l3lab/ntp-mathlib huggingface dataset:
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  

Proof state:
case this.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
⊢ s ∈ 𝓝[>] z

Next tactic:
have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
  

Proof state:
case this.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
⊢ s ∈ 𝓝[>] z

Next tactic:
have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
    rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2]
    exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
  have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
    

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
⊢ tᶜ ∩ Ioc z y ∈ 𝓝[>] z

Next tactic:
rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2]
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
  have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
    rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2]
    

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
⊢ tᶜ ∩ Ioc z y ∈ 𝓝[Ioc z y] z

Next tactic:
exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
  have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
    rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2]
    exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩
  

Proof state:
case this.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
this : tᶜ ∩ Ioc z y ∈ 𝓝[>] z
⊢ s ∈ 𝓝[>] z

Next tactic:
apply mem_of_superset this
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
  have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
    rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2]
    exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩
  apply mem_of_superset this
  

Proof state:
case this.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
this : tᶜ ∩ Ioc z y ∈ 𝓝[>] z
⊢ tᶜ ∩ Ioc z y ⊆ s

Next tactic:
have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩)
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty := by
  have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
  by_contra hst
  suffices : Icc x y ⊆ s
  exact hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
  apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
  rintro z ⟨zs, hz⟩
  have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
  have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by
    rw [← nhdsWithin_Ioc_eq_nhdsWithin_Ioi hz.2]
    exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩
  apply mem_of_superset this
  have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩)
  

Proof state:
case this.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬Set.Nonempty (Icc a b ∩ (s ∩ t))
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
this✝ : tᶜ ∩ Ioc z y ∈ 𝓝[>] z
this : Ioc z y ⊆ s ∪ t
⊢ tᶜ ∩ Ioc z y ⊆ s

Next tactic:
exact fun w ⟨wt, wzy⟩ => (this wzy).elim id fun h => (wt h).elim
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
theorem isPreconnected_Icc : IsPreconnected (Icc a b) :=
  isPreconnected_closed_iff.2
    (by
      

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
⊢ ∀ (t t' : Set α),
    IsClosed t →
      IsClosed t' →
        Icc a b ⊆ t ∪ t' → Set.Nonempty (Icc a b ∩ t) → Set.Nonempty (Icc a b ∩ t') → Set.Nonempty (Icc a b ∩ (t ∩ t'))

Next tactic:
rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
theorem isPreconnected_Icc : IsPreconnected (Icc a b) :=
  isPreconnected_closed_iff.2
    (by
      rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
      -- This used to use `wlog`, but it was causing timeouts.
      

Proof state:
case intro.intro
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
⊢ Set.Nonempty (Icc a b ∩ (s ∩ t))

Next tactic:
rcases le_total x y with h | h
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
theorem isPreconnected_Icc : IsPreconnected (Icc a b) :=
  isPreconnected_closed_iff.2
    (by
      rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
      -- This used to use `wlog`, but it was causing timeouts.
      rcases le_total x y with h | h
      · 

Proof state:
case intro.intro.inl
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : x ≤ y
⊢ Set.Nonempty (Icc a b ∩ (s ∩ t))

Next tactic:
exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
theorem isPreconnected_Icc : IsPreconnected (Icc a b) :=
  isPreconnected_closed_iff.2
    (by
      rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
      -- This used to use `wlog`, but it was causing timeouts.
      rcases le_total x y with h | h
      · exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
      · 

Proof state:
case intro.intro.inr
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ Set.Nonempty (Icc a b ∩ (s ∩ t))

Next tactic:
rw [inter_comm s t]
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
theorem isPreconnected_Icc : IsPreconnected (Icc a b) :=
  isPreconnected_closed_iff.2
    (by
      rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
      -- This used to use `wlog`, but it was causing timeouts.
      rcases le_total x y with h | h
      · exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
      · rw [inter_comm s t]
        

Proof state:
case intro.intro.inr
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ s ∪ t
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ Set.Nonempty (Icc a b ∩ (t ∩ s))

Next tactic:
rw [union_comm s t] at hab
/-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/
theorem isPreconnected_Icc : IsPreconnected (Icc a b) :=
  isPreconnected_closed_iff.2
    (by
      rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩
      -- This used to use `wlog`, but it was causing timeouts.
      rcases le_total x y with h | h
      · exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy
      · rw [inter_comm s t]
        rw [union_comm s t] at hab
        

Proof state:
case intro.intro.inr
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
s t : Set α
hs : IsClosed s
ht : IsClosed t
hab : Icc a b ⊆ t ∪ s
x : α
hx : x ∈ Icc a b ∩ s
y : α
hy : y ∈ Icc a b ∩ t
h : y ≤ x
⊢ Set.Nonempty (Icc a b ∩ (t ∩ s))

Next tactic:
exact isPreconnected_Icc_aux y x t s h ht hs hab hy hx
/-- In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(sInf s, sInf s)`, we include it into the list of
possible cases to improve readability. -/
theorem setOf_isPreconnected_eq_of_ordered :
    { s : Set α | IsPreconnected s } =
      -- bounded intervals
      range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      -- unbounded intervals and `univ`
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by
  

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
⊢ {s | IsPreconnected s} =
    range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})

Next tactic:
refine' Subset.antisymm setOf_isPreconnected_subset_of_ordered _
/-- In a dense conditionally complete linear order, the set of preconnected sets is exactly
the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`,
or `∅`. Though one can represent `∅` as `(sInf s, sInf s)`, we include it into the list of
possible cases to improve readability. -/
theorem setOf_isPreconnected_eq_of_ordered :
    { s : Set α | IsPreconnected s } =
      -- bounded intervals
      range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      -- unbounded intervals and `univ`
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by
  refine' Subset.antisymm setOf_isPreconnected_subset_of_ordered _
  

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
⊢ range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) ⊆
    {s | IsPreconnected s}

Next tactic:
simp only [subset_def, forall_range_iff, uncurry, or_imp, forall_and, mem_union,
    mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true_iff,
    isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Ioo,
    isPreconnected_Ioi, isPreconnected_Iio, isPreconnected_Ici, isPreconnected_Iic,
    isPreconnected_univ, isPreconnected_empty]
/-- **Intermediate Value Theorem** for continuous functions on closed intervals, unordered case. -/
theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f (uIcc a b)) :
    uIcc (f a) (f b) ⊆ f '' uIcc a b := by
  

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hf : ContinuousOn f (uIcc a b)
⊢ uIcc (f a) (f b) ⊆ f '' uIcc a b

Next tactic:
cases le_total (f a) (f b)
/-- **Intermediate Value Theorem** for continuous functions on closed intervals, unordered case. -/
theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f (uIcc a b)) :
    uIcc (f a) (f b) ⊆ f '' uIcc a b := by
  cases le_total (f a) (f b) <;> 

Proof state:
case inl
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hf : ContinuousOn f (uIcc a b)
h✝ : f a ≤ f b
⊢ uIcc (f a) (f b) ⊆ f '' uIcc a b

Next tactic:
simp [*, isPreconnected_uIcc.intermediate_value]
/-- **Intermediate Value Theorem** for continuous functions on closed intervals, unordered case. -/
theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f (uIcc a b)) :
    uIcc (f a) (f b) ⊆ f '' uIcc a b := by
  cases le_total (f a) (f b) <;> 

Proof state:
case inr
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a b : α
f : α → δ
hf : ContinuousOn f (uIcc a b)
h✝ : f b ≤ f a
⊢ uIcc (f a) (f b) ⊆ f '' uIcc a b

Next tactic:
simp [*, isPreconnected_uIcc.intermediate_value]
/-- **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. -/
theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
    (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (uIcc (f a) (f b)) :=
  by 

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
s : Set α
hs : OrdConnected s
f : α → δ
hf : ContinuousOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
⊢ SurjOn f s (uIcc (f a) (f b))

Next tactic:
rcases le_total (f a) (f b) with hab | hab
/-- **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. -/
theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
    (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (uIcc (f a) (f b)) :=
  by rcases le_total (f a) (f b) with hab | hab <;> 

Proof state:
case inl
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
s : Set α
hs : OrdConnected s
f : α → δ
hf : ContinuousOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
hab : f a ≤ f b
⊢ SurjOn f s (uIcc (f a) (f b))

Next tactic:
simp [hf.surjOn_Icc, *]
/-- **Intermediate value theorem**: if `f` is continuous on an order-connected set `s` and `a`,
`b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. -/
theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
    (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (uIcc (f a) (f b)) :=
  by rcases le_total (f a) (f b) with hab | hab <;> 

Proof state:
case inr
α : Type u
β : Type v
γ : Type w
inst✝¹⁰ : ConditionallyCompleteLinearOrder α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : OrderTopology α
inst✝⁷ : ConditionallyCompleteLinearOrder β
inst✝⁶ : TopologicalSpace β
inst✝⁵ : OrderTopology β
inst✝⁴ : Nonempty γ
inst✝³ : DenselyOrdered α
a✝ b✝ : α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
s : Set α
hs : OrdConnected s
f : α → δ
hf : ContinuousOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
hab : f b ≤ f a
⊢ SurjOn f s (uIcc (f a) (f b))

Next tactic:
simp [hf.surjOn_Icc, *]
/-- Given a `ContDiff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
at `a`, the inverse function (produced by `ContDiff.toPartialHomeomorph`) is
also `ContDiff`. -/
theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) := by
  

Proof state:
𝕂 : Type u_1
inst✝⁵ : IsROrC 𝕂
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕂 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕂 F
inst✝ : CompleteSpace E
f : E → F
f' : E ≃L[𝕂] F
a : E
n : ℕ∞
hf : ContDiffAt 𝕂 n f a
hf' : HasFDerivAt f (↑f') a
hn : 1 ≤ n
⊢ ContDiffAt 𝕂 n (localInverse hf hf' hn) (f a)

Next tactic:
have := hf.localInverse_apply_image hf' hn