mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit

Goal state before:
pp: case h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
l : Products I
gl : Products.isGood (π C fun x ↦ ord I x < o₁) l
x : ↑(π C fun x ↦ ord I x < o₂)
⊢ (Products.eval (π C fun x ↦ ord I x < o₂) ↑⟨l, ⋯⟩) x = ((πs' C h) (eval (π C fun x ↦ ord I x < o₁) ⟨l, gl⟩)) x

Tactic:
← Products.eval_πs' _ h (Products.prop_of_isGood C _ gl),

Goal state after:
case h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
l : Products I
gl : Products.isGood (π C fun x ↦ ord I x < o₁) l
x : ↑(π C fun x ↦ ord I x < o₂)
⊢ ((πs' C h) (Products.eval (π C fun x ↦ ord I x < o₁) l)) x = ((πs' C h) (eval (π C fun x ↦ ord I x < o₁) ⟨l, gl⟩)) x
Module: Mathlib.Topology.Category.Profinite.Nobeling.Successor

Goal state before:
pp: case neg
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this : Inhabited I
a : I
ha : a ∈ ↑q
hM : ¬↑(↑l).Tail = []
⊢ (↑(↑l).Tail).head! < term I ho

Tactic:
have := l.val.prop

Goal state after:
case neg
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this✝ : Inhabited I
a : I
ha : a ∈ ↑q
hM : ¬↑(↑l).Tail = []
this : List.IsChain (fun x1 x2 ↦ x1 > x2) ↑↑l
⊢ (↑(↑l).Tail).head! < term I ho
Module: Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : CompactSpace β
A : Set (α →ᵇ β)
closed : IsClosed A
H : Equicontinuous fun x ↦ ⇑↑x
⊢ IsCompact A

Tactic:
simp_rw [Equicontinuous, Metric.equicontinuousAt_iff_pair] at H

Goal state after:
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : CompactSpace β
A : Set (α →ᵇ β)
closed : IsClosed A
H : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε
⊢ IsCompact A
Module: Mathlib.Topology.Filter

Goal state before:
pp: α : Type u_2
l : Filter α
⊢ ⋂₀ {s | s ∈ 𝓝 l} = Iic l

Tactic:
nhds_eq,

Goal state after:
α : Type u_2
l : Filter α
⊢ ⋂₀ {s | s ∈ l.lift' (Iic ∘ 𝓟)} = Iic l
Module: Mathlib.Topology.Homotopy.LocallyContractible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
ι : Type u_3
p : X → ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h' : ∀ (x : X) (i : ι), p x i → ContractibleSpace ↑(s x i)
x : X
t : Set X
ht : t ∈ 𝓝 x
⊢ ∃ r ∈ 𝓝 x, ContractibleSpace ↑r ∧ r ⊆ t

Tactic:
obtain ⟨i, hpi, hi⟩ := (h x).mem_iff.mp ht

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
ι : Type u_3
p : X → ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h' : ∀ (x : X) (i : ι), p x i → ContractibleSpace ↑(s x i)
x : X
t : Set X
ht : t ∈ 𝓝 x
i : ι
hpi : p x i
hi : s x i ⊆ t
⊢ ∃ r ∈ 𝓝 x, ContractibleSpace ↑r ∧ r ⊆ t
Module: Mathlib.Topology.Instances.CantorSet

Goal state before:
pp: ⊢ cantorSet = (fun x ↦ x / 3) '' cantorSet ∪ (fun x ↦ (2 + x) / 3) '' cantorSet

Tactic:
simp only [cantorSet]
  rw [Set.image_iInter, Set.image_iInter]
  rotate_left
  · exact (mulRight_bijective₀ 3⁻¹ (by simp)).comp (AddGroup.addLeft_bijective 2)
  · exact mulRight_bijective₀ 3⁻¹ (by simp)
  simp_rw [← Function.comp_def,
    ← Set.iInter_union_of_antitone
      (Set.monotone_image.comp_antitone preCantorSet_antitone)
      (Set.monotone_image.comp_antitone preCantorSet_antitone),
    Function.comp_def, ← preCantorSet_succ]
  exact (preCantorSet_antitone.iInter_nat_add _).symm

Goal state after:
Module: Mathlib.Topology.Instances.CantorSet

Goal state before:
pp: i : ℕ
ih : ∀ {a : ℕ → Fin 3}, (∀ (n : ℕ), a n ≠ 1) → ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
⊢ ∀ (n : ℕ), a (n + 1) ≠ 1

Tactic:
by solve_by_elim

Goal state after:
Module: Mathlib.Topology.Instances.CantorSet

Goal state before:
pp: case h
i : ℕ
ih : ∀ {a : ℕ → Fin 3}, (∀ (n : ℕ), a n ≠ 1) → ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
this : (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i
⊢ (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i ∧ (ofDigits fun i ↦ a (i + 1)) / 3 = ofDigits a ∨
    (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i ∧ (2 + ofDigits fun i ↦ a (i + 1)) / 3 = ofDigits a

Tactic:
simp only [this, ofDigits_eq_sum_add_ofDigits a 1, Finset.range_one, ofDigitsTerm,
      Nat.cast_ofNat, Finset.sum_singleton, zero_add, pow_one, true_and, field]

Goal state after:
case h
i : ℕ
ih : ∀ {a : ℕ → Fin 3}, (∀ (n : ℕ), a n ≠ 1) → ofDigits a ∈ preCantorSet i
a : ℕ → Fin 3
h : ∀ (n : ℕ), a n ≠ 1
this : (ofDigits fun i ↦ a (i + 1)) ∈ preCantorSet i
⊢ ((ofDigits fun i ↦ a (i + 1)) = ↑↑(a 0) + ofDigits fun i ↦ a (i + 1)) ∨
    (2 + ofDigits fun i ↦ a (i + 1)) = ↑↑(a 0) + ofDigits fun i ↦ a (i + 1)
Module: Mathlib.Topology.List

Goal state before:
pp: case cons
α : Type u_1
inst✝ : TopologicalSpace α
l : List α
u : List (Set α)
a✝¹ : α
b✝ : Set α
l₁✝ : List α
l₂✝ : List (Set α)
ht : b✝ ∈ 𝓝 a✝¹
a✝ : List.Forall₂ (fun b s ↦ s ∈ 𝓝 b) l₁✝ l₂✝
ih :
  ∀ (s : Set (List α)), sequence l₂✝ ⊆ s → ∃ v, List.Forall₂ (fun a s ↦ IsOpen[inst✝] s ∧ a ∈ s) l₁✝ v ∧ sequence v ⊆ s
s : Set (List α)
hus : sequence (b✝ :: l₂✝) ⊆ s
⊢ ∃ v, List.Forall₂ (fun a s ↦ IsOpen[inst✝] s ∧ a ∈ s) (a✝¹ :: l₁✝) v ∧ sequence v ⊆ s

Tactic:
| cons ht _ ih =>

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

Goal state before:
pp: case inr
X : Type u_1
R : Type u_2
Y : Type u_3
inst✝³ : AddCommMonoid R
inst✝² : LinearOrder R
inst✝¹ : AddLeftStrictMono R
inst✝ : IsOrderedAddMonoid R
f : Y → PseudoMetric X R
s : Finset Y
h✝ : ∀ d ∈ s, (f d).IsUltra
x y z : X
hs : s.Nonempty
i : Y
hi : i ∈ s
h : (f i) x z ≤ (f i) x y ∨ (f i) x z ≤ (f i) y z
⊢ (∃ b ∈ s, (f i) x z ≤ (f b) x y) ∨ ∃ b ∈ s, (f i) x z ≤ (f b) y z

Tactic:
refine h.imp ?_ ?_ <;>
  intro H <;>
  exact ⟨i, hi, H⟩

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

Goal state before:
pp: case refine_1
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a : ℝ≥0∞
n : ℕ
J : Finset T
hJ_card : ↑(#J) ≤ a ^ n
c : ℝ≥0∞
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha1 : a ≤ 1
hJ : Nonempty ↥J
⊢ #J ≤ 1

Tactic:
suffices (#J : ENNReal) ≤ 1 by norm_cast at this

Goal state after:
case refine_1
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a : ℝ≥0∞
n : ℕ
J : Finset T
hJ_card : ↑(#J) ≤ a ^ n
c : ℝ≥0∞
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha1 : a ≤ 1
hJ : Nonempty ↥J
⊢ ↑(#J) ≤ 1
Module: Mathlib.Topology.MetricSpace.Kuratowski

Goal state before:
pp: case inr
α : Type u
inst✝¹ : MetricSpace α
inst✝ : SeparableSpace α
basepoint : α
h✝ : basepoint ∈ univ
⊢ ∃ f, Isometry f

Tactic:
have : Inhabited α := ⟨basepoint⟩

Goal state after:
case inr
α : Type u
inst✝¹ : MetricSpace α
inst✝ : SeparableSpace α
basepoint : α
h✝ : basepoint ∈ univ
this : Inhabited α
⊢ ∃ f, Isometry f
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoMetricSpace α
s : Set α
d : ℝ
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
hd : d ≤ s.infsep
hs : s.Subsingleton
⊢ d ≤ dist x y

Tactic:
hs.infsep_zero

Goal state after:
case neg
α : Type u_1
inst✝ : PseudoMetricSpace α
s : Set α
d : ℝ
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
hd : d ≤ 0
hs : s.Subsingleton
⊢ d ≤ dist x y
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
d : ℝ
h : d ∈ uncurry dist '' s.offDiag
⊢ 0 ≤ d

Tactic:
simp_rw [mem_image, Prod.exists, uncurry_apply_pair] at h

Goal state after:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
d : ℝ
h : ∃ a b, (a, b) ∈ s.offDiag ∧ dist a b = d
⊢ 0 ≤ d
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
w✝¹ w✝ : α
left✝ : (w✝¹, w✝) ∈ s.offDiag
⊢ 0 ≤ dist w✝¹ w✝

Tactic:
exact dist_nonneg

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

Goal state before:
pp: X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝¹ : IsCompact s
this✝ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
⊢ ghDist X Y ≤ hausdorffDist (range Φ') (range Ψ')

Tactic:
let F := kuratowskiEmbedding s

Goal state after:
X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝¹ : IsCompact s
this✝ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
F : ↑s → ↥(lp (fun x ↦ ℝ) ∞) := kuratowskiEmbedding ↑s
⊢ ghDist X Y ≤ hausdorffDist (range Φ') (range Ψ')
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
⊢ scottHausdorff α univ ≤ scott α univ

Tactic:
scott

Goal state after:
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
⊢ scottHausdorff α univ ≤ upperSet α ⊔ scottHausdorff α univ
Module: Mathlib.Topology.Sheaves.Skyscraper

Goal state before:
pp: case e_a
X : TopCat
p₀ : ↑X
inst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝¹ : Category.{u, v} C
A : C
inst✝ : HasTerminal C
y : ↑X
h : ¬p₀ ⤳ y
h1 : ∃ U, p₀ ∉ ↑U :=
  match not_specializes_iff_exists_open.mp h with
  | Exists.intro U ⟨ho, ⟨h₀, hy⟩⟩ => Exists.intro ⟨{ carrier := U, is_open' := ho }, h₀⟩ hy
c : Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A)
U : (OpenNhds y)ᵒᵖ
⊢ p₀ ∉ unop ((OpenNhds.inclusion y).op.obj (op (h1.choose ⊓ unop U)))

Tactic:
exact fun h => h1.choose_spec h.1

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
G : Presheaf C X
α : F ≅ G
b : ι
⊢ (α.hom.app (op (iSup U)) ≫ Fork.ι ((Cone.postcompose (diagram.isoOfIso U α).inv).obj (fork G U))) ≫
      Pi.π (fun i ↦ F.obj (op (U i))) b =
    res F U ≫ Pi.π (fun i ↦ F.obj (op (U i))) b

Tactic:
dsimp only [Fork.ι]

Goal state after:
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
G : Presheaf C X
α : F ≅ G
b : ι
⊢ (α.hom.app (op (iSup U)) ≫
        ((Cone.postcompose (diagram.isoOfIso U α).inv).obj (fork G U)).π.app WalkingParallelPair.zero) ≫
      Pi.π (fun i ↦ F.obj (op (U i))) b =
    res F U ≫ Pi.π (fun i ↦ F.obj (op (U i))) b
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case op.single
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c c' : Cone (SheafConditionEqualizerProducts.diagram F U)
f : c ⟶ c'
i : ι
⊢ f.hom ≫ Fork.ι c' ≫ Pi.π (fun i ↦ F.obj (op (U i))) i = Fork.ι c ≫ Pi.π (fun i ↦ F.obj (op (U i))) i

Tactic:
dsimp only [Fork.ι]

Goal state after:
case op.single
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c c' : Cone (SheafConditionEqualizerProducts.diagram F U)
f : c ⟶ c'
i : ι
⊢ f.hom ≫ c'.π.app WalkingParallelPair.zero ≫ Pi.π (fun i ↦ F.obj (op (U i))) i =
    c.π.app WalkingParallelPair.zero ≫ Pi.π (fun i ↦ F.obj (op (U i))) i
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_2
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : Cb X Y
h : f ∈ candidatesB X Y
r : ℝ
hr : HD (optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
⊢ ∀ x ∈ range (optimalGHInjr X Y), ∃ y ∈ range (optimalGHInjl X Y), dist x y ≤ r

Tactic:
rintro _ ⟨z, rfl⟩
    have I1 : (⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y)) < r :=
      lt_of_le_of_lt (le_max_right _ _) hr
    have I2 :
        ⨅ x, optimalGHDist X Y (inl x, inr z) ≤ ⨆ y, ⨅ x, optimalGHDist X Y (inl x, inr y) :=
      le_csSup (by simpa using HD_bound_aux2 _ 0) (mem_range_self _)
    have I : ⨅ x, optimalGHDist X Y (inl x, inr z) < r := lt_of_le_of_lt I2 I1
    rcases exists_lt_of_csInf_lt (range_nonempty _) I with ⟨r', ⟨z', rfl⟩, hr'⟩
    refine ⟨optimalGHInjl X Y z', mem_range_self _, le_of_lt ?_⟩
    rwa [dist_comm]

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
ha' : IsMinOn (fun x ↦ max (f x ↑y1) (f x ↑y2)) X a
ht' : t' < max (f a ↑y1) (f a ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : J1 ∩ J2 = ∅
hJ1_union_J2 : Subtype.val '' (J1 ∪ J2) = segment ℝ ↑y1 ↑y2
this : IsPreconnected univ
hJ1 : IsClosed J1
⊢ IsClosed J2

Tactic:
simp only [sup_comm (f _ y1)] at ha' ht'

Goal state after:
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
t : β
hinfi_le : ∀ y0 ∈ Y, ∃ x ∈ X, f x y0 ≤ t
a : E
ha : a ∈ X
t' : β
htt' : t < t'
y1 y2 : { x // x ∈ Y }
ht : ∀ x ∈ X, t < max (f x ↑y1) (f x ↑y2)
J1 : Set ↑(segment ℝ ↑y1 ↑y2) := setOfPred_sublevelLeft_subset X Y f t t' y1 y2
mem_J1_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J1 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y1
φ : ↑(segment ℝ ↑y1 ↑y2) ≃ₜ ↑(segment ℝ ↑y2 ↑y1) := Homeomorph.setCongr ⋯
J2 : Set ↑(segment ℝ ↑y1 ↑y2) := ⇑φ ⁻¹' setOfPred_sublevelLeft_subset X Y f t t' y2 y1
mem_J2_iff : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), z ∈ J2 ↔ sublevelLeft X f t ↑z ⊆ sublevelLeft X f t' ↑y2
h_mem_Y : ∀ (z : ↑(segment ℝ ↑y1 ↑y2)), ↑z ∈ Y
hJ1J2 : J1 ∩ J2 = ∅
hJ1_union_J2 : Subtype.val '' (J1 ∪ J2) = segment ℝ ↑y1 ↑y2
this : IsPreconnected univ
hJ1 : IsClosed J1
ha' : IsMinOn (fun x ↦ max (f x ↑y2) (f x ↑y1)) X a
ht' : t' < max (f a ↑y2) (f a ↑y1)
⊢ IsClosed J2
Module: Mathlib.Topology.UniformSpace.Dini

Goal state before:
pp: ι : Type u_1
α : Type u_2
G : Type u_3
inst✝⁵ : Preorder ι
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup G
inst✝² : Lattice G
inst✝¹ : HasSolidNorm G
inst✝ : IsOrderedAddMonoid G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F i)
hF_mono : Monotone F
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
x✝ : atTop.NeBot
F_le_f : ∀ (x : α) (n : ι), F n x ≤ f x
ε : ℝ
ε_pos : ε > 0
x : α
h_tendsto : ∀ (x : α), Tendsto (fun e ↦ ‖F e x - f x‖) atTop (𝓝 0)
n : ι
hn : ‖F n x - f x‖ < ε
m : ι
hnm : n ≤ m
z : α
hz : ‖F n z - f z‖ < ε
⊢ ‖F m z - f z‖ < ε

Tactic:
refine norm_le_norm_of_abs_le_abs ?_ |>.trans_lt hz

Goal state after:
ι : Type u_1
α : Type u_2
G : Type u_3
inst✝⁵ : Preorder ι
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup G
inst✝² : Lattice G
inst✝¹ : HasSolidNorm G
inst✝ : IsOrderedAddMonoid G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F i)
hF_mono : Monotone F
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
x✝ : atTop.NeBot
F_le_f : ∀ (x : α) (n : ι), F n x ≤ f x
ε : ℝ
ε_pos : ε > 0
x : α
h_tendsto : ∀ (x : α), Tendsto (fun e ↦ ‖F e x - f x‖) atTop (𝓝 0)
n : ι
hn : ‖F n x - f x‖ < ε
m : ι
hnm : n ≤ m
z : α
hz : ‖F n z - f z‖ < ε
⊢ |F m z - f z| ≤ |F n z - f z|
Module: Mathlib.Topology.UniformSpace.OfCompactT2

Goal state before:
pp: γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
x : γ
⊢ 𝓝 x =
    (⨆ i, ⨆ (_ : Inseparable x i), comap (fun x_2 ↦ x) (𝓝 i) ⊓ 𝓝 i) ⊔
      ⨆ i, ⨆ (_ : ¬Inseparable x i), comap (fun x_2 ↦ x) (𝓝 i) ⊓ 𝓝 i

Tactic:
conv_rhs =>
      congr
      · enter [1, i, 1, hi]
        rw [← hi.nhds_eq, comap_const_of_mem fun V => mem_of_mem_nhds]
      · enter [1, i, 1, hi]
        rw [(not_specializes_iff_exists_open.1 (mt Specializes.inseparable hi)).elim
          fun S hS => comap_const_of_notMem (hS.1.mem_nhds hS.2.1) hS.2.2]

Goal state after:
γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
x : γ
⊢ 𝓝 x = (⨆ i, ⨆ (_ : Inseparable x i), ⊤ ⊓ 𝓝 x) ⊔ ⨆ i, ⨆ (_ : ¬Inseparable x i), ⊥ ⊓ 𝓝 i
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : CompleteLinearOrder β
inst✝¹⁰ : DenselyOrdered β
X : Set E
Y : Set F
f : E → F → β
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul ℝ E
ne_X : X.Nonempty
cX : Convex ℝ X
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommGroup F
inst✝² : Module ℝ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cY : Convex ℝ Y
ne_Y : Y.Nonempty
kY : IsCompact Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
x : E
hx : x ∈ X
⊢ SemicontinuousWithinAt (fun x1 x2 ↦ (fun x ↦ ⨆ y ∈ Y, f x y) x1 > x2) X x

Tactic:
by
    apply lowerSemicontinuousWithinAt_iSup
    intro i
    exact lowerSemicontinuousWithinAt_iSup fun i_1 ↦ hfy i i_1 x hx

Goal state after: