/-- The variation of a function on the union of two sets `s` and `t`, with `s` to the left of `t`,
bounds the sum of the variations along `s` and `t`. -/
theorem add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
    eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) := by
  by_cases hs : s = ∅
  · simp [hs]
  have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } :=
    nonempty_monotone_mem (nonempty_iff_ne_empty.2 hs)
  by_cases ht : t = ∅
  · simp [ht]
  have : Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ t } :=
    nonempty_monotone_mem (nonempty_iff_ne_empty.2 ht)
  refine' ENNReal.iSup_add_iSup_le _
  /- We start from two sequences `u` and `v` along `s` and `t` respectively, and we build a new
    sequence `w` along `s ∪ t` by juxtaposing them. Its variation is larger than the sum of the
    variations. -/
  rintro ⟨n, ⟨u, hu, us⟩⟩ ⟨m, ⟨v, hv, vt⟩⟩
  let w i := if i ≤ n then u i else v (i - (n + 1))
  have wst : ∀ i, w i ∈ s ∪ t := by
    intro i
    by_cases hi : i ≤ n
    · simp [hi, us]
    · simp [hi, vt]
  have hw : Monotone w := by
    intro i j hij
    dsimp only
    split_ifs with h_1 h_2 h_2
    · exact hu hij
    · apply h _ (us _) _ (vt _)
    · exfalso; exact h_1 (hij.trans h_2)
    · apply hv (tsub_le_tsub hij le_rfl)
  

Proof state:
case neg.mk.mk.intro.mk.mk.intro
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
m : ℕ
v : ℕ → α
hv : Monotone v
vt : ∀ (i : ℕ), v i ∈ t
w : ℕ → α := fun i => if i ≤ n then u i else v (i - (n + 1))
wst : ∀ (i : ℕ), w i ∈ s ∪ t
hw : Monotone w
⊢ ∑ i in Finset.range (n, { val := u, property := (_ : Monotone u ∧ ∀ (i : ℕ), u i ∈ s) }).1,
        edist (f (↑(n, { val := u, property := (_ : Monotone u ∧ ∀ (i : ℕ), u i ∈ s) }).2 (i + 1)))
          (f (↑(n, { val := u, property := (_ : Monotone u ∧ ∀ (i : ℕ), u i ∈ s) }).2 i)) +
      ∑ i in Finset.range (m, { val := v, property := (_ : Monotone v ∧ ∀ (i : ℕ), v i ∈ t) }).1,
        edist (f (↑(m, { val := v, property := (_ : Monotone v ∧ ∀ (i : ℕ), v i ∈ t) }).2 (i + 1)))
          (f (↑(m, { val := v, property := (_ : Monotone v ∧ ∀ (i : ℕ), v i ∈ t) }).2 i)) ≤
    eVariationOn f (s ∪ t)

Next tactic:
calc
    ((∑ i in Finset.range n, edist (f (u (i + 1))) (f (u i))) +
          ∑ i : ℕ in Finset.range m, edist (f (v (i + 1))) (f (v i))) =
        (∑ i in Finset.range n, edist (f (w (i + 1))) (f (w i))) +
          ∑ i : ℕ in Finset.range m, edist (f (w (n + 1 + i + 1))) (f (w (n + 1 + i))) := by
      dsimp only
      congr 1
      · refine Finset.sum_congr rfl fun i hi => ?_
        simp only [Finset.mem_range] at hi
        have : i + 1 ≤ n := Nat.succ_le_of_lt hi
        simp [hi.le, this]
      · refine Finset.sum_congr rfl fun i hi => ?_
        simp only [Finset.mem_range] at hi
        have B : ¬n + 1 + i ≤ n := by linarith
        have A : ¬n + 1 + i + 1 ≤ n := fun h => B ((n + 1 + i).le_succ.trans h)
        have C : n + 1 + i - n = i + 1 := by
          rw [tsub_eq_iff_eq_add_of_le]
          · abel
          · exact n.le_succ.trans (n.succ.le_add_right i)
        simp only [A, B, C, Nat.succ_sub_succ_eq_sub, if_false, add_tsub_cancel_left]
    _ = (∑ i in Finset.range n, edist (f (w (i + 1))) (f (w i))) +
          ∑ i : ℕ in Finset.Ico (n + 1) (n + 1 + m), edist (f (w (i + 1))) (f (w i)) := by
      congr 1
      rw [Finset.range_eq_Ico]
      convert Finset.sum_Ico_add (fun i : ℕ => edist (f (w (i + 1))) (f (w i))) 0 m (n + 1)
        using 3 <;> abel
    _ ≤ ∑ i in Finset.range (n + 1 + m), edist (f (w (i + 1))) (f (w i)) := by
      rw [← Finset.sum_union]
      · apply Finset.sum_le_sum_of_subset _
        rintro i hi
        simp only [Finset.mem_union, Finset.mem_range, Finset.mem_Ico] at hi ⊢
        cases' hi with hi hi
        · exact lt_of_lt_of_le hi (n.le_succ.trans (n.succ.le_add_right m))
        · exact hi.2
      · refine Finset.disjoint_left.2 fun i hi h'i => ?_
        simp only [Finset.mem_Ico, Finset.mem_range] at hi h'i
        exact hi.not_lt (Nat.lt_of_succ_le h'i.left)
    _ ≤ eVariationOn f (s ∪ t) := sum_le f _ hw wst