mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
Ψ : X → ↥(lp (fun n ↦ ℝ) ∞)
isomΨ : Isometry Ψ
rangeΨ : range Ψ = ↑p
f : ↑(range Ψ) ≃ᵢ ↑(range (kuratowskiEmbedding X))
⊢ (↑(range Ψ) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding X)) = (↥p ≃ᵢ ↑(range (kuratowskiEmbedding X)))

Tactic:
dsimp only [NonemptyCompacts.kuratowskiEmbedding]

Goal state after:
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
Ψ : X → ↥(lp (fun n ↦ ℝ) ∞)
isomΨ : Isometry Ψ
rangeΨ : range Ψ = ↑p
f : ↑(range Ψ) ≃ᵢ ↑(range (kuratowskiEmbedding X))
⊢ (↑(range Ψ) ≃ᵢ ↥{ carrier := range (kuratowskiEmbedding X), isCompact' := ⋯, nonempty' := ⋯ }) =
    (↥p ≃ᵢ ↑(range (kuratowskiEmbedding X)))
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝² : PseudoMetricSpace α
s : Set α
inst✝¹ : Decidable s.Nontrivial
inst✝ : Fintype ↑s
⊢ s.infsep = if hs : s.Nontrivial then s.offDiag.toFinset.inf' ⋯ (uncurry dist) else 0

Tactic:
split_ifs with hs
  · refine eq_of_forall_le_iff fun _ => ?_
    simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, mem_toFinset, mem_offDiag,
      Prod.forall, uncurry_apply_pair, and_imp]
  · rw [not_nontrivial_iff] at hs
    exact hs.infsep_zero

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

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
e : ↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y)
⊢ (↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y)) =
    (↑(range (kuratowskiEmbedding X)) ≃ᵢ ↑(range (kuratowskiEmbedding Y)))

Tactic:
dsimp only [NonemptyCompacts.kuratowskiEmbedding]

Goal state after:
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
e : ↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y)
⊢ (↥{ carrier := range (kuratowskiEmbedding X), isCompact' := ⋯, nonempty' := ⋯ } ≃ᵢ
      ↥{ carrier := range (kuratowskiEmbedding Y), isCompact' := ⋯, nonempty' := ⋯ }) =
    (↑(range (kuratowskiEmbedding X)) ≃ᵢ ↑(range (kuratowskiEmbedding Y)))
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
e : X ≃ᵢ Y
f : ↑(range (kuratowskiEmbedding X)) ≃ᵢ X
g : Y ≃ᵢ ↑(range (kuratowskiEmbedding Y))
⊢ (↑(range (kuratowskiEmbedding X)) ≃ᵢ ↑(range (kuratowskiEmbedding Y))) =
    (↥(NonemptyCompacts.kuratowskiEmbedding X) ≃ᵢ ↥(NonemptyCompacts.kuratowskiEmbedding Y))

Tactic:
dsimp only [NonemptyCompacts.kuratowskiEmbedding]

Goal state after:
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
e : X ≃ᵢ Y
f : ↑(range (kuratowskiEmbedding X)) ≃ᵢ X
g : Y ≃ᵢ ↑(range (kuratowskiEmbedding Y))
⊢ (↑(range (kuratowskiEmbedding X)) ≃ᵢ ↑(range (kuratowskiEmbedding Y))) =
    (↥{ carrier := range (kuratowskiEmbedding X), isCompact' := ⋯, nonempty' := ⋯ } ≃ᵢ
      ↥{ carrier := range (kuratowskiEmbedding Y), isCompact' := ⋯, nonempty' := ⋯ })
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝ : PseudoMetricSpace α
s : Finset α
⊢ (↑s).infsep = if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0

Tactic:
by
  have H : (s : Set α).Nontrivial ↔ s.offDiag.Nonempty := by
    rw [← Set.offDiag_nonempty, ← Finset.coe_offDiag, Finset.coe_nonempty]
  split_ifs with hs
  · classical simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
  · exact (not_nontrivial_iff.mp (H.mp.mt hs)).infsep_zero

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

Goal state before:
pp: X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
s : Set X
⊢ ediam (id ⁻¹' s) ^ α = ediam s ^ α

Tactic:
Set.preimage_id

Goal state after:
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
s : Set X
⊢ ediam s ^ α = ediam s ^ α
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: α : Type u_1
s : Set α
D : Set (Set α)
inst✝¹ : Preorder α
t : TopologicalSpace α
inst✝ : IsScottHausdorff α D
hDL : IsLowerSet D
⊢ IsOpen[t] s ↔ DirSupInaccOn D s

Tactic:
isOpen_iff (D := D),

Goal state after:
α : Type u_1
s : Set α
D : Set (Set α)
inst✝¹ : Preorder α
t : TopologicalSpace α
inst✝ : IsScottHausdorff α D
hDL : IsLowerSet D
⊢ (∀ ⦃d : Set α⦄,
      d ∈ D → d.Nonempty → DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s) ↔
    DirSupInaccOn D s
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : CompleteLattice α
inst✝⁴ : CompleteLattice β
inst✝³ : TopologicalSpace α
inst✝² : IsLower α
inst✝¹ : TopologicalSpace β
inst✝ : IsLower β
f : sInfHom α β
b : β
⊢ IsClosed[inst✝³] (⇑f ⁻¹' Ici b)

Tactic:
convert! isClosed_Ici (a := sInf <| f ⁻¹' Ici b)

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : CompleteLattice α
inst✝⁴ : CompleteLattice β
inst✝³ : TopologicalSpace α
inst✝² : IsLower α
inst✝¹ : TopologicalSpace β
inst✝ : IsLower β
f : sInfHom α β
b : β
⊢ ⇑f ⁻¹' Ici b = Ici (sInf (⇑f ⁻¹' Ici b))
Module: Mathlib.Topology.Order.Completion

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
⊢ ∀ (a₁ a₂ : Fill α), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂

Tactic:
by
    simp only [ofLex_toLex, Subtype.forall, Prod.Lex.lt_iff, Subtype.mk_lt_mk,
      Lex.forall, Prod.forall]
    rintro x q ⟨hx₁, hx₂⟩ y r ⟨hy₁, hy₂⟩ (h | ⟨rfl, h⟩)
    · by_cases hx : IsPredPrelimit x
      · obtain ⟨z, hz, hz'⟩ := hx.lt_iff_exists_lt.1 h
        use some z
        simp [some, Prod.Lex.lt_iff, hz', hz]
      obtain ⟨s, hs⟩ := exists_gt (max 0 q)
      rw [max_lt_iff] at hs
      refine ⟨⟨toLex (x, s), ?_⟩, ?_⟩
      · simp [hx, hs.1.le]
      · simp [Prod.Lex.lt_iff, hs.2, h]
    · obtain ⟨s, hs, hs'⟩ := exists_between h
      refine ⟨⟨toLex (x, s), ?_⟩, ?_⟩
      · grind [ofLex_toLex]
      · simp [Prod.Lex.lt_iff, hs, hs']

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

Goal state before:
pp: α : Type u_1
inst✝² : CompleteLattice α
T : Set α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
hT : ∀ p ∈ T, InfPrime p
S : Set ↑T
⊢ IsClosed[instTopologicalSpaceSubtype] S ↔ ∃ a, S = hull T a

Tactic:
by
  simp only [← isOpen_compl_iff, isOpen_iff hT, compl_inj_iff]

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
⊢ Continuous ⇑some

Tactic:
by
  simp only [OrderTopology.continuous_iff, ofLex_toLex, Subtype.forall, Lex.forall, Prod.forall]
  refine fun x q ⟨hx₁, hx₂⟩ ↦ ⟨?_, ?_⟩
  · obtain hq | hq := le_or_gt 0 q
    · convert isOpen_Ioi (a := x)
      ext
      simp [some, Prod.Lex.lt_iff, hq.not_gt]
    · obtain ⟨y, hy⟩ := not_isSuccPrelimit_iff.1 <| mt hx₁ hq.not_ge
      convert isOpen_Ioi (a := y)
      ext
      simpa [some, Prod.Lex.lt_iff, hq, le_iff_lt_or_eq] using hy.le_iff_lt_right
  · obtain hq | hq := le_or_gt q 0
    · convert isOpen_Iio (a := x)
      ext
      simp [some, Prod.Lex.lt_iff, hq.not_gt]
    · obtain ⟨y, hy⟩ := not_isPredPrelimit_iff.1 <| mt hx₂ hq.not_ge
      convert isOpen_Iio (a := y)
      ext
      simpa [some, Prod.Lex.lt_iff, hq, le_iff_lt_or_eq] using hy.le_iff_lt_left

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

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n ↦ ℝ) ∞)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n ↦ ℝ) ∞)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n ↦ ℝ) ∞) :=
  fun x ↦
    match x with
    | inl y => Φ y
    | inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p ↦ dist (f p.1) (f p.2)
Fgood : F ∈ candidates X Y
Fb : GromovHausdorff.Cb✝ X Y := candidatesBOfCandidates F Fgood
this✝¹ : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb
r : ℝ
hr : hausdorffDist ↑p ↑q < r
x : X
this✝ : f (inl x) ∈ ↑p
z : ↥(lp (fun n ↦ ℝ) ∞)
zq : z ∈ ↑q
hz : dist (f (inl x)) z < r
this : z ∈ range Ψ
y : Y
hy : Ψ y = z
⊢ dist (Φ x) (Ψ y) = dist (f (inl x)) z

Tactic:
by rw [hy]

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

Goal state before:
pp: X : Type u
inst✝ : MetricSpace X
p q : NonemptyCompacts X
⊢ dist p.toGHSpace q.toGHSpace ≤ dist p q

Tactic:
have ha : Isometry ((↑) : p → X) := isometry_subtype_coe

Goal state after:
X : Type u
inst✝ : MetricSpace X
p q : NonemptyCompacts X
ha : Isometry Subtype.val
⊢ dist p.toGHSpace q.toGHSpace ≤ dist p q
Module: Mathlib.Topology.Separation.PerfectlyNormal

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
h : ∀ (s : Set X), IsClosed[inst✝] s → ∃ f, s = ⇑f ⁻¹' {0} ∧ ∀ (x : X), f x ∈ Icc 0 1
s t : Set X
hs : IsClosed[inst✝] s
ht : IsClosed[inst✝] t
hst : Disjoint s t
f : C(X, ℝ)
hf : s = ⇑f ⁻¹' {0}
hfr : ∀ (x : X), f x ∈ Icc 0 1
g : C(X, ℝ)
hg : t = ⇑g ⁻¹' {0}
hgr : ∀ (x : X), g x ∈ Icc 0 1
hsn : SeparatedNhds {0} {1}
hfg : ∀ (x : X), f x + g x ≠ 0
hp : s = (fun a ↦ f a / (f a + g a)) ⁻¹' {0}
this : t = (fun a ↦ f a / (f a + g a)) ⁻¹' {1}
⊢ SeparatedNhds ((fun a ↦ f a / (f a + g a)) ⁻¹' {0}) ((fun a ↦ f a / (f a + g a)) ⁻¹' {1})

Tactic:
exact hsn.preimage (f.continuous.div₀ (f.continuous.add g.continuous) hfg)

Goal state after:
Module: Mathlib.Topology.Sheaves.Skyscraper

Goal state before:
pp: X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝² : Category.{u, v} C
inst✝¹ : HasTerminal C
inst✝ : HasColimits C
𝓕 : Presheaf C X
c : C
f : 𝓕.stalk p₀ ⟶ c
⊢ fromStalk p₀ (toSkyscraperPresheaf p₀ f) = f

Tactic:
refine 𝓕.stalk_hom_ext fun U hxU ↦ ?_
  rw [germ_fromStalk, toSkyscraperPresheaf_app, dif_pos hxU, Category.assoc, Category.assoc,
    eqToHom_trans, eqToHom_refl, Category.comp_id, Presheaf.germ]

Goal state after:
Module: Mathlib.Topology.Sheaves.Points

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
U : Opens X
S : Sieve U
hS :
  ∀ (Φ : (pointsGrothendieckTopology X).FullSubcategory) (x : Φ.obj.fiber.obj U),
    ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
x : X
hx : x ∈ U
⊢ ∃ U_1 f, S.arrows f ∧ x ∈ U_1

Tactic:
obtain ⟨V, f, hf, ⟨⟨hV⟩⟩, _⟩ := hS ⟨_, ⟨x⟩⟩ ⟨⟨hx⟩⟩
    exact ⟨V, f, hf, hV⟩

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: case right
t₀ t₁ : ↑I
⊢ ∀ b ∈ uIcc t₀ t₁, ∃ a, Icc.convexComb t₀ t₁ a = b

Tactic:
intro t (ht : (t : ℝ) ∈ uIcc (t₀ : ℝ) (t₁ : ℝ))

Goal state after:
case right
t₀ t₁ : ↑I
t : ↑(Icc 0 1)
ht : ↑t ∈ uIcc ↑t₀ ↑t₁
⊢ ∃ a, Icc.convexComb t₀ t₁ a = t
Module: Mathlib.Topology.Subpath

Goal state before:
pp: case zero
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
n : ℕ
γ : Path a b
t : Fin (0 + 1) → ↑I
⊢ (concat (⇑γ ∘ t) fun k ↦ γ.subpath (t k.castSucc) (t k.succ)).Homotopy (γ.subpath (t 0) (t (last 0)))

Tactic:
simp only [concat_zero, reduceLast, subpath_self]

Goal state after:
case zero
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
n : ℕ
γ : Path a b
t : Fin (0 + 1) → ↑I
⊢ (Path.refl ((⇑γ ∘ t) 0)).Homotopy (Path.refl (γ (t 0)))
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: case refine_1
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
⊢ ∀ (i : SetRel X X), i ∈ 𝓤 X ∧ i.IsSymm ∧ i.IsTrans → ∃ i', (x ∈ i' ∧ IsClopen i') ∧ id i' ⊆ ball x (id i)

Tactic:
intro V ⟨hV, h_symm, h_trans⟩

Goal state after:
case refine_1
X : Type u_1
inst✝¹ : UniformSpace X
inst✝ : IsUltraUniformity X
x : X
V : SetRel X X
hV : V ∈ 𝓤 X
h_symm : V.IsSymm
h_trans : V.IsTrans
⊢ ∃ i', (x ∈ i' ∧ IsClopen i') ∧ id i' ⊆ ball x (id V)
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
| (⨆ 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:
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:
Module: Mathlib.Topology.Sion

Goal state before:
pp: case a.inr
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
cX : Convex ℝ X
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
ne_Y : Y.Nonempty
t : β
ht : t < inf_sup
s : Finset ↑Y
hs : ∀ x ∈ X, ∃ i ∈ s, t < f x ↑i
hs' : ∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y
y0 : (∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y) → F
hy0 : ∀ (ht : ∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y), y0 ht ∈ Y
ht0 : ∀ (ht : ∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y), ∀ x ∈ X, t < f x (y0 ht)
⊢ t < sup_inf

Tactic:
simp only [lt_isLUB_iff hsup_inf, mem_ofPred_eq, exists_exists_and_eq_and]

Goal state after:
case a.inr
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
cX : Convex ℝ X
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
ne_Y : Y.Nonempty
t : β
ht : t < inf_sup
s : Finset ↑Y
hs : ∀ x ∈ X, ∃ i ∈ s, t < f x ↑i
hs' : ∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y
y0 : (∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y) → F
hy0 : ∀ (ht : ∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y), y0 ht ∈ Y
ht0 : ∀ (ht : ∀ x ∈ X, ∃ y ∈ Subtype.val '' ↑s, t < f x y), ∀ x ∈ X, t < f x (y0 ht)
⊢ ∃ a ∈ Y, t < inf_x a