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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X →. Y
h : PContinuous f
⊢ IsOpen[inst✝¹] (f.preimage Set.univ)

Tactic:
exact h _ isOpen_univ

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

Goal state before:
pp: α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
a : α
⊢ IsClosed[lawson α] {a}

Tactic:
rw [← (Set.OrdConnected.upperClosure_inter_lowerClosure ordConnected_singleton),
      ← WithLawson.isClosed_preimage_ofLawson]

Goal state after:
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
a : α
⊢ IsClosed[WithLawson.instTopologicalSpace] (⇑WithLawson.ofLawson ⁻¹' (↑(upperClosure {a}) ∩ ↑(lowerClosure {a})))
Module: Mathlib.Topology.Separation.LinearUpperLowerSetTopology

Goal state before:
pp: case inr.inr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : Topology.IsUpperSet α
s t : Set α
hcst : Disjoint (closure s) t
hsct : Disjoint s (closure t)
a : α
ha : a ∈ s
b : α
hb : b ∈ t
⊢ Disjoint (nhdsSet s) (nhdsSet t)

Tactic:
exfalso

Goal state after:
case inr.inr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : Topology.IsUpperSet α
s t : Set α
hcst : Disjoint (closure s) t
hsct : Disjoint s (closure t)
a : α
ha : a ∈ s
b : α
hb : b ∈ t
⊢ False
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : toGlueL hΦ hΨ ∘ optimalGHInjr X Y = toGlueR hΦ hΨ ∘ optimalGHInjl Y Z
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (toGlueR hΦ hΨ '' range (optimalGHInjl Y Z)) (toGlueR hΦ hΨ '' range (optimalGHInjr Y Z)) =
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))

Tactic:
hausdorffDist_image (toGlueR_isometry hΦ hΨ)

Goal state after:
x y z : GHSpace
X : Type := x.Rep
Y : Type := y.Rep
Z : Type := z.Rep
γ1 : Type := OptimalGHCoupling X Y
γ2 : Type := OptimalGHCoupling Y Z
Φ : Y → γ1 := optimalGHInjr X Y
hΦ : Isometry Φ
Ψ : Y → γ2 := optimalGHInjl Y Z
hΨ : Isometry Ψ
Comm : toGlueL hΦ hΨ ∘ optimalGHInjr X Y = toGlueR hΦ hΨ ∘ optimalGHInjl Y Z
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z)) =
    hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) +
      hausdorffDist (range (optimalGHInjl Y Z)) (range (optimalGHInjr Y Z))
Module: Mathlib.Topology.Sheaves.Skyscraper

Goal state before:
pp: case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝² : Category.{u, v} C
A : C
inst✝¹ : HasTerminal C
inst✝ : HasColimits C
𝓕 𝓖 : Presheaf C X
f : 𝓕 ⟶ 𝓖
U : Opens ↑X
h : p₀ ∉ U
⊢ f.app (op U) ≫ (⋯ ▸ terminalIsTerminal).from (𝓖.obj (op U)) =
    (⋯ ▸ terminalIsTerminal).from (𝓕.obj (op U)) ≫ (⋯ ▸ terminalIsTerminal).from (if p₀ ∈ U then 𝓕.stalk p₀ else ⊤_ C)

Tactic:
apply ((if_neg h).symm.ndrec terminalIsTerminal).hom_ext

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

Goal state before:
pp: case neg
X✝ : TopCat
p₀ : ↑X✝
inst✝³ : (U : Opens ↑X✝) → Decidable (p₀ ∈ U)
C : Type v
inst✝² : Category.{u, v} C
A : C
inst✝¹ : HasTerminal C
inst✝ : HasColimits C
X : Presheaf C X✝
j✝ : (OpenNhds p₀)ᵒᵖ
h : p₀ ∉ (OpenNhds.inclusion p₀).obj (unop j✝)
⊢ (⋯ ▸ terminalIsTerminal).from (X.obj (op ((OpenNhds.inclusion p₀).obj (unop j✝)))) =
    colimit.ι ((OpenNhds.inclusion p₀).op ⋙ X) j✝ ≫ eqToHom ⋯

Tactic:
apply ((if_neg h).symm.ndrec terminalIsTerminal).hom_ext

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

Goal state before:
pp: case refine_3
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
this : (F p).snd ((E p) x) ((E p) y) = ⌊ε⁻¹ * dist x y⌋
⊢ |dist x y - dist (Φ x) (Φ y)| ≤ ε

Tactic:
have Ap : (F p).2 ⟨i, hip⟩ ⟨j, hjp⟩ = ⌊ε⁻¹ * dist x y⌋ := by rw [← this]

Goal state after:
case refine_3
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
this : (F p).snd ((E p) x) ((E p) y) = ⌊ε⁻¹ * dist x y⌋
Ap : (F p).snd ⟨i, hip⟩ ⟨j, hjp⟩ = ⌊ε⁻¹ * dist x y⌋
⊢ |dist x y - dist (Φ x) (Φ y)| ≤ ε
Module: Mathlib.Topology.Sheaves.LocallySurjective

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X : TopCat
ℱ 𝒢 : Presheaf C X
inst✝¹ : Limits.HasColimits C
inst✝ : Limits.PreservesFilteredColimits (forget C)
T : ℱ ⟶ 𝒢
hT : ∀ (x : ↑X), Function.Surjective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map T))
U : Opens ↑X
t : ToType (𝒢.obj (op U))
x : ↑X
hxU : x ∈ U
t_x : CC (𝒢.stalk x) := (ConcreteCategory.hom (𝒢.germ U x hxU)) t
ht_x : t_x = (ConcreteCategory.hom (𝒢.germ U x hxU)) t
V : Opens ↑X
hxV : x ∈ V
s : ToType (ℱ.obj (op V))
hs_x : (ConcreteCategory.hom ((stalkFunctor C x).map T)) ((ConcreteCategory.hom (ℱ.germ V x hxV)) s) = t_x
⊢ (ConcreteCategory.hom (𝒢.germ V x hxV)) ((ConcreteCategory.hom (T.app (op V))) s) =
    (ConcreteCategory.hom (𝒢.germ U x hxU)) t

Tactic:
convert! hs_x using 1

Goal state after:
case e'_2
C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X : TopCat
ℱ 𝒢 : Presheaf C X
inst✝¹ : Limits.HasColimits C
inst✝ : Limits.PreservesFilteredColimits (forget C)
T : ℱ ⟶ 𝒢
hT : ∀ (x : ↑X), Function.Surjective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map T))
U : Opens ↑X
t : ToType (𝒢.obj (op U))
x : ↑X
hxU : x ∈ U
t_x : CC (𝒢.stalk x) := ⋯
ht_x : t_x = (ConcreteCategory.hom (𝒢.germ U x hxU)) t
V : Opens ↑X
hxV : x ∈ V
s : ToType (ℱ.obj (op V))
hs_x : (ConcreteCategory.hom ((stalkFunctor C x).map T)) ((ConcreteCategory.hom (ℱ.germ V x hxV)) s) = t_x
e_1✝ : CC (𝒢.stalk x) = CC ((stalkFunctor C x).obj 𝒢)
⊢ (ConcreteCategory.hom (𝒢.germ V x hxV)) ((ConcreteCategory.hom (T.app (op V))) s) =
    (ConcreteCategory.hom ((stalkFunctor C x).map T)) ((ConcreteCategory.hom (ℱ.germ V x hxV)) s)
Module: Mathlib.Topology.Sheaves.LocallySurjective

Goal state before:
pp: case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X : TopCat
ℱ 𝒢 : Presheaf C X
inst✝¹ : Limits.HasColimits C
inst✝ : Limits.PreservesFilteredColimits (forget C)
T : ℱ ⟶ 𝒢
hT : ∀ (x : ↑X), Function.Surjective ⇑(ConcreteCategory.hom ((stalkFunctor C x).map T))
⊢ IsLocallySurjective T

Tactic:
· /- human proof:
        Let U be an open set, t ∈ Γ ℱ U a section, x ∈ U a point.
        By surjectivity on stalks, the germ of t is the image of
        some germ f ∈ Γₛₜ ℱ x. Represent f on some open set V ⊆ X as ⟨s, V⟩.
        Then there is some possibly smaller open set x ∈ W ⊆ V ∩ U on which
        we have T(s) |_ W = t |_ W. -/
    constructor
    intro U t x hxU
    set t_x := 𝒢.germ _ x hxU t with ht_x
    obtain ⟨s_x, hs_x : ((stalkFunctor C x).map T) s_x = t_x⟩ := hT x t_x
    obtain ⟨V, hxV, s, rfl⟩ := ℱ.exists_germ_eq s_x
    -- rfl : ℱ.germ x s = s_x
    have key_W := 𝒢.germ_eq x hxV hxU (T.app _ s) t <| by
      convert! hs_x using 1
      symm
      convert! stalkFunctor_map_germ_apply _ _ _ _ s
    obtain ⟨W, hxW, hWV, hWU, h_eq⟩ := key_W
    refine ⟨W, hWU, ⟨ℱ.map hWV.op s, ?_⟩, hxW⟩
    convert! h_eq using 1
    simp only [← ConcreteCategory.comp_apply, T.naturality]

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
⊢ Epi (S.g.hom.app (op U))

Tactic:
refine (AddCommGrpCat.epi_iff_surjective _).mpr (fun s ↦ ?_)

Goal state after:
X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
⊢ ∃ a, (ConcreteCategory.hom (S.g.hom.app (op U))) a = s
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_2
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace X
inst✝ : CompactSpace Y
⊢ Equicontinuous fun x ↦ ⇑↑x

Tactic:
refine equicontinuous_of_continuity_modulus (fun t => 2 * maxVar X Y * t) ?_ _ ?_

Goal state after:
case refine_2.refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace X
inst✝ : CompactSpace Y
⊢ Tendsto (fun t ↦ 2 * ↑(maxVar X Y) * t) (𝓝 0) (𝓝 0)

case refine_2.refine_2
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace X
inst✝ : CompactSpace Y
⊢ ∀ (x y : (X ⊕ Y) × (X ⊕ Y)) (i : ↑(candidatesB X Y)), dist (↑i x) (↑i y) ≤ 2 * ↑(maxVar X Y) * dist x y
Module: Mathlib.Topology.Subpath

Goal state before:
pp: case right
t₀ t₁ : ↑I
t : ↑(Icc 0 1)
⊢ t ∈ uIcc t₀ t₁ → ∃ a, Icc.convexComb t₀ t₁ a = t

Tactic:
(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.Spectral.ConstructibleTopology

Goal state before:
pp: case neg
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : ¬IsPreirreducible Z
⊢ False

Tactic:
obtain ⟨Y₁, Y₂, hY₁_cl, hY₂_cl, hc₁, hc₂, hsub, ⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩ :=
      PrespectralSpace.exists_isClosed_of_not_isPreirreducible _ hZ_irred

Goal state after:
case neg
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : ¬IsPreirreducible Z
Y₁ Y₂ : Set X
hY₁_cl : IsClosed[inst✝⁴] Y₁
hY₂_cl : IsClosed[inst✝⁴] Y₂
hc₁ : IsCompact Y₁ᶜ
hc₂ : IsCompact Y₂ᶜ
hsub : Z ⊆ Y₁ ∪ Y₂
x₁ : X
hx₁ : x₁ ∈ Z ∩ Y₁ᶜ
x₂ : X
hx₂ : x₂ ∈ Z ∩ Y₂ᶜ
⊢ False
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 = ∅
z : { x // x ∈ segment ℝ ↑y1 ↑y2 }
⊢ ↑z ∈ Subtype.val '' (J1 ∪ J2)

Tactic:
refine ⟨z, ?_, rfl⟩

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)
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 = ∅
z : { x // x ∈ segment ℝ ↑y1 ↑y2 }
⊢ z ∈ J1 ∪ J2
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
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
⊢ ∀ (x : E), x ∉ X ∩ ⋂ i ∈ Y, (fun x ↦ f x i) ⁻¹' Iic t

Tactic:
rintro x ⟨hx, hx'⟩

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
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
x : E
hx : x ∈ X
hx' : x ∈ ⋂ i ∈ Y, (fun x ↦ f x i) ⁻¹' Iic t
⊢ False