mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Unramified.LocalRing

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal R
inst✝³ : p.IsPrime
q : Ideal S
inst✝² : q.IsPrime
hq : p.primesOver S = {q}
inst✝¹ : Module.Finite R S
inst✝ : q.LiesOver p
hRS : (RingHom.ker (algebraMap R S)).FG
H✝ : Function.Bijective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset S
hs : Algebra.adjoin R ↑s = ⊤
a b : S → R
hbp : ∀ (x : S), b x ∉ p
e : ∀ (x : S), (algebraMap R S) (a x) = x * (algebraMap R S) (b x)
r : R
hrp : r ∉ p
hr : ∀ (r' : R), r ∣ r' → Function.Injective ⇑(awayMap (algebraMap R S) r')
r' : R
hr' : r * ∏ i ∈ s, b i ∣ r'
H : (IsScalarTower.toAlgHom R S (Away ((Algebra.ofId R S) r'))).range ≤ (awayMapₐ (Algebra.ofId R S) r').range
x : S
n : ℕ
y : Away r'
hy : (awayMap (algebraMap R S) r') y = (algebraMap S (Away ((algebraMap R S) r'))) x
⊢ ∃ a,
    (awayMap (algebraMap R S) r') a =
      IsLocalization.mk' (Away ((algebraMap R S) r')) x ⟨(fun x ↦ (algebraMap R S) r' ^ x) n, ⋯⟩

Tactic:
refine ⟨y * Localization.Away.invSelf _ ^ n, ?_⟩

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
p : Ideal R
inst✝³ : p.IsPrime
q : Ideal S
inst✝² : q.IsPrime
hq : p.primesOver S = {q}
inst✝¹ : Module.Finite R S
inst✝ : q.LiesOver p
hRS : (RingHom.ker (algebraMap R S)).FG
H✝ : Function.Bijective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset S
hs : Algebra.adjoin R ↑s = ⊤
a b : S → R
hbp : ∀ (x : S), b x ∉ p
e : ∀ (x : S), (algebraMap R S) (a x) = x * (algebraMap R S) (b x)
r : R
hrp : r ∉ p
hr : ∀ (r' : R), r ∣ r' → Function.Injective ⇑(awayMap (algebraMap R S) r')
r' : R
hr' : r * ∏ i ∈ s, b i ∣ r'
H : (IsScalarTower.toAlgHom R S (Away ((Algebra.ofId R S) r'))).range ≤ (awayMapₐ (Algebra.ofId R S) r').range
x : S
n : ℕ
y : Away r'
hy : (awayMap (algebraMap R S) r') y = (algebraMap S (Away ((algebraMap R S) r'))) x
⊢ (awayMap (algebraMap R S) r') (y * Away.invSelf r' ^ n) =
    IsLocalization.mk' (Away ((algebraMap R S) r')) x ⟨(fun x ↦ (algebraMap R S) r' ^ x) n, ⋯⟩
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: case inl
ι : Sort u_1
h✝ : IsEmpty ι
⊢ ⨅ x, 0 = 0

pp: case inr
ι : Sort u_1
h✝ : Nonempty ι
⊢ ⨅ x, 0 = 0

Tactic:
· exact Real.iInf_of_isEmpty _

Goal state after:
case inr
ι : Sort u_1
h✝ : Nonempty ι
⊢ ⨅ x, 0 = 0
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: a : ℝ
ha : 1 < a
k : ℕ
posk : 0 < k
hk : 1 / ↑k + 1 < a
⊢ (1 / ↑k + 1) ^ (2 * k ^ 2) < a ^ (2 * k ^ 2)

Tactic:
by gcongr

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.PrimaryComponent

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDedekindDomain A
M₁ : Type u_5
M₂ : Type u_6
inst✝³ : AddCommGroup M₁
inst✝² : AddCommGroup M₂
inst✝¹ : Module A M₁
inst✝ : Module A M₂
hM₁ : IsTorsion A M₁
P : HeightOneSpectrum A
φ : M₁ →ₗ[A] M₂
hf✝ : Surjective ⇑φ
b : M₁
hy : φ b ∈ primaryComponent M₂ P.asIdeal
f : Π₀ (i : HeightOneSpectrum A), ↥(primaryComponent M₁ i.asIdeal)
hf : ((DFinsupp.lsum ℕ) fun i ↦ (primaryComponent M₁ i.asIdeal).subtype) f = b
⊢ φ b - φ ↑(f P) ∈ ⨆ j, ⨆ (_ : j ≠ P), (fun P ↦ primaryComponent M₂ P.asIdeal) j

Tactic:
· have hdiff : φ b - φ ↑(f P) = ∑ Q ∈ f.support \ {P}, φ ↑(f Q) := by
      rw [sub_eq_iff_eq_add',
        ← Finset.sum_eq_add_sum_sdiff_singleton P (fun P ↦ φ (f P)) (by aesop)]
      simpa [DFinsupp.sumAddHom_apply, DFinsupp.sum] using congr(φ $hf).symm
    rw [hdiff]
    exact Submodule.sum_mem _ fun Q hQ ↦ Submodule.mem_iSup_of_mem Q <|
        Submodule.mem_iSup_of_mem (by grind) (primaryComponent_map_mem _ _ _)

Goal state after:
Module: Mathlib.Order.Interval.Set.OrdConnectedComponent

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x : α
⊢ (s ∩ t).ordConnectedComponent x = s.ordConnectedComponent x ∩ t.ordConnectedComponent x

Tactic:
simp [ordConnectedComponent, ofPred_and]

Goal state after:
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: case inr
ι : Sort u_1
inst✝ : Nonempty ι
f : ι → ℝ
h : ∀ (i : ι), 0 ≤ f i
⊢ ENNReal.ofReal (⨅ i, f i) = ⨅ i, ENNReal.ofReal (f i)

Tactic:
refine eq_of_forall_le_iff fun a ↦ ?_

Goal state after:
case inr
ι : Sort u_1
inst✝ : Nonempty ι
f : ι → ℝ
h : ∀ (i : ι), 0 ≤ f i
a : ℝ≥0∞
⊢ a ≤ ENNReal.ofReal (⨅ i, f i) ↔ a ≤ ⨅ i, ENNReal.ofReal (f i)
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: case inr
ι : Sort u_1
a : ℝ≥0∞
inst✝ : Nonempty ι
f : ι → ℝ≥0∞
ha : a ≠ ∞
⊢ a + ⨆ i, f i ≤ ⨆ i, a + f i

Tactic:
refine add_le_of_le_tsub_left_of_le (le_iSup_of_le (Classical.arbitrary _) le_self_add) ?_

Goal state after:
case inr
ι : Sort u_1
a : ℝ≥0∞
inst✝ : Nonempty ι
f : ι → ℝ≥0∞
ha : a ≠ ∞
⊢ ⨆ i, f i ≤ (⨆ i, a + f i) - a
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: case mpr
x : EReal
h : x ≠ ⊤
⊢ ∃ y, x ≤ ↑y

Tactic:
exact ⟨x.toReal, le_coe_toReal h⟩

Goal state after:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: x y : ℝ
⊢ WithBot.some '' WithTop.some '' Ioo x y = Ioo ↑x ↑y

Tactic:
WithTop.image_coe_Ioo,

Goal state after:
x y : ℝ
⊢ WithBot.some '' Ioo ↑x ↑y = Ioo ↑x ↑y
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case neg
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 : FractionalIdeal R⁰ K
n : ℕ
h : count K v (I ^ n) = ↑n * count K v I
hI : ¬I = 0
⊢ ↑n * count K v I + count K v I = (↑n + ↑1) * count K v I

Tactic:
Nat.cast_one

Goal state after:
case neg
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 : FractionalIdeal R⁰ K
n : ℕ
h : count K v (I ^ n) = ↑n * count K v I
hI : ¬I = 0
⊢ ↑n * count K v I + count K v I = (↑n + 1) * count K v I
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
⊢ eball x 0 = ∅

Tactic:
rw [eball_eq_empty_iff]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝¹ : PseudoMetricSpace α
β : Type u_3
p : β → Prop
f : β → ℝ
l : Filter β
inst✝ : l.NeBot
hf₀ : ∀ (i : β), p i → 0 < f i
hf₁ : ∀ᶠ (i : β) in l, p i
hf : ∀ (i : ℝ), 0 < i → ∀ᶠ (x : β) in l, f x ∈ closedBall 0 i
⊢ ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε

Tactic:
refine fun ε hε ↦ hf₁.and (hf ε hε) |>.exists.imp fun i ↦ and_imp.mpr fun hp hi ↦ ?_

Goal state after:
α : Type u
inst✝¹ : PseudoMetricSpace α
β : Type u_3
p : β → Prop
f : β → ℝ
l : Filter β
inst✝ : l.NeBot
hf₀ : ∀ (i : β), p i → 0 < f i
hf₁ : ∀ᶠ (i : β) in l, p i
hf : ∀ (i : ℝ), 0 < i → ∀ᶠ (x : β) in l, f x ∈ closedBall 0 i
ε : ℝ
hε : 0 < ε
i : β
hp : p i
hi : f i ∈ closedBall 0 ε
⊢ p i ∧ f i ≤ ε
Module: Mathlib.Topology.MetricSpace.Pseudo.Basic

Goal state before:
pp: α : Type u
β : Type v
ι : Type u_1
inst✝ : PseudoMetricSpace α
F : ι → β → α
f : β → α
p : Filter ι
p' : Filter β
⊢ TendstoUniformlyOnFilter F f p p' ↔ ∀ ε > 0, ∀ᶠ (n : ι × β) in p ×ˢ p', dist (f n.2) (F n.1 n.2) < ε

Tactic:
by
  refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩
  rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
  exact (H ε εpos).mono fun n hn => hε hn

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
f : Filter α
⊢ Cauchy f ↔ f ≠ ⊥ ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, edist x y < ε

Tactic:
rw [← neBot_iff]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
f : Filter α
⊢ Cauchy f ↔ f.NeBot ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, edist x y < ε
Module: Mathlib.Topology.Order.DenselyOrdered

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a : α
ha : (Iio a).Nonempty
⊢ interior (Ici a) = Ioi a

Tactic:
by
  rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ Ioc a b ⊆ closure[inst✝³] (interior (Ioc a b))

Tactic:
rcases eq_or_ne a b with (rfl | h)

Goal state after:
case inl
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a : α
⊢ Ioc a a ⊆ closure[inst✝³] (interior (Ioc a a))

case inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
h : a ≠ b
⊢ Ioc a b ⊆ closure[inst✝³] (interior (Ioc a b))
Module: Mathlib.Topology.Order.IsLUB

Goal state before:
pp: case inr
γ : Type u_2
α : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : ClosedIicTopology α
f : γ → α
inst✝ : TopologicalSpace γ
S : Set γ
hS : Dense S
hf : Continuous[inst✝, inst✝³] f
h : BddAbove (range f)
h✝ : Nonempty γ
⊢ IsLUB (range f) (sSup (range fun s ↦ f ↑s))

Tactic:
refine (isLUB_congr (hS.upperBounds_image hf)).mp (isLUB_ciSup_set ?_ hS.nonempty)

Goal state after:
case inr
γ : Type u_2
α : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : ClosedIicTopology α
f : γ → α
inst✝ : TopologicalSpace γ
S : Set γ
hS : Dense S
hf : Continuous[inst✝, inst✝³] f
h : BddAbove (range f)
h✝ : Nonempty γ
⊢ BddAbove (f '' S)
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
this :
  ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : Field K] [inst_2 : Algebra R K]
    [inst_3 : IsFractionRing R K] [IsDedekindDomain R] {a b c : FractionalIdeal R⁰ K},
    a ≤ c → a ≠ 0 → b ≠ 0 → b = 1 → ∃ x, a + FractionalIdeal.spanSingleton R⁰ x * b = c
hb' : ¬b = 1
⊢ b⁻¹ * a ≤ b⁻¹ * c

Tactic:
by gcongr

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
t : Set α
x : α
inst✝ : (𝓝 x).IsCountablyGenerated
htx : IsLUB t x
notMem : x ∉ t
ht : t.Nonempty
v : ℕ → α
hvt : ∀ (n : ℕ), v n ∈ t
hvx : Tendsto v atTop (𝓝 x)
hvx' : ∀ {n : ℕ}, v n < x
this : ∀ (k : ℕ), ∀ᶠ (l : ℕ) in atTop, v k < v l
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t

Tactic:
choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
t : Set α
x : α
inst✝ : (𝓝 x).IsCountablyGenerated
htx : IsLUB t x
notMem : x ∉ t
ht : t.Nonempty
v : ℕ → α
hvt : ∀ (n : ℕ), v n ∈ t
hvx : Tendsto v atTop (𝓝 x)
hvx' : ∀ {n : ℕ}, v n < x
this : ∀ (k : ℕ), ∀ᶠ (l : ℕ) in atTop, v k < v l
N : ℕ → ℕ
hN : ∀ (k : ℕ), k < N k
hvN : ∀ (k : ℕ), v k < v (N k)
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t
Module: Mathlib.Topology.MetricSpace.Basic

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
H : ∀ ε > 0, ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε
⊢ SecondCountableTopology α

Tactic:
by
  refine secondCountable_of_almost_dense_set fun ε ε0 => ?_
  rcases H ε ε0 with ⟨β, fβ, F, hF⟩
  let Finv := rangeSplitting F
  refine ⟨range Finv, ⟨countable_range _, fun x => ?_⟩⟩
  let x' := Finv ⟨F x, mem_range_self _⟩
  have : F x' = F x := apply_rangeSplitting F _
  exact ⟨x', mem_range_self _, hF _ _ this.symm⟩

Goal state after:
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 : y < x
u : ℕ → α
hu : StrictMono u
huyxf : ∀ (n : ℕ), u n ∈ Ioo y x ∩ range f
hlim : Tendsto u atTop (𝓝 x)
huyx : ∀ (n : ℕ), u n ∈ Ioo y x
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x)

Tactic:
have huf (n : ℕ) : u n ∈ range f := (huyxf n).2

Goal state after:
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 : y < x
u : ℕ → α
hu : StrictMono u
huyxf : ∀ (n : ℕ), u n ∈ Ioo y x ∩ range f
hlim : Tendsto u atTop (𝓝 x)
huyx : ∀ (n : ℕ), u n ∈ Ioo y x
huf : ∀ (n : ℕ), u n ∈ range f
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x)
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 : y < x
u : ℕ → α
hu : StrictMono u
huyxf : ∀ (n : ℕ), u n ∈ Ioo y x ∩ range f
hlim : Tendsto u atTop (𝓝 x)
huyx : ∀ (n : ℕ), u n ∈ Ioo y x
v : ℕ → β
hv : ∀ (n : ℕ), f (v n) = u n
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x)

Tactic:
obtain rfl : f ∘ v = u := funext hv

Goal state after:
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 : y < x
v : ℕ → β
hu : StrictMono (f ∘ v)
huyxf : ∀ (n : ℕ), (f ∘ v) n ∈ Ioo y x ∩ range f
hlim : Tendsto (f ∘ v) atTop (𝓝 x)
huyx : ∀ (n : ℕ), (f ∘ v) n ∈ Ioo y x
hv : ∀ (n : ℕ), f (v n) = (f ∘ v) n
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x)
Module: Mathlib.Topology.Order.IsLUB

Goal state before:
pp: α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : LinearOrder α
inst✝⁴ : OrderTopology α
β : Type u_3
inst✝³ : LinearOrder β
inst✝² : DenselyOrdered α
inst✝¹ : NoMinOrder α
inst✝ : FirstCountableTopology α
f : β → α
hf : DenseRange f
hmono : Monotone f
x : α
u : ℕ → α
hu : StrictMono u
huxf : ∀ (n : ℕ), u n ∈ Iio x ∩ range f
hlim : Tendsto u atTop (𝓝 x)
hux : ∀ (n : ℕ), u n ∈ Iio x
v : ℕ → β
hv : ∀ (n : ℕ), f (v n) = u n
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Iio x) ∧ Tendsto (f ∘ u) atTop (𝓝 x)

Tactic:
obtain rfl : f ∘ v = u := funext hv

Goal state after:
α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : LinearOrder α
inst✝⁴ : OrderTopology α
β : Type u_3
inst✝³ : LinearOrder β
inst✝² : DenselyOrdered α
inst✝¹ : NoMinOrder α
inst✝ : FirstCountableTopology α
f : β → α
hf : DenseRange f
hmono : Monotone f
x : α
v : ℕ → β
hu : StrictMono (f ∘ v)
huxf : ∀ (n : ℕ), (f ∘ v) n ∈ Iio x ∩ range f
hlim : Tendsto (f ∘ v) atTop (𝓝 x)
hux : ∀ (n : ℕ), (f ∘ v) n ∈ Iio x
hv : ∀ (n : ℕ), f (v n) = (f ∘ v) n
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), f (u n) ∈ Iio x) ∧ Tendsto (f ∘ u) atTop (𝓝 x)
Module: Mathlib.Topology.Order.Bornology

Goal state before:
pp: α : Type u_1
inst✝² : Bornology α
inst✝¹ : LinearOrder α
inst✝ : IsOrderBornology α
this : Nonempty α
s : Set α
⊢ ((∃ i, True ∧ Iic i ⊆ sᶜᶜ) ∧ ∃ i, True ∧ Ici i ⊆ sᶜᶜ) → BddBelow sᶜ ∧ BddAbove sᶜ

Tactic:
compl_compl s

Goal state after:
α : Type u_1
inst✝² : Bornology α
inst✝¹ : LinearOrder α
inst✝ : IsOrderBornology α
this : Nonempty α
s : Set α
⊢ ((∃ i, True ∧ Iic i ⊆ s) ∧ ∃ i, True ∧ Ici i ⊆ s) → BddBelow sᶜ ∧ BddAbove sᶜ
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
⊢ (∃ i, True ∧ Ici i ⊆ sᶜᶜ) → BddBelow sᶜ ∧ BddAbove sᶜ

Tactic:
compl_compl s

Goal state after:
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
⊢ (∃ i, True ∧ Ici i ⊆ s) → BddBelow sᶜ ∧ BddAbove sᶜ
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case neg.inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s = Ici (sInf s)
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case neg.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s ∈ {Ioi (sInf s)}
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
· exact Or.inl hs

Goal state after:
case neg.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s ∈ {Ioi (sInf s)}
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case pos.inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : BddAbove s
hs : s = Iic (sSup s)
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case pos.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : BddAbove s
hs : s ∈ {Iio (sSup s)}
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
· exact Or.inl hs

Goal state after:
case pos.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : BddAbove s
hs : s ∈ {Iio (sSup s)}
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}
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:
rintro ⟨x', y'⟩ ⟨(rfl : x' = x), hy'⟩

Goal state after:
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
y : β
hyK : y ∈ K
z : α
x' : γ
y' : β
hy' : (x', y').2 ∈ 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'
hz : z > f x' y
⊢ (x', y') ∈ ↿f ⁻¹' Iio z
Module: Mathlib.Topology.MetricSpace.Cauchy

Goal state before:
pp: case h
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 : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
N : β
hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
b : β × β
hb : (N, N) ≤ b
⊢ ∀ x ∈ s, (F b.1 x, F b.2 x) ∈ u

Tactic:
x

Goal state after:
case h
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 : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
u : Set (α × α)
hu : u ∈ 𝓤 α
ε : ℝ
hε : ε > 0
hab : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ u
N : β
hN : ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
b : β × β
hb : (N, N) ≤ b
x : γ
⊢ x ∈ s → (F b.1 x, F b.2 x) ∈ u
Module: Mathlib.Analysis.Normed.Group.Basic

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

Tactic:
by
  simpa using norm_mul_le' (u / v) (v / w)

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

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝¹ : SeminormedGroup E
inst✝ : SeminormedGroup F
x : E
y : F
h : ‖x‖ ≤ ‖y‖
⊢ ENNReal.ofReal ‖x‖ ≤ ENNReal.ofReal ‖y‖

Tactic:
by gcongr

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

Goal state before:
pp: α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
n : ℕ
hn : 1 ≤ r * ↑n
x : α
hx : ↑n * max b 0 ≤ f x
⊢ 1 * max b 0 ≤ r * ↑n * max b 0

Tactic:
by gcongr

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

Goal state before:
pp: α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
n : ℕ
hn : 1 ≤ r * ↑n
x : α
hx : ↑n * max b 0 ≤ f x
⊢ r * (↑n * max b 0) ≤ r * f x

Tactic:
by gcongr

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : TopologicalSpace β
k s : Set β
f : β → α
hk : IsCompact k
hf : ∀ x ∈ k, ContinuousWithinAt f s x
⊢ ∃ t, k ⊆ t ∧ IsOpen[inst✝] t ∧ Bornology.IsBounded (f '' (t ∩ s))

Tactic:
have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by
    rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left]
    exact fun x hx ↦ disjoint_left_comm.2 <|
      tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx)
  rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this
    with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩
  refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩
  rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_right]

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

Goal state before:
pp: α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
n : ℕ
hn : 1 ≤ n • r
hn' : 1 ≤ ↑n * r
x : α
hx : max b 0 * ↑n ≤ f x
⊢ max b 0 * 1 ≤ max b 0 * (↑n * r)

Tactic:
by gcongr

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

Goal state before:
pp: α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
n : ℕ
hn : 1 ≤ n • r
hn' : 1 ≤ ↑n * r
x : α
hx : max b 0 * ↑n ≤ f x
⊢ max b 0 * ↑n * r ≤ f x * r

Tactic:
by gcongr

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : TopologicalSpace β
k : Set β
f : β → α
hk : IsCompact k
hf : ∀ x ∈ k, ContinuousAt f x
⊢ ∃ t, k ⊆ t ∧ IsOpen[inst✝] t ∧ Bornology.IsBounded (f '' t)

Tactic:
by
  simp_rw [← continuousWithinAt_univ] at hf
  simpa only [inter_univ] using
    exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf

Goal state after:
Module: Mathlib.GroupTheory.Archimedean

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
H : Subgroup G
hbot : H ≠ ⊥
a : G
h₀ : 1 < a
hd : Disjoint (↑H) (Ioo 1 a)
hex : ∀ g > 1, ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))
this : ∃ n, (↑H ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (a ^ m) (a ^ (m + 1))).Nonempty
x : G
hxH : x ∈ ↑H
hnx : a ^ n < x
hxn : x ≤ a ^ (n + 1)
hxmin : ∀ (x : G), x ∈ H ∧ 1 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 1 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 1 < y
m : ℕ
hm : a ^ m < y
hya : y ≤ a ^ (m + 1)
hnm : n ≤ m
⊢ a ^ m * a < y * a

Tactic:
by gcongr

Goal state after: