l3lab/ntp-mathlib huggingface dataset:
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      

Proof state:
case mk.mk.mk.inl.inr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
z : E i
hij : i ≠ j
⊢ dist x z ≤
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) y +
      (dist y (Nonempty.some (_ : Nonempty (E j))) + 1 + dist (Nonempty.some (_ : Nonempty (E i))) z)

Next tactic:
calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
z : E i
hij : i ≠ j
⊢ dist x z ≤ dist x (Nonempty.some (_ : Nonempty (E i))) + 0 + 0 + (0 + 0 + dist (Nonempty.some (_ : Nonempty (E i))) z)

Next tactic:
simpa only [zero_add, add_zero] using dist_triangle _ _ _
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by 

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
z : E i
hij : i ≠ j
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 0 + 0 + (0 + 0 + dist (Nonempty.some (_ : Nonempty (E i))) z) ≤
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) y +
      (dist y (Nonempty.some (_ : Nonempty (E j))) + 1 + dist (Nonempty.some (_ : Nonempty (E i))) z)

Next tactic:
apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · 

Proof state:
case mk.mk.mk.inr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
⊢ dist { fst := i, snd := x } { fst := k, snd := z } ≤
    dist { fst := i, snd := x } { fst := j, snd := y } + dist { fst := j, snd := y } { fst := k, snd := z }

Next tactic:
rcases eq_or_ne i j with (rfl | hij)
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · 

Proof state:
case mk.mk.mk.inr.inl
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
k : ι
z : E k
hik : i ≠ k
y : E i
⊢ dist { fst := i, snd := x } { fst := k, snd := z } ≤
    dist { fst := i, snd := x } { fst := i, snd := y } + dist { fst := i, snd := y } { fst := k, snd := z }

Next tactic:
simp only [Sigma.dist_ne hik, Sigma.dist_same]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      

Proof state:
case mk.mk.mk.inr.inl
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
k : ι
z : E k
hik : i ≠ k
y : E i
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z ≤
    dist x y + (dist y (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z)

Next tactic:
calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
k : ι
z : E k
hik : i ≠ k
y : E i
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z ≤
    dist x y + dist y (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z

Next tactic:
apply_rules [add_le_add, le_rfl, dist_triangle]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by 

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
k : ι
z : E k
hik : i ≠ k
y : E i
⊢ dist x y + dist y (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z =
    dist x y + (dist y (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z)

Next tactic:
abel
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · 

Proof state:
case mk.mk.mk.inr.inr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
⊢ dist { fst := i, snd := x } { fst := k, snd := z } ≤
    dist { fst := i, snd := x } { fst := j, snd := y } + dist { fst := j, snd := y } { fst := k, snd := z }

Next tactic:
rcases eq_or_ne j k with (rfl | hjk)
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · 

Proof state:
case mk.mk.mk.inr.inr.inl
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist { fst := i, snd := x } { fst := j, snd := z } ≤
    dist { fst := i, snd := x } { fst := j, snd := y } + dist { fst := j, snd := y } { fst := j, snd := z }

Next tactic:
simp only [Sigma.dist_ne hij, Sigma.dist_same]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · simp only [Sigma.dist_ne hij, Sigma.dist_same]
        

Proof state:
case mk.mk.mk.inr.inr.inl
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) z ≤
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) y + dist y z

Next tactic:
calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            apply_rules [add_le_add, le_rfl, dist_triangle]
          _ = _ := by abel
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · simp only [Sigma.dist_ne hij, Sigma.dist_same]
        calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) z ≤
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + (dist (Nonempty.some (_ : Nonempty (E j))) y + dist y z)

Next tactic:
apply_rules [add_le_add, le_rfl, dist_triangle]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · simp only [Sigma.dist_ne hij, Sigma.dist_same]
        calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            apply_rules [add_le_add, le_rfl, dist_triangle]
          _ = _ := by 

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + (dist (Nonempty.some (_ : Nonempty (E j))) y + dist y z) =
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) y + dist y z

Next tactic:
abel
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · simp only [Sigma.dist_ne hij, Sigma.dist_same]
        calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            apply_rules [add_le_add, le_rfl, dist_triangle]
          _ = _ := by abel
      · 

Proof state:
case mk.mk.mk.inr.inr.inr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
hjk : j ≠ k
⊢ dist { fst := i, snd := x } { fst := k, snd := z } ≤
    dist { fst := i, snd := x } { fst := j, snd := y } + dist { fst := j, snd := y } { fst := k, snd := z }

Next tactic:
simp only [hik, hij, hjk, Sigma.dist_ne, Ne.def, not_false_iff]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · simp only [Sigma.dist_ne hij, Sigma.dist_same]
        calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            apply_rules [add_le_add, le_rfl, dist_triangle]
          _ = _ := by abel
      · simp only [hik, hij, hjk, Sigma.dist_ne, Ne.def, not_false_iff]
        

Proof state:
case mk.mk.mk.inr.inr.inr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
hjk : j ≠ k
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z ≤
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E j))) y +
      (dist y (Nonempty.some (_ : Nonempty (E j))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z)

Next tactic:
calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z =
              dist x (Nonempty.some ⟨x⟩) + 1 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
            simp only [add_zero, zero_add]
          _ ≤ _ := by apply_rules [add_le_add, zero_le_one, dist_nonneg, le_rfl]
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z := by
  rcases x with ⟨i, x⟩; rcases y with ⟨j, y⟩; rcases z with ⟨k, z⟩
  rcases eq_or_ne i k with (rfl | hik)
  · rcases eq_or_ne i j with (rfl | hij)
    · simpa using dist_triangle x y z
    · simp only [Sigma.dist_same, Sigma.dist_ne hij, Sigma.dist_ne hij.symm]
      calc
        dist x z ≤ dist x (Nonempty.some ⟨x⟩) + 0 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
          simpa only [zero_add, add_zero] using dist_triangle _ _ _
        _ ≤ _ := by apply_rules [add_le_add, le_rfl, dist_nonneg, zero_le_one]
  · rcases eq_or_ne i j with (rfl | hij)
    · simp only [Sigma.dist_ne hik, Sigma.dist_same]
      calc
        dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
            dist x y + dist y (Nonempty.some ⟨y⟩) + 1 + dist (Nonempty.some ⟨z⟩) z := by
          apply_rules [add_le_add, le_rfl, dist_triangle]
        _ = _ := by abel
    · rcases eq_or_ne j k with (rfl | hjk)
      · simp only [Sigma.dist_ne hij, Sigma.dist_same]
        calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            apply_rules [add_le_add, le_rfl, dist_triangle]
          _ = _ := by abel
      · simp only [hik, hij, hjk, Sigma.dist_ne, Ne.def, not_false_iff]
        calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z =
              dist x (Nonempty.some ⟨x⟩) + 1 + 0 + (0 + 0 + dist (Nonempty.some ⟨z⟩) z) := by
            

Proof state:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
hjk : j ≠ k
⊢ dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + dist (Nonempty.some (_ : Nonempty (E k))) z =
    dist x (Nonempty.some (_ : Nonempty (E i))) + 1 + 0 + (0 + 0 + dist (Nonempty.some (_ : Nonempty (E k))) z)

Next tactic:
simp only [add_zero, zero_add]