l3lab/ntp-mathlib huggingface dataset:
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · 

Proof state:
case h.left
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
⊢ ∀ i ≤ Nat.succ n, p' i ∈ range ⇑γ

Next tactic:
rintro i hi
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        

Proof state:
case h.left
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
⊢ p' i ∈ range ⇑γ

Next tactic:
by_cases hi' : i ≤ n
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · 

Proof state:
case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : i ≤ n
⊢ p' i ∈ range ⇑γ

Next tactic:
rw [range_eq]
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          

Proof state:
case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : i ≤ n
⊢ p' i ∈ range ⇑γ₀ ∪ range ⇑γ₁

Next tactic:
left
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          

Proof state:
case pos.h
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : i ≤ n
⊢ p' i ∈ range ⇑γ₀

Next tactic:
exact hγ₀.1 i hi'
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · 

Proof state:
case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : ¬i ≤ n
⊢ p' i ∈ range ⇑γ

Next tactic:
rw [not_le, ← Nat.succ_le_iff] at hi'
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          

Proof state:
case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : Nat.succ n ≤ i
⊢ p' i ∈ range ⇑γ

Next tactic:
have : i = n.succ := le_antisymm hi hi'
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          have : i = n.succ := le_antisymm hi hi'
          

Proof state:
case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : Nat.succ n ≤ i
this : i = Nat.succ n
⊢ p' i ∈ range ⇑γ

Next tactic:
rw [this]
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          have : i = n.succ := le_antisymm hi hi'
          rw [this]
          

Proof state:
case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : Nat.succ n ≤ i
this : i = Nat.succ n
⊢ p' (Nat.succ n) ∈ range ⇑γ

Next tactic:
use 1
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          have : i = n.succ := le_antisymm hi hi'
          rw [this]
          use 1
          

Proof state:
case h
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
i : ℕ
hi : i ≤ Nat.succ n
hi' : Nat.succ n ≤ i
this : i = Nat.succ n
⊢ γ 1 = p' (Nat.succ n)

Next tactic:
exact γ.target
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          have : i = n.succ := le_antisymm hi hi'
          rw [this]
          use 1
          exact γ.target
      · 

Proof state:
case h.right
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
⊢ range ⇑γ ⊆ s

Next tactic:
rw [range_eq]
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          have : i = n.succ := le_antisymm hi hi'
          rw [this]
          use 1
          exact γ.target
      · rw [range_eq]
        

Proof state:
case h.right
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
⊢ range ⇑γ₀ ∪ range ⇑γ₁ ⊆ s

Next tactic:
apply union_subset hγ₀.2
theorem IsPathConnected.exists_path_through_family {n : ℕ}
    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
    ∃ γ : Path (p 0) (p n), range γ ⊆ s ∧ ∀ i, p i ∈ range γ := by
  let p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, n.zero_lt_succ⟩
  obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
    have hp' : ∀ i ≤ n, p' i ∈ s := by
      intro i hi
      simp [Nat.lt_succ_of_le hi, hp]
    clear_value p'
    clear hp p
    induction' n with n hn
    · use Path.refl (p' 0)
      · constructor
        · rintro i hi
          rw [le_zero_iff.mp hi]
          exact ⟨0, rfl⟩
        · rw [range_subset_iff]
          rintro _x
          exact hp' 0 le_rfl
    · rcases hn fun i hi => hp' i <| Nat.le_succ_of_le hi with ⟨γ₀, hγ₀⟩
      rcases h.joinedIn (p' n) (hp' n n.le_succ) (p' <| n + 1) (hp' (n + 1) <| le_rfl) with
        ⟨γ₁, hγ₁⟩
      let γ : Path (p' 0) (p' <| n + 1) := γ₀.trans γ₁
      use γ
      have range_eq : range γ = range γ₀ ∪ range γ₁ := γ₀.trans_range γ₁
      constructor
      · rintro i hi
        by_cases hi' : i ≤ n
        · rw [range_eq]
          left
          exact hγ₀.1 i hi'
        · rw [not_le, ← Nat.succ_le_iff] at hi'
          have : i = n.succ := le_antisymm hi hi'
          rw [this]
          use 1
          exact γ.target
      · rw [range_eq]
        apply union_subset hγ₀.2
        

Proof state:
case h.right
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F s : Set X
h : IsPathConnected s
p' : ℕ → X
n : ℕ
hn : (∀ i ≤ n, p' i ∈ s) → ∃ γ, (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hp' : ∀ i ≤ Nat.succ n, p' i ∈ s
γ₀ : Path (p' 0) (p' n)
hγ₀ : (∀ i ≤ n, p' i ∈ range ⇑γ₀) ∧ range ⇑γ₀ ⊆ s
γ₁ : Path (p' n) (p' (n + 1))
hγ₁ : ∀ (t : ↑I), γ₁ t ∈ s
γ : Path (p' 0) (p' (n + 1)) := Path.trans γ₀ γ₁
range_eq : range ⇑γ = range ⇑γ₀ ∪ range ⇑γ₁
⊢ range ⇑γ₁ ⊆ s

Next tactic:
rw [range_subset_iff]