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]
        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 _
  rintro ⟨i, hi⟩
  

Proof state:
case h.mk
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 : ℕ
hi : i < n + 1
⊢ p { val := i, isLt := hi } ∈ range ⇑γ

Next tactic:
suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ 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'
          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 _
  rintro ⟨i, hi⟩
  suffices p ⟨i, hi⟩ = p' i 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
hpp' : ∀ k < n + 1, p ↑k = p' k
i : ℕ
hi : i < n + 1
this : p { val := i, isLt := hi } = p' i
⊢ p { val := i, isLt := hi } ∈ range ⇑γ

Next tactic:
convert hγ.1 i (Nat.le_of_lt_succ 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'
          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 _
  rintro ⟨i, hi⟩
  suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ hi)
  

Proof state:
case h.mk
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 : ℕ
hi : i < n + 1
⊢ p { val := i, isLt := hi } = p' i

Next tactic:
rw [← hpp' 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'
        · 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 _
  rintro ⟨i, hi⟩
  suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ hi)
  rw [← hpp' i hi]
  

Proof state:
case h.mk
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 : ℕ
hi : i < n + 1
⊢ p { val := i, isLt := hi } = p ↑i

Next tactic:
suffices i = i % n.succ by 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
    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 _
  rintro ⟨i, hi⟩
  suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ hi)
  rw [← hpp' i hi]
  suffices i = i % n.succ 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
hpp' : ∀ k < n + 1, p ↑k = p' k
i : ℕ
hi : i < n + 1
this : i = i % Nat.succ n
⊢ p { val := i, isLt := hi } = p ↑i

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
    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 _
  rintro ⟨i, hi⟩
  suffices p ⟨i, hi⟩ = p' i by convert hγ.1 i (Nat.le_of_lt_succ hi)
  rw [← hpp' i hi]
  suffices i = i % n.succ by congr
  

Proof state:
case h.mk
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 : ℕ
hi : i < n + 1
⊢ i = i % Nat.succ n

Next tactic:
rw [Nat.mod_eq_of_lt 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)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := 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
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Next tactic:
rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
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)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := by
  rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
  

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
γ : Path (p 0) (p ↑n)
hγ : range ⇑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ range ⇑γ
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Next tactic:
rcases hγ with ⟨h₁, h₂⟩
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)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := by
  rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
  rcases hγ with ⟨h₁, h₂⟩
  

Proof state:
case intro.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
γ : Path (p 0) (p ↑n)
h₁ : range ⇑γ ⊆ s
h₂ : ∀ (i : Fin (n + 1)), p i ∈ range ⇑γ
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Next tactic:
simp only [range, mem_setOf_eq] at h₂
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)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := by
  rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
  rcases hγ with ⟨h₁, h₂⟩
  simp only [range, mem_setOf_eq] at h₂
  

Proof state:
case intro.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
γ : Path (p 0) (p ↑n)
h₁ : range ⇑γ ⊆ s
h₂ : ∀ (i : Fin (n + 1)), ∃ y, γ y = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Next tactic:
rw [range_subset_iff] at h₁
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)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := by
  rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
  rcases hγ with ⟨h₁, h₂⟩
  simp only [range, mem_setOf_eq] at h₂
  rw [range_subset_iff] at h₁
  

Proof state:
case intro.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
γ : Path (p 0) (p ↑n)
h₁ : ∀ (y : ↑I), γ y ∈ s
h₂ : ∀ (i : Fin (n + 1)), ∃ y, γ y = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Next tactic:
choose! t ht using h₂
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)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := by
  rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
  rcases hγ with ⟨h₁, h₂⟩
  simp only [range, mem_setOf_eq] at h₂
  rw [range_subset_iff] at h₁
  choose! t ht using h₂
  

Proof state:
case intro.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
γ : Path (p 0) (p ↑n)
h₁ : ∀ (y : ↑I), γ y ∈ s
t : Fin (n + 1) → ↑I
ht : ∀ (i : Fin (n + 1)), γ (t i) = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Next tactic:
exact ⟨γ, t, h₁, ht⟩
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := 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
⊢ PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X)

Next tactic:
letI := pathSetoid X
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  

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
this : Setoid X := pathSetoid X
⊢ PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X)

Next tactic:
constructor
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · 

Proof state:
case mp
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
this : Setoid X := pathSetoid X
⊢ PathConnectedSpace X → Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X)

Next tactic:
intro h
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    

Proof state:
case mp
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
this : Setoid X := pathSetoid X
h : PathConnectedSpace X
⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X)

Next tactic:
refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
    

Proof state:
case mp
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
this : Setoid X := pathSetoid X
h : PathConnectedSpace X
⊢ ∀ (a b : ZerothHomotopy X), a = b

Next tactic:
rintro ⟨x⟩ ⟨y⟩
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
    rintro ⟨x⟩ ⟨y⟩
    

Proof state:
case mp.mk.mk
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
this : Setoid X := pathSetoid X
h : PathConnectedSpace X
a✝ : ZerothHomotopy X
x : X
b✝ : ZerothHomotopy X
y : X
⊢ Quot.mk Setoid.r x = Quot.mk Setoid.r y

Next tactic:
exact Quotient.sound (PathConnectedSpace.joined x y)
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
    rintro ⟨x⟩ ⟨y⟩
    exact Quotient.sound (PathConnectedSpace.joined x y)
  · 

Proof state:
case mpr
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
this : Setoid X := pathSetoid X
⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) → PathConnectedSpace X

Next tactic:
unfold ZerothHomotopy
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
    rintro ⟨x⟩ ⟨y⟩
    exact Quotient.sound (PathConnectedSpace.joined x y)
  · unfold ZerothHomotopy
    

Proof state:
case mpr
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
this : Setoid X := pathSetoid X
⊢ Nonempty (Quotient (pathSetoid X)) ∧ Subsingleton (Quotient (pathSetoid X)) → PathConnectedSpace X

Next tactic:
rintro ⟨h, h'⟩
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
    rintro ⟨x⟩ ⟨y⟩
    exact Quotient.sound (PathConnectedSpace.joined x y)
  · unfold ZerothHomotopy
    rintro ⟨h, h'⟩
    

Proof state:
case mpr.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
this : Setoid X := pathSetoid X
h : Nonempty (Quotient (pathSetoid X))
h' : Subsingleton (Quotient (pathSetoid X))
⊢ PathConnectedSpace X

Next tactic:
skip
theorem pathConnectedSpace_iff_zerothHomotopy :
    PathConnectedSpace X ↔ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) := by
  letI := pathSetoid X
  constructor
  · intro h
    refine' ⟨(nonempty_quotient_iff _).mpr h.1, ⟨_⟩⟩
    rintro ⟨x⟩ ⟨y⟩
    exact Quotient.sound (PathConnectedSpace.joined x y)
  · unfold ZerothHomotopy
    rintro ⟨h, h'⟩
    skip
    

Proof state:
case mpr.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
this : Setoid X := pathSetoid X
h : Nonempty (Quotient (pathSetoid X))
h' : Subsingleton (Quotient (pathSetoid X))
⊢ PathConnectedSpace X

Next tactic:
exact ⟨(nonempty_quotient_iff _).mp h, fun x y => Quotient.exact <| Subsingleton.elim ⟦x⟧ ⟦y⟧⟩