mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
hf : BddBelow (f '' Ioi x)
hf_mono : Monotone f
⊢ Nonempty { r' // x < ↑r' }

Tactic:
obtain ⟨r, hrx⟩ := exists_rat_gt x

Goal state after:
f : ℝ → ℝ
x : ℝ
hf : BddBelow (f '' Ioi x)
hf_mono : Monotone f
r : ℚ
hrx : x < ↑r
⊢ Nonempty { r' // x < ↑r' }
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) =
    ∏ᶠ (a : HeightOneSpectrum R), ↑(a.asIdeal ^ (Associates.mk a.asIdeal).count (Associates.mk I).factors)

Tactic:
FractionalIdeal.coeIdeal_pow,

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) =
    ∏ᶠ (a : HeightOneSpectrum R), ↑a.asIdeal ^ (Associates.mk a.asIdeal).count (Associates.mk I).factors
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: E : Type u_3
inst✝ : Group E
s : Set (GroupSeminorm E)
hs : ¬BddAbove s
⊢ sSup s = 0

Tactic:
by
  simp [SupSet.sSup, hs]

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

Goal state before:
pp: E : Type u_3
inst✝ : AddGroup E
s : Set (NonarchAddGroupSeminorm E)
hs : ¬BddAbove s
⊢ sSup s = 0

Tactic:
by
  simp [SupSet.sSup, hs]

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

Goal state before:
pp: case inr.inr
α✝ : Type u_1
inst✝¹ : LinearOrder α✝
s✝ t✝ : Set α✝
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
a : α
has : a ∈ s
b : α
hbt : b ∈ t
hab : a ≤ b
hsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
x : { x // x ∈ s.ordSeparatingSet t }
ha : [[a, ↑x]] ⊆ tᶜ
hb : [[b, ↑x]] ⊆ sᶜ
hax : a ≤ ↑x
hxb : ↑x ≤ b
y : α
hy : y ∈ (s.ordSeparatingSet t).ordConnectedComponent ↑x
⊢ y ∈ [[a, b]]

Tactic:
rw [uIcc_of_le hab, mem_Icc, ← not_lt, ← not_lt]

Goal state after:
case inr.inr
α✝ : Type u_1
inst✝¹ : LinearOrder α✝
s✝ t✝ : Set α✝
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
a : α
has : a ∈ s
b : α
hbt : b ∈ t
hab : a ≤ b
hsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
x : { x // x ∈ s.ordSeparatingSet t }
ha : [[a, ↑x]] ⊆ tᶜ
hb : [[b, ↑x]] ⊆ sᶜ
hax : a ≤ ↑x
hxb : ↑x ≤ b
y : α
hy : y ∈ (s.ordSeparatingSet t).ordConnectedComponent ↑x
⊢ ¬y < a ∧ ¬b < y
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: case pos
x : EReal
h : x = ⊤
⊢ x.toENNReal = ∞ ↔ x = ⊤

Tactic:
simp [h]

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

Goal state before:
pp: a b : ℝ≥0∞
ha₀ : a ≠ 0
ha : a ≠ ∞
⊢ a = 0 → b = 0

Tactic:
simp [ha₀]

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

Goal state before:
pp: case pos
x : EReal
hx : 0 ≤ x
h : x = ⊤
⊢ x.toENNReal.toReal = x.toReal

Tactic:
simp [h]

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

Goal state before:
pp: a b : ℝ≥0∞
ha₀ : a = 0 → b = 0
ha : a = ∞ → b = 0
⊢ a * (b / a) = b

Tactic:
rw [mul_comm, ENNReal.div_mul_cancel' ha₀ ha]

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

Goal state before:
pp: case coe.inl
x✝ : ℝ≥0
h : 0 < x✝
⊢ (↑x✝)⁻¹ < (↑0)⁻¹

Tactic:
simp [h]

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

Goal state before:
pp: a b c : ℝ≥0∞
h : a ≤ b * c
h0 : c = 0
⊢ a = 0

Tactic:
simpa [h0] using h

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

Goal state before:
pp: a : ℝ≥0∞
h : a ≠ 0
⊢ ∃ n, (↑n)⁻¹ < a⁻¹⁻¹

Tactic:
simp only [ENNReal.inv_lt_inv, ENNReal.exists_nat_gt (inv_ne_top.2 h)]

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

Goal state before:
pp: ι : Sort u_1
f : ι → ℝ≥0∞
a : ℝ≥0∞
ha₀ : a ≠ 0
ha : a ≠ ∞
⊢ a = 0 → Nonempty ι

Tactic:
simp [ha₀]

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

Goal state before:
pp: ι : Sort u_1
f : ι → ℝ≥0∞
a : ℝ≥0∞
ha₀ : a ≠ 0
ha : a ≠ ∞
⊢ a = 0 → ⨅ i, f i = 0 → ∃ i, f i = 0

Tactic:
simp [ha₀]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x₁ y₁ x₂ y₂ : α
⊢ dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂)

Tactic:
rw [add_left_comm, dist_comm x₁, ← add_assoc]

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x₁ y₁ x₂ y₂ : α
⊢ dist x₂ y₂ ≤ dist x₂ x₁ + dist x₁ y₁ + dist y₁ y₂
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✝¹ : PseudoMetricSpace α✝
α : Type u
inst✝ : PseudoEMetricSpace α
h : ∀ (x y : α), edist x y ≠ ∞
⊢ ∀ (x y : α), edist x y = ENNReal.ofReal (edist x y).toReal

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Data.Real.Pointwise

Goal state before:
pp: case neg
α : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Module α ℝ
inst✝ : IsOrderedModule α ℝ
a : α
ha : a ≤ 0
s : Set ℝ
hs : s.Nonempty
ha' : a < 0
h : ¬BddBelow s
⊢ sSup (a • s) = a • sInf s

Tactic:
· rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_neg ha').1 h),
        Real.sInf_of_not_bddBelow h, smul_zero]

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

Goal state before:
pp: ι : Sort u_3
inst✝ : Nonempty ι
a : ℝ≥0
g : ι → ℝ≥0
h : ℝ≥0
H : ∀ (i : ι), a ≤ g i * h
⊢ a ≤ iInf g * h

Tactic:
rw [iInf_mul]
  exact le_ciInf H

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

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
H : ∀ ε > 0, ∃ β x F, ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε
hs : s.Nonempty
⊢ TotallyBounded s

Tactic:
rcases hs with ⟨x0, hx0⟩

Goal state after:
case inr
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
H : ∀ ε > 0, ∃ β x F, ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε
x0 : α
hx0 : x0 ∈ s
⊢ TotallyBounded s
Module: Mathlib.Topology.MetricSpace.Pseudo.Basic

Goal state before:
pp: β : Type v
α : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetrizableSpace α
f : β → α
inst✝ : TopologicalSpace β
hf : IsInducing f
s : Set α
hs : IsSeparable s
⊢ IsSeparable (f ⁻¹' s)

Tactic:
by
  let : UniformSpace α := TopologicalSpace.pseudoMetrizableSpaceUniformity α
  have := pseudoMetrizableSpaceUniformity_countably_generated
  have : SeparableSpace s := hs.separableSpace
  have : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _
  have : IsInducing ((mapsTo_preimage f s).restrict _ _ _) :=
    (hf.comp IsInducing.subtypeVal).codRestrict _
  have := this.secondCountableTopology
  exact .of_subtype _

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

Goal state before:
pp: c : ℝ
c_nn : 0 ≤ c
⊢ closedBall 0 c = Icc 0 c.toNNReal

Tactic:
by
  convert! NNReal.closedBall_zero_eq_Icc' (NNReal.mk c c_nn)
  simp [Real.toNNReal, c_nn]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
r : ℝ
⊢ sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r

Tactic:
obtain hr | rfl | hr := lt_trichotomy r 0

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
r : ℝ
hr : r < 0
⊢ sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r

case inr.inl
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
⊢ sphere x 0 = sphere x.1 0 ×ˢ closedBall x.2 0 ∪ closedBall x.1 0 ×ˢ sphere x.2 0

case inr.inr
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α × β
r : ℝ
hr : 0 < r
⊢ sphere x r = sphere x.1 r ×ˢ closedBall x.2 r ∪ closedBall x.1 r ×ˢ sphere x.2 r
Module: Mathlib.Topology.MetricSpace.Pseudo.Pi

Goal state before:
pp: case pos
β : Type u_2
X : β → Type u_3
inst✝² : Fintype β
inst✝¹ : (b : β) → PseudoMetricSpace (X b)
inst✝ : Nonempty β
f g : (b : β) → X b
r : ℝ
hr : 0 ≤ r
⊢ dist f g ≤ r ↔ ∀ (b : β), dist (f b) (g b) ≤ r

pp: case neg
β : Type u_2
X : β → Type u_3
inst✝² : Fintype β
inst✝¹ : (b : β) → PseudoMetricSpace (X b)
inst✝ : Nonempty β
f g : (b : β) → X b
r : ℝ
hr : ¬0 ≤ r
⊢ dist f g ≤ r ↔ ∀ (b : β), dist (f b) (g b) ≤ r

Tactic:
· exact dist_pi_le_iff hr

Goal state after:
case neg
β : Type u_2
X : β → Type u_3
inst✝² : Fintype β
inst✝¹ : (b : β) → PseudoMetricSpace (X b)
inst✝ : Nonempty β
f g : (b : β) → X b
r : ℝ
hr : ¬0 ≤ r
⊢ dist f g ≤ r ↔ ∀ (b : β), dist (f b) (g b) ≤ r
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
x : R
hx : x ≠ 0
y : R
hxI : x ∈ Ideal.span {x} ⊔ Ideal.span {y}
⊢ ∃ y_1, Ideal.span {y, x} = Ideal.span {y_1, x}

Tactic:
use y

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

Goal state before:
pp: β : Type u_2
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
x : (b : β) → X b
r : ℝ
h : 0 < r ∨ Nonempty β
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r

Tactic:
obtain hr | rfl | hr := lt_trichotomy r 0

Goal state after:
case inl
β : Type u_2
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
x : (b : β) → X b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : r < 0
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r

case inr.inl
β : Type u_2
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
x : (b : β) → X b
h : 0 < 0 ∨ Nonempty β
⊢ sphere x 0 = (⋃ i, Function.eval i ⁻¹' sphere (x i) 0) ∩ closedBall x 0

case inr.inr
β : Type u_2
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
x : (b : β) → X b
r : ℝ
h : 0 < r ∨ Nonempty β
hr : 0 < r
⊢ sphere x r = (⋃ i, Function.eval i ⁻¹' sphere (x i) r) ∩ closedBall x r
Module: Mathlib.Topology.MetricSpace.Basic

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
inst✝² : PseudoMetricSpace α✝
γ : Type w
inst✝¹ : MetricSpace γ
α : Type u
inst✝ : EMetricSpace α
h : ∀ (x y : α), edist x y ≠ ⊤
⊢ ∀ (x y : α), edist x y = ENNReal.ofReal (edist x y).toReal

Tactic:
simp [h]

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

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

Tactic:
← Set.compl_Iio,

Goal state after:
α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
a d : α
h : ∀ᶠ (x : γ) in l, f x < d
hd : IsGLB (Ioi a) d
H0 : (Iic a)ᶜ = (Iio d)ᶜ
x : γ
hx : f x < d
⊢ f x ≤ a
Module: Mathlib.Topology.Order.Bornology

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

Tactic:
by
  refine atTop_le_cobounded.antisymm' fun s ↦ ?_
  rw [Filter.atTop_basis.mem_iff,
    ← compl_compl s, ← isBounded_def, isBounded_iff_bddBelow_bddAbove, compl_compl s]
  refine fun ⟨b, _, hb⟩ ↦ ⟨⟨⊥, fun x hx ↦ by simp⟩, ⟨b, fun x hx ↦ ?_⟩⟩
  by_contra! hx'
  exact hx (hb hx'.le)

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

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := ⋯
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := ⋯
⊢ (x '' t).PairwiseDisjoint fun x ↦ Ioo x (y (f x))

pp: case h'
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := ⋯
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := ⋯
⊢ ∀ x_1 ∈ x '' t, x_1 < y (f x_1)

Tactic:
swap

Goal state after:
case h'
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := ⋯
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := ⋯
⊢ ∀ x_1 ∈ x '' t, x_1 < y (f x_1)

case h
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := ⋯
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := ⋯
⊢ (x '' t).PairwiseDisjoint fun x ↦ Ioo x (y (f x))
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: case inl
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 = ∅
⊢ Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x)))

Tactic:
simp [h]

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

Goal state before:
pp: case inr
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
⊢ Tendsto (fun x ↦ sSup (f x '' K)) (𝓝 x) (𝓝 (sSup (f x '' K)))

Tactic:
h2y,

Goal state after:
case inr
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
⊢ Tendsto (fun x ↦ sSup (f x '' K)) (𝓝 x) (𝓝 (f x y))
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inr.refine_1
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
⊢ ∀ᶠ (b : γ) in 𝓝 x, z < sSup (f b '' K)

Tactic:
refine (this.1 z hz).mono fun x' hx' =>
      hx'.trans_le <| le_csSup ?_ <| mem_image_of_mem (f x') hyK

Goal state after:
case inr.refine_1
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' : γ
hx' : z < f x' y
⊢ BddAbove (f x' '' K)
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:
rw [dist_eq_norm_inv_mul, Set.apply_mulIndicator_symmDiff norm_inv']

Goal state after:
α : Type u_2
E : Type u_5
inst✝ : SeminormedGroup E
s t : Set α
f : α → E
x : α
⊢ ‖(s.mulIndicator f x)⁻¹ * t.mulIndicator f x‖ = ‖s.mulIndicator f x / t.mulIndicator f x‖
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
ha : a ∈ s
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty
⊢ b ∈ s

Tactic:
let S := s ∩ Icc a b

Goal state after:
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
ha : a ∈ s
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty
S : Set α := s ∩ Icc a b
⊢ b ∈ s
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
⊢ ∏ i, (↑i).maxPowDividing (map (algebraMap S R) p) = ∏ i, ↑i ^ (↑i).ramificationIdx S

pp: case h
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
⊢ (mulSupport fun v ↦ v.maxPowDividing (map (algebraMap S R) p)) ⊆ ↑{v | v.asIdeal ∣ map (algebraMap S R) p}.toFinset

Tactic:
· let _ : Fintype {v : HeightOneSpectrum R // v.asIdeal ∣ map (algebraMap S R) p} := hF
    refine Fintype.prod_equiv (equivPrimesOver _ hp) _ _ fun ⟨v, _⟩ ↦ ?_
    have : v.asIdeal.LiesOver p := by rwa [Ideal.liesOver_iff_dvd_map v.2.ne_top]
    simp [maxPowDividing_eq_pow_multiset_count _ h, ramificationIdx_eq_factors_count p v h]

Goal state after:
case h
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
⊢ (mulSupport fun v ↦ v.maxPowDividing (map (algebraMap S R) p)) ⊆ ↑{v | v.asIdeal ∣ map (algebraMap S R) p}.toFinset
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: case inr.refine_2
α : Type u
inst✝ : PseudoMetricSpace α
U : Set α
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : α
⊢ ⋃ i, (fun i ↦ U ∩ ball x ↑i) i = U

Tactic:
simp [← inter_iUnion]

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

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

Goal state before:
pp: α : Type u_1
m n : ℤ
⊢ dist m n = ‖↑(-m + n)‖

Tactic:
by
    simp only [dist_eq, norm, cast_add, cast_neg]
    rw [abs_sub_comm, neg_add_eq_sub]

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

Goal state before:
pp: z : ℤ
c : ℝ≥0
⊢ |z| ≤ ↑⌊c⌋₊ ↔ ‖z‖₊ ≤ c

Tactic:
Int.abs_eq_natAbs,

Goal state after:
z : ℤ
c : ℝ≥0
⊢ ↑z.natAbs ≤ ↑⌊c⌋₊ ↔ ‖z‖₊ ≤ c
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]

Goal state after:
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
⊢ (⨆ i, f a i) + ⨆ i, ∑ a ∈ s, f a i = ⨆ i, f a i + ∑ a ∈ s, f a i
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Preorder α
f : Filter β
u : β → α
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u ↔ ∃ s, BddAbove (u '' s) ∧ ∀ᶠ (x : β) in f, x ∈ s

Tactic:
by
  constructor
  · rintro ⟨b, hb⟩
    exact ⟨{a | u a ≤ b}, ⟨b, by rintro _ ⟨a, ha, rfl⟩; exact ha⟩, hb⟩
  · rintro ⟨s, ⟨b, hb⟩, hs⟩
    exact ⟨b, hs.mono <| by simpa [upperBounds] using hb⟩

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

Goal state before:
pp: case mp
x : EReal
h : ⊤ + x = ⊤
⊢ x ≠ ⊥

pp: case mpr
x : EReal
h : x ≠ ⊥
⊢ ⊤ + x = ⊤

Tactic:
· rintro rfl
    rw [add_bot] at h
    exact bot_ne_top h

Goal state after:
case mpr
x : EReal
h : x ≠ ⊥
⊢ ⊤ + x = ⊤
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: case inl
α : Type u_1
f : Filter α
inst✝¹ : LinearOrder α
inst✝ : f.NeBot
t : α
ht : ∀ (a : α), (∀ᶠ (x : α) in f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) t a
tbot : IsBot t
⊢ ∃ l, ∃ᶠ (x : α) in f, l ≤ x

pp: case inr
α : Type u_1
f : Filter α
inst✝¹ : LinearOrder α
inst✝ : f.NeBot
t : α
ht : ∀ (a : α), (∀ᶠ (x : α) in f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) t a
t' : α
ht' : t' < t
⊢ ∃ l, ∃ᶠ (x : α) in f, l ≤ x

Tactic:
· exact ⟨t, .of_forall fun r ↦ tbot r⟩

Goal state after:
case inr
α : Type u_1
f : Filter α
inst✝¹ : LinearOrder α
inst✝ : f.NeBot
t : α
ht : ∀ (a : α), (∀ᶠ (x : α) in f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) t a
t' : α
ht' : t' < t
⊢ ∃ l, ∃ᶠ (x : α) in f, l ≤ x
Module: Mathlib.Topology.Order.MonotoneConvergence

Goal state before:
pp: case inl
α : Type u_1
ι : Type u_3
inst✝³ : Preorder ι
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompletePartialOrderSup α
inst✝ : SupConvergenceClass α
f : ι → α
h_mono : Monotone f
hbdd : BddAbove (range f)
h : atTop = ⊥
⊢ Tendsto f atTop (𝓝 (⨆ i, f i))

Tactic:
simp [h]

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

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝³ : Preorder ι
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompletePartialOrderSup α
inst✝ : SupConvergenceClass α
f : ι → α
h_anti : Antitone f
hbdd : BddAbove (range f)
⊢ Tendsto f atBot (𝓝 (⨆ i, f i))

Tactic:
by convert! tendsto_atTop_ciSup h_anti.dual hbdd.dual using 1

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

Goal state before:
pp: case left
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : Preorder α
inst✝³ : OrderClosedTopology α
inst✝² : Preorder β
inst✝¹ : IsDirectedOrder β
inst✝ : Nonempty β
f : β → α
a : α
hf : Monotone f
ha : Tendsto f atTop (𝓝 a)
⊢ a ∈ upperBounds (range f)

pp: case right
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : Preorder α
inst✝³ : OrderClosedTopology α
inst✝² : Preorder β
inst✝¹ : IsDirectedOrder β
inst✝ : Nonempty β
f : β → α
a : α
hf : Monotone f
ha : Tendsto f atTop (𝓝 a)
⊢ a ∈ lowerBounds (upperBounds (range f))

Tactic:
· rintro _ ⟨b, rfl⟩
    exact hf.ge_of_tendsto ha b

Goal state after:
case right
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : Preorder α
inst✝³ : OrderClosedTopology α
inst✝² : Preorder β
inst✝¹ : IsDirectedOrder β
inst✝ : Nonempty β
f : β → α
a : α
hf : Monotone f
ha : Tendsto f atTop (𝓝 a)
⊢ a ∈ lowerBounds (upperBounds (range f))
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: β : Type u_2
α : Type u_6
inst✝ : ConditionallyCompleteLattice β
f : Filter α
u v : α → β
h : ∀ᶠ (a : α) in f, u a = v a
b : β
⊢ (∀ᶠ (n : α) in f, u n ≤ b) ↔ ∀ᶠ (n : β) in map v f, n ≤ b

Tactic:
exact eventually_congr (h.mono fun x hx => by simp [hx])

Goal state after: