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
      · 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
        rw [range_subset_iff]
        

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
⊢ ∀ (y : ↑I), γ₁ y ∈ s

Next tactic:
exact hγ₁
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
        rw [range_subset_iff]
        exact hγ₁
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
⊢ ∃ γ, range ⇑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ⇑γ

Next tactic:
have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    ext
    rw [Fin.val_cast_of_lt hk]
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
⊢ ∀ k < n + 1, p ↑k = p' k

Next tactic:
intro k hk
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
k : ℕ
hk : k < n + 1
⊢ p ↑k = p' k

Next tactic:
simp only [hk, dif_pos]
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
k : ℕ
hk : k < n + 1
⊢ p ↑k = p { val := k, isLt := (_ : k < n + 1) }

Next tactic:
congr
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    

Proof state:
case e_a
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
k : ℕ
hk : k < n + 1
⊢ ↑k = { val := k, isLt := (_ : k < n + 1) }

Next tactic:
ext
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    ext
    

Proof state:
case e_a.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 : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
k : ℕ
hk : k < n + 1
⊢ ↑↑k = ↑{ val := k, isLt := (_ : k < n + 1) }

Next tactic:
rw [Fin.val_cast_of_lt hk]
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    ext
    rw [Fin.val_cast_of_lt hk]
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hpp' : ∀ k < n + 1, p ↑k = p' k
⊢ ∃ γ, range ⇑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ⇑γ

Next tactic:
use γ.cast (hpp' 0 n.zero_lt_succ) (hpp' n n.lt_succ_self)
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    ext
    rw [Fin.val_cast_of_lt hk]
  use γ.cast (hpp' 0 n.zero_lt_succ) (hpp' n n.lt_succ_self)
  

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 : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hpp' : ∀ k < n + 1, p ↑k = p' k
⊢ range ⇑(Path.cast γ (_ : p ↑0 = p' 0) (_ : p ↑n = p' n)) ⊆ s ∧
    ∀ (i : Fin (n + 1)), p i ∈ range ⇑(Path.cast γ (_ : p ↑0 = p' 0) (_ : p ↑n = p' n))

Next tactic:
simp only [γ.cast_coe]
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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    ext
    rw [Fin.val_cast_of_lt hk]
  use γ.cast (hpp' 0 n.zero_lt_succ) (hpp' n n.lt_succ_self)
  simp only [γ.cast_coe]
  

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 : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hpp' : ∀ k < n + 1, p ↑k = p' k
⊢ range ⇑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ⇑γ

Next tactic:
refine' And.intro hγ.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
        rw [range_subset_iff]
        exact hγ₁
  have hpp' : ∀ k < n + 1, p k = p' k := by
    intro k hk
    simp only [hk, dif_pos]
    congr
    ext
    rw [Fin.val_cast_of_lt hk]
  use γ.cast (hpp' 0 n.zero_lt_succ) (hpp' n n.lt_succ_self)
  simp only [γ.cast_coe]
  refine' And.intro hγ.2 _
  

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 : Set X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
p' : ℕ → X := fun k => if h : k < n + 1 then p { val := k, isLt := h } else p { val := 0, isLt := (_ : 0 < Nat.succ n) }
γ : Path (p' 0) (p' n)
hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s
hpp' : ∀ k < n + 1, p ↑k = p' k
⊢ ∀ (i : Fin (n + 1)), p i ∈ range ⇑γ

Next tactic:
rintro ⟨i, hi⟩