝ b✝)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : CompactIccSpace α
⊢ ∀ {a b : αᵒᵈ}, IsCompact (Icc a b)

Tactic:
by

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

Goal state before:
pp: α✝ : Type u_1
α : Type u_2
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
⊢ IsCompact univ

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsCompact (Ico a b)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsCompact (Ioc a b)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsCompact (Ioo a b)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
s : Set α
hs : IsCompact s
ne_s : s.Nonempty
⊢ ∃ x, IsLeast s x

Tactic:
by

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

Goal state before:
pp: X : Type u
α : Type v
inst✝⁴ : TopologicalSpace X
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderClosedTopology α
s : Set X
hs : IsPreconnected s
a : X
l : Filter X
ha : a ∈ s
inst✝ : l.NeBot
hl : l ≤ 𝓟 s
f g : X → α
hf : ContinuousOn f s
hg : ContinuousOn g s
ha' : f a ≤ g a
he : g ≤ᶠ[l] f
⊢ ∃ x ∈ s, f x = g x

Tactic:
by

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

Goal state before:
pp: X : Type u
α : Type v
inst✝⁵ : TopologicalSpace X
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderClosedTopology α
s : Set X
hs : IsPreconnected s
l₁ l₂ : Filter X
inst✝¹ : l₁.NeBot
inst✝ : l₂.NeBot
hl₁ : l₁ ≤ 𝓟 s
hl₂ : l₂ ≤ 𝓟 s
f g : X → α
hf : ContinuousOn f s
hg : ContinuousOn g s
he₁ : f ≤ᶠ[l₁] g
he₂ : g ≤ᶠ[l₂] f
⊢ ∃ x ∈ s, f x = g x

Tactic:
by

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

Goal state before:
pp: α : Type v
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
s : Set α
hs : IsPreconnected s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ Icc a b ⊆ s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : CompactIccSpace α
⊢ cocompact α ≤ atBot ⊔ atTop

Tactic:
by

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

Goal state before:
pp: α : Type v
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
s : Set α
hs : IsPreconnected s
hb : ¬BddBelow s
ha : ¬BddAbove s
⊢ s = univ

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : PreconnectedSpace α
x y : α
hxy : x < y
this : (Iio y ∩ Ioi x).Nonempty
⊢ ∃ a, x < a ∧ a < y

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : PreconnectedSpace α
x y : α
hxy : x < y
⊢ ∃ a, x < a ∧ a < y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTop α
inst✝ : CompactIccSpace α
⊢ cocompact α ≤ atBot

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : NoMinOrder α
inst✝ : ClosedIicTopology α
⊢ atBot ≤ cocompact α

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIicTopology α
s : Set β
hs : IsCompact s
ne_s : s.Nonempty
f : β → α
hf : ContinuousOn f s
⊢ ∃ x ∈ s, IsMinOn f s x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
s : Set β
hs : IsCompact s
hf : ContinuousOn f s
a : α
hf' : ∀ b ∈ s, a < f b
⊢ ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIicTopology α
s : Set β
f : β → α
hf : ContinuousOn f s
hsc : IsClosed[inst✝¹] s
x₀ : β
h₀ : x₀ ∈ s
hc : ∀ᶠ (x : β) in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x
⊢ ∃ x ∈ s, IsMinOn f s x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIicTopology α
f : β → α
hf : Continuous[inst✝¹, inst✝²] f
x₀ : β
h : ∀ᶠ (x : β) in cocompact β, f x₀ ≤ f x
⊢ ∀ᶠ (x : β) in cocompact β ⊓ 𝓟 univ, f x₀ ≤ f x

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty β
f : β → α
hf : Continuous[inst✝², inst✝³] f
hlim : Tendsto f (cocompact β) atTop
⊢ ∃ x, ∀ (y : β), f x ≤ f y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : ClosedIicTopology α
inst✝¹ : Nonempty β
inst✝ : One α
f : β → α
hf : Continuous[inst✝³, inst✝⁴] f
h : HasCompactMulSupport f
⊢ ∃ x, ∀ (y : β), f x ≤ f y

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty α
s : Set α
hs : IsCompact s
⊢ BddBelow s

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology α
f : β → α
K : Set β
hK : IsCompact K
h0K : K.Nonempty
hf : ContinuousOn f K
y : α
h : ∀ x ∈ K, f x < y
x : β
hx : x ∈ K
h2x : IsMaxOn f K x
x' : β
hx' : x' ∈ K
⊢ f x' ≤ f x

Tactic:
exact h2x hx'

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIciTopology α
f : β → α
K : Set β
hK : IsCompact K
h0K : K.Nonempty
hf : ContinuousOn f K
y : α
⊢ sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y

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' : I'.divMod I J' = 0
⊢ J' = I'

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
K : Set β
hK : IsCompact K
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
h0K : K.Nonempty
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z > f x y
⊢ {x} ×ˢ K ⊆ ↿f ⁻¹' Iio z

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' : ¬I'.divMod I J' = 0
⊢ spanSingleton R⁰ (I'.divMod I 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
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 :
  (spanSingleton R⁰ (I'.divMod I J'))⁻¹ * J' ⊓
      (spanSingleton R⁰ (I'.divMod I J'))⁻¹ * (spanSingleton R⁰ (I'.divMod I J') * I) =
    J
H' : ¬I'.divMod I J' = 0
⊢ spanSingleton R⁰ (I'.divMod I J') ≠ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
K : Set β
hK : IsCompact K
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
⊢ Continuous[inst✝, inst✝³] fun x ↦ sSup (f x '' K)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : TopologicalSpace β
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b : α
h : ContinuousOn f [[a, b]]
⊢ f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ dist a b = ‖b⁻¹ * a‖

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : TopologicalSpace β
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f (Icc a b)
hc : c ∈ Icc a b
⊢ sInf (f '' Icc a b) ≤ f c

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
r : ℝ
hpos : 0 < r
hr : ∀ (x : E), x ≠ 1 → r ≤ ‖x‖
x y : E
hne : x ≠ y
⊢ x⁻¹ * y ≠ 1

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
r : ℝ
hpos : 0 < r
hr : ∀ (x : E), x ≠ 1 → r ≤ ‖x‖
x y : E
hne : x ≠ y
⊢ (fun x1 x2 ↦ r ≤ dist x1 x2) x y

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ dist a 1 = ‖a‖

Tactic:
by

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : TopologicalSpace β
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f (Icc a b)
hc : c ∈ Icc a b
⊢ f c ≤ sSup (f '' Icc a b)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ dist 1 a = ‖a‖

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ ‖a⁻¹‖ = ‖a‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
n : ℤ
⊢ ‖a ^ |n|‖ = ‖a ^ n‖

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
⊢ SecondCountableTopology α

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
f : Filter α
hf : Cauchy f
⊢ ∃ x, f ≤ nhds x

Tactic:
by

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

Goal state before:
pp: α✝ : Type u
β✝ : Type v
X : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α✝
α : Type u_3
β : Type u_4
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : ProperSpace α
inst✝ : ProperSpace β
⊢ ∀ (x : α × β) (r : ℝ), IsCompact (closedBall x r)

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
ι : Type u_2
inst✝² : PseudoMetricSpace α
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
h : ∀ (b : β), ProperSpace (X b)
⊢ ProperSpace ((b : β) → X b)

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
n : ℤ
⊢ ‖a ^ n.natAbs‖ = ‖a ^ n‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
n : ℤ
hn : IsUnit n
⊢ ‖a ^ n‖ = ‖a‖

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
h : UniformCauchySeqOn F atTop s
ε : ℝ
hε : ε > 0
u : Set (α × α) := {a | dist a.1 a.2 < ε}
⊢ ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u

Tactic:
by

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

Goal state before:
pp: α : Type u_2
E : Type u_5
inst✝ : SeminormedGroup E
s t : Set α
f : α → E
x : α
⊢ dist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ ‖a * b‖ ≤ ‖a‖ + ‖b‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b c : E
⊢ ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ ‖a‖ ≤ ‖a / b‖ + ‖b‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ 0 ≤ ‖a‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
⊢ ‖1‖ = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ a = 1 → ‖a‖ = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝¹ : SeminormedGroup E
inst✝ : Subsingleton E
a : E
⊢ ‖a‖ = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x : E
⊢ 0 < 1 + ‖x‖ ^ 2

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ ‖a / b‖ ≤ ‖a‖ + ‖b‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ dist a b ≤ ‖a‖ + ‖b‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ |‖a‖ - ‖b‖| ≤ ‖a⁻¹ * b‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ ‖a‖ - ‖b‖ ≤ ‖a * b‖

Tactic:
by

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

Goal state before:
pp: α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ UniformCauchySeqOn F atTop s ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖u‖ ≤ ‖v‖ + ‖u / v‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖u‖ ≤ ‖v‖ + ‖u⁻¹ * v‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖v‖ ≤ ‖u‖ + ‖u / v‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖u‖ = ‖u * v / v‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖v‖ = ‖u⁻¹ * (u * v)‖

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖u⁻¹‖ + ‖u * v‖ = ‖u * v‖ + ‖u‖

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖x * y‖ = ‖y‖

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology 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.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
⊢ Submodule.map ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I = ↑(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✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x * y‖ = ‖x‖

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : ℕ → α
hs : CauchySeq s
S : ℕ → Set ℝ := fun N ↦ (fun p ↦ dist (s p.1) (s p.2)) '' {p | p.1 ≥ N ∧ p.2 ≥ N}
⊢ ∀ (N : ℕ), ∃ x, ∀ y ∈ S N, y ≤ x

Tactic:
by

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖x / y‖ = ‖y‖

Tactic:
by

Goal state after: