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

Goal state before:
pp: case basic
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
s u : Set α
h' : u ∈ {t | ∃ a, (Ici a)ᶜ = t}
⊢ IsLowerSet u

Tactic:
obtain ⟨a, rfl⟩ := h'

Goal state after:
case basic
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
s : Set α
a : α
⊢ IsLowerSet (Ici a)ᶜ
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
⊢ {s | ∃ t, t.Finite ∧ (↑(upperClosure t))ᶜ = s} = (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ {s | ∃ a, (Ici a)ᶜ = s}}

Tactic:
coe_upperClosure,

Goal state after:
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
⊢ {s | ∃ t, t.Finite ∧ (⋃ a ∈ t, Ici a)ᶜ = s} = (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ {s | ∃ a, (Ici a)ᶜ = s}}
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case mpr
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
F : Set (Set α)
hF : F.Finite
hs : F ⊆ {s | ∃ a, (Ici a)ᶜ = s}
this : Finite ↑F
⊢ (fun f ↦ ⋂₀ f) F ∈ {s | ∃ t, t.Finite ∧ ⋂ i ∈ t, (Ici i)ᶜ = s}

Tactic:
subset_def,

Goal state after:
case mpr
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
F : Set (Set α)
hF : F.Finite
hs : ∀ x ∈ F, x ∈ {s | ∃ a, (Ici a)ᶜ = s}
this : Finite ↑F
⊢ (fun f ↦ ⋂₀ f) F ∈ {s | ∃ t, t.Finite ∧ ⋂ i ∈ t, (Ici i)ᶜ = s}
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
⊢ ∀ ⦃s : Set α⦄,
    s ∈ {s | ∃ a, (Ici a)ᶜ = s} → ∀ ⦃t : Set α⦄, t ∈ {s | ∃ a, (Ici a)ᶜ = s} → s ∩ t ∈ {s | ∃ a, (Ici a)ᶜ = s}

Tactic:
rintro _ ⟨b, rfl⟩ _ ⟨c, rfl⟩

Goal state after:
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
b c : α
⊢ (Ici b)ᶜ ∩ (Ici c)ᶜ ∈ {s | ∃ a, (Ici a)ᶜ = s}
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous[inst✝³, inst✝²] f
a b : α
hab : a ∈ closure[inst✝³] {b}
⊢ f a ∈ closure[inst✝²] {f b}

Tactic:
apply Continuous.closure_preimage_subset hf {f b}

Goal state after:
case mpr
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Continuous[inst✝³, inst✝²] f
a b : α
hab : a ∈ closure[inst✝³] {b}
⊢ a ∈ closure[inst✝³] (f ⁻¹' {f b})
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
⊢ IsOpen[lower (α × β)] ((↑(upperClosure s))ᶜ ×ˢ (↑(upperClosure t))ᶜ)

Tactic:
coe_upperClosure,

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
⊢ IsOpen[lower (α × β)] ((⋃ a ∈ s, Ici a)ᶜ ×ˢ (⋃ a ∈ t, Ici a)ᶜ)
Module: Mathlib.Topology.Order.HullKernel

Goal state before:
pp: case mpr
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
h : ∃ a, S = (hull T a)ᶜ
⊢ IsOpen[instTopologicalSpaceSubtype] S

Tactic:
· obtain ⟨a, ha⟩ := h
    exact ⟨(Ici a)ᶜ, isClosed_Ici.isOpen_compl, ha.symm⟩

Goal state after:
Module: Mathlib.Topology.Separation.PerfectlyNormal

Goal state before:
pp: case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
h✝ : PerfectlyNormalSpace X
s : Set X
hs : IsClosed[inst✝] s
U : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen[inst✝] (U n)
hu : s = ⋂ n, U n
this : ∀ (n : ℕ), Disjoint s (U n)ᶜ
f : ℕ → C(X, ℝ)
hfs : ∀ (n : ℕ), EqOn (⇑(f n)) 0 s
hfu : ∀ (n : ℕ), EqOn (⇑(f n)) 1 (U n)ᶜ
hfr : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
hsb : ∀ (x : X) (n : ℕ), ‖(f n) x * (1 / 2 / 2 ^ n)‖ ≤ 1 / 2 / 2 ^ n
hsx : ∀ (x : X), Summable fun n ↦ (f n) x * (1 / 2 / 2 ^ n)
h : C(X, ℝ) := { toFun := fun x ↦ ∑' (n : ℕ), (f n) x * (1 / 2 / 2 ^ n), continuous_toFun := ⋯ }
x : X
⊢ x ∈ s ↔ x ∈ ⇑h ⁻¹' {0}

Tactic:
refine ⟨fun hp => ?_, fun hp => ?_⟩

Goal state after:
case refine_1.refine_1
X : Type u_1
inst✝ : TopologicalSpace X
h✝ : PerfectlyNormalSpace X
s : Set X
hs : IsClosed[inst✝] s
U : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen[inst✝] (U n)
hu : s = ⋂ n, U n
this : ∀ (n : ℕ), Disjoint s (U n)ᶜ
f : ℕ → C(X, ℝ)
hfs : ∀ (n : ℕ), EqOn (⇑(f n)) 0 s
hfu : ∀ (n : ℕ), EqOn (⇑(f n)) 1 (U n)ᶜ
hfr : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
hsb : ∀ (x : X) (n : ℕ), ‖(f n) x * (1 / 2 / 2 ^ n)‖ ≤ 1 / 2 / 2 ^ n
hsx : ∀ (x : X), Summable fun n ↦ (f n) x * (1 / 2 / 2 ^ n)
h : C(X, ℝ) := { toFun := fun x ↦ ∑' (n : ℕ), (f n) x * (1 / 2 / 2 ^ n), continuous_toFun := ⋯ }
x : X
hp : x ∈ s
⊢ x ∈ ⇑h ⁻¹' {0}

case refine_1.refine_2
X : Type u_1
inst✝ : TopologicalSpace X
h✝ : PerfectlyNormalSpace X
s : Set X
hs : IsClosed[inst✝] s
U : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen[inst✝] (U n)
hu : s = ⋂ n, U n
this : ∀ (n : ℕ), Disjoint s (U n)ᶜ
f : ℕ → C(X, ℝ)
hfs : ∀ (n : ℕ), EqOn (⇑(f n)) 0 s
hfu : ∀ (n : ℕ), EqOn (⇑(f n)) 1 (U n)ᶜ
hfr : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
hsb : ∀ (x : X) (n : ℕ), ‖(f n) x * (1 / 2 / 2 ^ n)‖ ≤ 1 / 2 / 2 ^ n
hsx : ∀ (x : X), Summable fun n ↦ (f n) x * (1 / 2 / 2 ^ n)
h : C(X, ℝ) := { toFun := fun x ↦ ∑' (n : ℕ), (f n) x * (1 / 2 / 2 ^ n), continuous_toFun := ⋯ }
x : X
hp : x ∈ ⇑h ⁻¹' {0}
⊢ x ∈ s
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case refine_1
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 : p.Rep), ∃ y ∈ s p, dist x y ≤ ε

Tactic:
show ∀ x : p.Rep, ∃ y ∈ s p, dist x y ≤ ε
      -- by construction, `s p` is `ε`-dense
      intro x
      have : x ∈ ⋃ y ∈ s p, ball y ε := (hs p).2 (mem_univ _)
      obtain ⟨y, ys, hy⟩ := mem_iUnion₂.1 this
      exact ⟨y, ys, hy.le⟩

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

Goal state before:
pp: 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
⊢ IsTerminal (if PUnit.unit ∈ ⊥ then A else ⊤_ C)

Tactic:
rw [if_neg]

Goal state after:
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
⊢ IsTerminal (⊤_ C)

case hnc
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
⊢ PUnit.unit ∉ ⊥
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: 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
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
⊢ ∀ (j : (CategoryTheory.Pairwise ι)ᵒᵖ),
    𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
        (Functor.mapCone F (cocone U).op).π.app j =
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app j

Tactic:
intro x
            induction x with | op x => ?_
            rcases x with ⟨⟩
            · simp
              rfl
            · dsimp [coneEquivInverse, SheafConditionEqualizerProducts.res,
                SheafConditionEqualizerProducts.leftRes]
              simp only [limit.lift_π, limit.lift_π_assoc, Category.id_comp, Fan.mk_π_app,
                Category.assoc]
              rw [← F.map_comp]
              rfl

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

Goal state before:
pp: 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
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
⊢ ∀ (j : (CategoryTheory.Pairwise ι)ᵒᵖ),
    𝟙 (Functor.mapCone F (cocone U).op).pt ≫
        ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app j =
      (Functor.mapCone F (cocone U).op).π.app j

Tactic:
intro x
            induction x with | op x => ?_
            rcases x with ⟨⟩
            · simp
              rfl
            · dsimp [coneEquivInverse, SheafConditionEqualizerProducts.res,
                SheafConditionEqualizerProducts.leftRes]
              simp only [limit.lift_π, limit.lift_π_assoc, Category.id_comp, Fan.mk_π_app,
                Category.assoc]
              rw [← F.map_comp]
              rfl

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

Goal state before:
pp: X : TopCat
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝¹ : S.X₁.IsFlasque
inst✝ : S.X₂.IsFlasque
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
⊢ Epi (S.g.hom.app U ≫ S.X₃.obj.map i)

Tactic:
rw [← S.g.hom.naturality i]

Goal state after:
X : TopCat
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝¹ : S.X₁.IsFlasque
inst✝ : S.X₂.IsFlasque
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
⊢ Epi (S.X₂.obj.map i ≫ S.g.hom.app 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
𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)
F : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s
V : Set (γ × γ)
V_in : V ∈ 𝓝Δ
H : ¬F ⊓ 𝓟 Vᶜ = ⊥
this : (F ⊓ 𝓟 Vᶜ).NeBot
x y : γ
hxy : ClusterPt (x, y) (F ⊓ 𝓟 Vᶜ)
clV : ClusterPt (x, y) (𝓟 Vᶜ)
ninsep_xy : ¬Inseparable x y
U₁ V₁ : Set γ
V₁_in : V₁ ∈ 𝓝 x
U₂ V₂ : Set γ
V₂_in : V₂ ∈ 𝓝 y
V₁_cl : IsClosed[inst✝²] V₁
V₂_cl : IsClosed[inst✝²] V₂
U₁_op : IsOpen[inst✝²] U₁
U₂_op : IsOpen[inst✝²] U₂
VU₁ : V₁ ⊆ U₁
VU₂ : V₂ ⊆ U₂
hU₁₂ : Disjoint U₁ U₂
U₃ : Set γ := (V₁ ∪ V₂)ᶜ
U₃_op : IsOpen[inst✝²] U₃
W : Set (γ × γ) := U₁ ×ˢ U₁ ∪ U₂ ×ˢ U₂ ∪ U₃ ×ˢ U₃
⊢ W ∈ 𝓝Δ

Tactic:
rw [mem_nhdsSet_iff_forall]

Goal state after:
γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)
F : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s
V : Set (γ × γ)
V_in : V ∈ 𝓝Δ
H : ¬F ⊓ 𝓟 Vᶜ = ⊥
this : (F ⊓ 𝓟 Vᶜ).NeBot
x y : γ
hxy : ClusterPt (x, y) (F ⊓ 𝓟 Vᶜ)
clV : ClusterPt (x, y) (𝓟 Vᶜ)
ninsep_xy : ¬Inseparable x y
U₁ V₁ : Set γ
V₁_in : V₁ ∈ 𝓝 x
U₂ V₂ : Set γ
V₂_in : V₂ ∈ 𝓝 y
V₁_cl : IsClosed[inst✝²] V₁
V₂_cl : IsClosed[inst✝²] V₂
U₁_op : IsOpen[inst✝²] U₁
U₂_op : IsOpen[inst✝²] U₂
VU₁ : V₁ ⊆ U₁
VU₂ : V₂ ⊆ U₂
hU₁₂ : Disjoint U₁ U₂
U₃ : Set γ := (V₁ ∪ V₂)ᶜ
U₃_op : IsOpen[inst✝²] U₃
W : Set (γ × γ) := U₁ ×ˢ U₁ ∪ U₂ ×ˢ U₂ ∪ U₃ ×ˢ U₃
⊢ ∀ x ∈ diagonal γ, W ∈ 𝓝 x
Module: Mathlib.Topology.UniformSpace.OfCompactT2

Goal state before:
pp: case a
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

Tactic:
enter [1, i, 1, hi]
        rw [← hi.nhds_eq, comap_const_of_mem fun V => mem_of_mem_nhds]

Goal state after:
case a
γ : Type u_1
inst✝² : TopologicalSpace γ
inst✝¹ : CompactSpace γ
inst✝ : R1Space γ
x i : γ
hi : Inseparable x i
| ⊤ ⊓ 𝓝 x
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 : ∃ c ∈ lowerBounds {x | ∃ x_1 ∈ X, sup_y x_1 = x}, t < c
x : E
hx : x ∈ X
hx' : ∀ i ∈ Y, f x i ≤ t
⊢ False

Tactic:
obtain ⟨c, hc, htc⟩ := 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
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
x : E
hx : x ∈ X
hx' : ∀ i ∈ Y, f x i ≤ t
c : β
hc : c ∈ lowerBounds {x | ∃ x_1 ∈ X, sup_y x_1 = x}
htc : t < c
⊢ False