mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: x y : EReal
h : x ≤ y
hx : x ≠ ⊥
hy : y ≠ ⊤
⊢ x.toReal ≤ y.toReal

Tactic:
lift x to ℝ using ⟨ne_top_of_le_ne_top hy h, hx⟩

Goal state after:
y : EReal
hy : y ≠ ⊤
x : ℝ
h : ↑x ≤ y
hx : ↑x ≠ ⊥
⊢ (↑x).toReal ≤ y.toReal
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case pos
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal R⁰ K
inst✝ : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'

Tactic:
exact count_mul K v h.1 h.2

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

Goal state before:
pp: case pos
x : EReal
hx : 0 ≤ x
h_top : x = ⊤
⊢ ↑(if x = ⊤ then ∞ else ENNReal.ofReal x.toReal) = x

Tactic:
if_pos h_top,

Goal state after:
case pos
x : EReal
hx : 0 ≤ x
h_top : x = ⊤
⊢ ↑∞ = x
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: case inr
x : EReal
hx : x ≤ 0
⊢ ↑x.toENNReal = max 0 x

Tactic:
rw [toENNReal_of_nonpos hx, max_eq_left hx, coe_ennreal_zero]

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

Goal state before:
pp: case pos
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
n : ℤ
I : FractionalIdeal R⁰ K
hI : I = 0
hn : n = 0
⊢ count K v (I ^ 0) = -count K v (I ^ 0)

Tactic:
zpow_zero,

Goal state after:
case pos
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
n : ℤ
I : FractionalIdeal R⁰ K
hI : I = 0
hn : n = 0
⊢ count K v 1 = -count K v 1
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case coe
a : ℝ≥0
⊢ (↑a)⁻¹.toNNReal = (↑a).toNNReal⁻¹

Tactic:
rcases eq_or_ne a 0 with (rfl | ha); · simp
    rw [← coe_inv ha, toNNReal_coe, toNNReal_coe]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ edist x y < ↑c ↔ nndist x y < c

Tactic:
edist_nndist,

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ ↑(nndist x y) < ↑c ↔ nndist x y < c
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ edist x y ≤ ↑c ↔ nndist x y ≤ c

Tactic:
edist_nndist,

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x y : α
c : ℝ≥0
⊢ ↑(nndist x y) ≤ ↑c ↔ nndist x y ≤ c
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ (∃ C, ∀ x ∈ sᶜᶜ, ∀ y ∈ sᶜᶜ, dist x y ≤ C) ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C

Tactic:
compl_compl

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ (∃ C, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α✝
α : Type ?u.9
B : Bornology α
m : PseudoMetricSpace α
H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
s : Set α
⊢ (∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C) ↔ ∃ C, ∀ x ∈ sᶜᶜ, ∀ y ∈ sᶜᶜ, dist x y ≤ C

Tactic:
compl_compl

Goal state after:
α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α✝
α : Type ?u.9
B : Bornology α
m : PseudoMetricSpace α
H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s
s : Set α
⊢ (∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C
Module: Mathlib.Data.NNReal.Basic

Goal state before:
pp: ι : Sort u_3
f : ι → ℝ≥0
a : ℝ≥0
⊢ (⨆ i, f i) * a = ⨆ i, f i * a

Tactic:
by
  rw [mul_comm, mul_iSup]
  simp_rw [mul_comm]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ edist x y = 0 ↔ nndist x y = 0

Tactic:
edist_nndist,

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ ↑(nndist x y) = 0 ↔ nndist x y = 0
Module: Mathlib.Topology.EMetricSpace.Basic

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α
x✝ y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
inst✝ : SigmaCompactSpace α
T : ℕ → Set α
h✝ : ∀ (n : ℕ), T n ⊆ compactCovering α n
hTc : ∀ (n : ℕ), (T n).Countable
hsubT : ∀ (n : ℕ), compactCovering α n ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (T n)
x : α
n : ℕ
hn : x ∈ compactCovering α n
⊢ x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ n, T n)

Tactic:
exact closure_mono (subset_iUnion _ n) (hsubT _ hn)

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 : α
ha : (Iio a).Nonempty
⊢ interior (Ici a) = Ioi a

Tactic:
← compl_Iio,

Goal state after:
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 (Iio a)ᶜ = Ioi a
Module: Mathlib.Topology.Order.IsLUB

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
s : Set α
hs : Dense s
x y : α
hy : y < x
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Ioo y x ∩ s) ∧ Tendsto u atTop (𝓝 x)

Tactic:
have hnonempty : (Ioo y x ∩ s).Nonempty := by
    obtain ⟨z, hyz, hzx⟩ := hs.exists_between hy
    exact ⟨z, mem_inter hzx hyz⟩

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
inst✝ : FirstCountableTopology α
s : Set α
hs : Dense s
x y : α
hy : y < x
hnonempty : (Ioo y x ∩ s).Nonempty
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Ioo y x ∩ s) ∧ Tendsto u atTop (𝓝 x)
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
  simp only [← dist_one_right, dist_pi_lt_iff hr, Pi.one_apply]

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

Goal state before:
pp: case pos
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
⊢ 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:
iterate 6 apply Or.inr

Goal state after:
case pos
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
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}
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
  simpa using norm_mul_le' (a / b) (b / c)

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

Goal state before:
pp: case inr
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
⊢ Icc a b ⊆ s

Tactic:
set t₁ := sSup A

Goal state after:
case inr
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
⊢ Icc a b ⊆ s
Module: Mathlib.Analysis.Normed.Group.Basic

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

Tactic:
edist_nndist,

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ ↑(nndist 1 a) = ↑‖a‖₊
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
⊢ edist (s.mulIndicator f x) (t.mulIndicator f x) = ↑‖(s ∆ t).mulIndicator f x‖₊

Tactic:
edist_nndist,

Goal state after:
α : Type u_2
E : Type u_5
inst✝ : SeminormedGroup E
s t : Set α
f : α → E
x : α
⊢ ↑(nndist (s.mulIndicator f x) (t.mulIndicator f x)) = ↑‖(s ∆ t).mulIndicator f x‖₊
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:
simp only [mem_ball, dist_eq_norm_inv_mul, mul_comm _ (b * c), mul_comm a⁻¹ b]

Goal state after:
E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
c : E
⊢ ‖b * c * (a * c)⁻¹‖ < r ↔ ‖b * a⁻¹‖ < r
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:
rcases le_total (f a) (f b) with hab | hab

Goal state after:
case inl
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
hab : f a ≤ f b
⊢ SurjOn f s [[f a, f b]]

case inr
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
hab : f b ≤ f a
⊢ SurjOn f s [[f a, f b]]
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 (comap Int.cast atTop)

Tactic:
tendsto_comap_iff

Goal state after:
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 (Int.cast ∘ f) l atTop
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case neg
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
a b : α
hab : a < b
h : f b ≤ f a
H : f b < f a
ha : f ⊥ < f a
hb : f b ≤ f ⊥
⊢ False

Tactic:
replace hb : f b < f ⊥ := lt_of_le_of_ne hb <| hf_i.ne (lt_of_lt_of_le' hab bot_le).ne'

Goal state after:
case neg
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
a b : α
hab : a < b
h : f b ≤ f a
H : f b < f a
ha : f ⊥ < f a
hb : f b < f ⊥
⊢ False
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 (comap Int.cast atBot)

Tactic:
tendsto_comap_iff

Goal state after:
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 (Int.cast ∘ f) l atBot
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
  rw [← comap_dist_right_atTop c, tendsto_comap_iff, Function.comp_def]

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 (comap Rat.cast atTop)

Tactic:
tendsto_comap_iff

Goal state after:
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 (Rat.cast ∘ f) l atTop
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 (comap Rat.cast atBot)

Tactic:
tendsto_comap_iff

Goal state after:
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 (Rat.cast ∘ f) l atBot
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 : Ring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : r < 0
hf : Tendsto f l atBot
⊢ Tendsto (fun x ↦ f x * r) l atTop

Tactic:
by
  simpa only [mul_neg, tendsto_neg_atBot_iff] using hf.atBot_mul_const' (neg_pos.2 hr)

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
a : G
a_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}
a_in : a ∈ H
a_pos : 1 < a
⊢ {a} ⊆ ↑H

Tactic:
by simp [a_in]

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

Goal state before:
pp: case refine_2
α : Type u
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
K s : Set α
hsd : IsDiscrete s
hK : Bornology.IsBounded K
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ IsDiscrete (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K ∩ s)

Tactic:
exact hsd.mono Set.inter_subset_right

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

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderClosedTopology α
inst✝¹ : PseudoMetricSpace β
inst✝ : ProperSpace β
f : β → α
hf : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, inst✝³] f
x₀ : β
h : Bornology.IsBounded {x | f x ≤ f x₀}
hU : {x | f x₀ < f x} ∈ cocompact β
⊢ ∀ᶠ (x : β) in cocompact β, f x₀ ≤ f x

Tactic:
filter_upwards [hU] with x hx using hx.le

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)
g : G
hg : g > 1
⊢ ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))

Tactic:
rcases existsUnique_mul_zpow_mem_Ico h₀ 1 (g / a) with ⟨m, ⟨hm, hm'⟩, -⟩
    simp only [one_mul, div_le_iff_le_mul, div_mul_cancel, ← zpow_add_one] at hm hm'
    lift m to ℕ
    · rw [← Int.lt_add_one_iff, ← zpow_lt_zpow_iff_right h₀, zpow_zero]
      exact hg.trans_le hm
    · simp only [← Nat.cast_succ, zpow_natCast] at hm hm'
      exact ⟨m, hm', hm⟩

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

Goal state before:
pp: case cons
ι : Type u_1
α : Type u_2
f : α → ι → ℝ≥0∞
hf : ∀ (i j : ι), ∃ k, ∀ (a : α), f a i ≤ f a k ∧ f a j ≤ f a k
a : α
s : Finset α
ha : a ∉ s
ihs : ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i
⊢ ∑ a ∈ Finset.cons a s ha, ⨆ i, f a i = ⨆ i, ∑ a ∈ Finset.cons a s ha, f a i

Tactic:
simp_rw [Finset.sum_cons, ihs]
    refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_
    gcongr
    exacts [(hk a).1, (hk _).2]

Goal state after:
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s : Set α
⊢ IsBounded r (𝓟 s) ↔ ∃ t, ∀ x ∈ s, r x t

Tactic:
by
  simp [IsBounded]

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

Goal state before:
pp: x y : EReal
h_top : x ≠ ⊤ ∨ y ≠ ⊤
h_bot : x ≠ ⊥ ∨ y ≠ ⊥
⊢ 0 ≤ x - y ↔ y ≤ x

Tactic:
cases x <;> cases y <;> simp_all [← EReal.coe_sub]

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

Goal state before:
pp: x y : EReal
h_top : x ≠ ⊤ ∨ y ≠ ⊤
h_bot : x ≠ ⊥ ∨ y ≠ ⊥
⊢ x - y < 0 ↔ x < y

Tactic:
cases x <;> cases y <;> simp_all [← EReal.coe_sub]

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

Goal state before:
pp: case pos
α : Type u_2
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
f : Filter α
inst✝ : f.NeBot
hc : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
hb : IsBounded (fun x1 x2 ↦ x1 ≤ x2) f
hn : Nontrivial α
⊢ ClusterPt f.limsSup f

pp: case neg
α : Type u_2
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
f : Filter α
inst✝ : f.NeBot
hc : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
hb : IsBounded (fun x1 x2 ↦ x1 ≤ x2) f
hn : Subsingleton α
⊢ ClusterPt f.limsSup f

Tactic:
· by_cases! htop : ∀ x, x ≤ f.limsSup
    · let : OrderTop α := { top := f.limsSup, le_top := htop }
      exact nhds_top_basis.clusterPt_iff_frequently |>.mpr fun a => frequently_lt_of_lt_limsSup hc
    · by_cases! hbot : ∀ x, f.limsSup ≤ x
      · let : OrderBot α := { bot := f.limsSup, bot_le := hbot }
        refine nhds_bot_basis.clusterPt_iff_frequently |>.mpr fun a h => ?_
        exact lt_mem_sets_of_limsSup_lt hb h |>.frequently
      · refine (nhds_basis_Ioo' hbot htop).clusterPt_iff_frequently |>.mpr fun a ⟨hl, hg⟩ => ?_
        exact frequently_lt_of_lt_limsSup hc hl |>.and_eventually <| lt_mem_sets_of_limsSup_lt hb hg

Goal state after:
case neg
α : Type u_2
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
f : Filter α
inst✝ : f.NeBot
hc : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
hb : IsBounded (fun x1 x2 ↦ x1 ≤ x2) f
hn : Subsingleton α
⊢ ClusterPt f.limsSup f
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b c d : EReal
h1 : c ≠ ⊥ ∨ d ≠ ⊤
h2 : c ≠ ⊤ ∨ d ≠ ⊥
⊢ a + b - (c + d) = a - c + (b - d)

Tactic:
rw [sub_eq_add_neg, sub_eq_add_neg, sub_eq_add_neg, EReal.neg_add h1 h2, sub_eq_add_neg]

Goal state after:
a b c d : EReal
h1 : c ≠ ⊥ ∨ d ≠ ⊤
h2 : c ≠ ⊤ ∨ d ≠ ⊥
⊢ a + b + (-c + -d) = a + -c + (b + -d)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b : EReal
⊢ ¬a * b = ⊤ ↔ (a ≠ ⊥ ∨ 0 ≤ b) ∧ (0 ≤ a ∨ b ≠ ⊥) ∧ (a ≠ ⊤ ∨ b ≤ 0) ∧ (a ≤ 0 ∨ b ≠ ⊤)

Tactic:
mul_eq_top

Goal state after:
a b : EReal
⊢ ¬(a = ⊥ ∧ b < 0 ∨ a < 0 ∧ b = ⊥ ∨ a = ⊤ ∧ 0 < b ∨ 0 < a ∧ b = ⊤) ↔
    (a ≠ ⊥ ∨ 0 ≤ b) ∧ (0 ≤ a ∨ b ≠ ⊥) ∧ (a ≠ ⊤ ∨ b ≤ 0) ∧ (a ≤ 0 ∨ b ≠ ⊤)
Module: Mathlib.Topology.Algebra.InfiniteSum.Defs

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
L : SummationFilter β
hL : ¬L.NeBot
f : β → α
this : L.LeAtTop
hf : (mulSupport f).Finite
⊢ (if L.HasSupport ∧ (mulSupport fun b ↦ f b).Finite then ∏ᶠ (b : β), f b
    else if HasProd (fun b ↦ f b) 1 L then 1 else Exists.choose ⋯) =
    ∏ᶠ (b : β), f b

Tactic:
eq_true_intro hf,

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
L : SummationFilter β
hL : ¬L.NeBot
f : β → α
this : L.LeAtTop
hf : (mulSupport f).Finite
⊢ (if L.HasSupport ∧ True then ∏ᶠ (b : β), f b else if HasProd (fun b ↦ f b) 1 L then 1 else Exists.choose ⋯) =
    ∏ᶠ (b : β), f b
Module: Mathlib.Topology.Algebra.InfiniteSum.Group

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : IsTopologicalGroup α
f : β → α
a₁ a₂ : α
s : Set β
hf : HasProd (f ∘ Subtype.val) a₁
⊢ HasProd (f ∘ Subtype.val) a₂ ↔ HasProd f (a₁ * a₂)

Tactic:
refine ⟨fun h ↦ hf.mul_compl h, fun h ↦ ?_⟩
  rw [hasProd_subtype_iff_mulIndicator] at hf ⊢
  rw [Set.mulIndicator_compl]
  simpa only [div_eq_mul_inv, mul_inv_cancel_comm] using! h.div hf

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Group

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : CommGroup α
inst✝ : IsUniformGroup α
f : β → α
⊢ (∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e) ↔
    ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), Disjoint t ↑s → ∏' (b : ↑t), f ↑b ∈ e

Tactic:
Set.disjoint_left,

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : CommGroup α
inst✝ : IsUniformGroup α
f : β → α
⊢ (∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e) ↔
    ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), (∀ ⦃a : β⦄, a ∈ t → a ∉ ↑s) → ∏' (b : ↑t), f ↑b ∈ e
Module: Mathlib.Topology.Algebra.InfiniteSum.Group

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : CommGroup α
inst✝ : IsUniformGroup α
f : β → α
⊢ (∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e) ↔
    ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), (∀ ⦃a : β⦄, a ∈ t → a ∉ ↑s) → ∏' (b : ↑t), f ↑b ∈ e

Tactic:
disjoint_left

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : CommGroup α
inst✝ : IsUniformGroup α
f : β → α
⊢ (∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), (∀ ⦃a : β⦄, a ∈ t → a ∉ s) → ∏ b ∈ t, f b ∈ e) ↔
    ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Set β), (∀ ⦃a : β⦄, a ∈ t → a ∉ ↑s) → ∏' (b : ↑t), f ↑b ∈ e
Module: Mathlib.Topology.Algebra.InfiniteSum.NatInt

Goal state before:
pp: case inr
M : Type u_1
inst✝³ : CommMonoid M
inst✝² : TopologicalSpace M
α : Type u_3
β : Type u_4
inst✝¹ : Encodable β
inst✝ : CompleteLattice α
m : α → M
m0 : m ⊥ = 1
s : β → α
n : ℕ
hn : decode₂ β n = none
⊢ m (⨆ b ∈ decode₂ β n, s b) = 1 n

Tactic:
simp [hn, m0]

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.NatInt

Goal state before:
pp: G : Type u_2
inst✝² : CommGroup G
g : G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
f : ℕ → G
k : ℕ
⊢ HasProd (fun n ↦ f (n + k)) (g / ∏ i ∈ range k, f i) ↔ HasProd f g

Tactic:
by
  simp [hasProd_nat_add_iff]

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
g : γ → β
f : β → α
hg : Injective g
⊢ ∏' (x : ↑Set.univ), f (g ↑x) = ∏' (x : γ), f (g x)

Tactic:
simp_rw [← comp_apply (g := g), tprod_univ (f ∘ g)]

Goal state after:
Module: Mathlib.Topology.Instances.NNReal.Lemmas

Goal state before:
pp: α : Type u_2
L : SummationFilter α
f : α → ℝ≥0
r : ℝ≥0
⊢ HasSum (fun a ↦ ↑(f a)) (↑r) L ↔ HasSum f r L

Tactic:
by
  simp only [HasSum, ← coe_sum, tendsto_coe]

Goal state after: