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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace X
inst✝ : CompactSpace Y
⊢ IsCompact (candidatesB X Y)

Tactic:
by
  refine arzela_ascoli₂
      (Icc 0 (maxVar X Y) : Set ℝ) isCompact_Icc (candidatesB X Y) closed_candidatesB ?_ ?_
  · rintro f ⟨x1, x2⟩ hf
    simp only [Set.mem_Icc]
    exact ⟨candidates_nonneg hf, candidates_le_maxVar hf⟩
  · refine equicontinuous_of_continuity_modulus (fun t => 2 * maxVar X Y * t) ?_ _ ?_
    · have : Tendsto (fun t : ℝ => 2 * (maxVar X Y : ℝ) * t) (𝓝 0) (𝓝 (2 * maxVar X Y * 0)) :=
        tendsto_const_nhds.mul tendsto_id
      simpa using this
    · rintro x y ⟨f, hf⟩
      exact (candidates_lipschitz hf).dist_le_mul _ _

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

Goal state before:
pp: X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ ∀ (t : F.stalk x), ∃ U f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = t

Tactic:
intro t

Goal state after:
X : TopCat
F : Presheaf (Type v) X
x : ↑X
t : F.stalk x
⊢ ∃ U f, ∃ (_ : (Sheafify.isLocallyGerm F).pred f), f ⟨x, ⋯⟩ = t
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
by
    by_cases h1 : p₀ ∈ unop U
    · by_cases h2 : p₀ ∈ unop V
      · simp_all only [skyscraperSheaf_obj_obj, skyscraperSheaf_obj_map, ↓reduceDIte]
        infer_instance
      · simp
        grind
    · have h2 : p₀ ∉ unop V := fun hV => h1 (r.unop.le hV)
      have := isIso_of_isTerminal (isTerminalSkyscraperSheafObjObjOfNotMem h1)
        (isTerminalSkyscraperSheafObjObjOfNotMem h2) ((skyscraperSheaf p₀ A).obj.map r)
      infer_instance

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

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
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)
main : ghDist p.Rep q.Rep ≤ ε + ε / 2 + ε
⊢ ε + ε / 2 + ε = δ / 2

Tactic:
simp only [ε, one_div]; ring

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: 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
⊢ False

Tactic:
have hZ_nonempty : Z.Nonempty :=
    CompactSpace.nonempty_sInter (fun _ ht ↦ ht.2) fun t ht hf ↦
      hB.prop.2.1 _ (subset_trans ht (by simp [B'])) hf

Goal state after:
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
⊢ False
Module: Mathlib.Topology.Subpath

Goal state before:
pp: case left
t₀ t₁ : ↑I
⊢ ∀ (a : ↑I), Icc.convexComb t₀ t₁ a ∈ uIcc t₀ t₁

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

Tactic:
· intro s
    exact convex_uIcc (t₀ : ℝ) t₁ left_mem_uIcc right_mem_uIcc
      (one_minus_nonneg s) (nonneg s) (sub_add_cancel _ _)

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

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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ : ↑I
γ₁ : (t : ↑I) → Path (γ t₀) (γ (Icc.convexComb t₁ t₂ t)) := fun t ↦ γ.subpath t₀ (Icc.convexComb t₁ t₂ t)
γ₂ : (t : ↑I) → Path (γ (Icc.convexComb t₁ t₂ t)) (γ t₂) := fun t ↦ γ.subpath (Icc.convexComb t₁ t₂ t) t₂
⊢ Continuous fun x ↦ ((γ.subpath t₀ (Icc.convexComb t₁ t₂ x.1)).trans (γ.subpath (Icc.convexComb t₁ t₂ x.1) t₂)) x.2

Tactic:
refine Path.trans_continuous_family γ₁ ?_ γ₂ ?_

Goal state after:
case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ : ↑I
γ₁ : (t : ↑I) → Path (γ t₀) (γ (Icc.convexComb t₁ t₂ t)) := fun t ↦ γ.subpath t₀ (Icc.convexComb t₁ t₂ t)
γ₂ : (t : ↑I) → Path (γ (Icc.convexComb t₁ t₂ t)) (γ t₂) := fun t ↦ γ.subpath (Icc.convexComb t₁ t₂ t) t₂
⊢ Continuous ↿γ₁

case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ : ↑I
γ₁ : (t : ↑I) → Path (γ t₀) (γ (Icc.convexComb t₁ t₂ t)) := fun t ↦ γ.subpath t₀ (Icc.convexComb t₁ t₂ t)
γ₂ : (t : ↑I) → Path (γ (Icc.convexComb t₁ t₂ t)) (γ t₂) := fun t ↦ γ.subpath (Icc.convexComb t₁ t₂ t) t₂
⊢ Continuous ↿γ₂
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 : F
hz : z ∈ segment ℝ ↑y1 ↑y2
⊢ z ∈ Subtype.val '' (J1 ∪ J2)

Tactic:
lift z to segment ℝ (y1 : F) y2 using hz

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 ∈ Subtype.val '' (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
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
hJ2 : IsClosed J2
⊢ Subtype.val '' univ ⊆ Subtype.val '' (J1 ∪ J2)

Tactic:
by simp [hJ1_union_J2]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Ultra.Completion

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : UniformSpace X
inst✝¹ : UniformSpace Y
inst✝ : IsUltraUniformity X
this :
  (map (Prod.map SeparationQuotient.mk SeparationQuotient.mk) (𝓤 X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans)
    fun i ↦ Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
⊢ IsUltraUniformity (SeparationQuotient X)

Tactic:
← SeparationQuotient.uniformity_eq

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : UniformSpace X
inst✝¹ : UniformSpace Y
inst✝ : IsUltraUniformity X
this :
  (𝓤 (SeparationQuotient X)).HasBasis (fun s ↦ s ∈ 𝓤 X ∧ s.IsSymm ∧ s.IsTrans) fun i ↦
    Prod.map SeparationQuotient.mk SeparationQuotient.mk '' id i
⊢ IsUltraUniformity (SeparationQuotient X)
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
⊢ ∀ ⦃c : β⦄, c < inf_sup → c < sup_inf

Tactic:
intro t

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
⊢ t < inf_sup → t < sup_inf