/-- Consider a function `d : X → X → ℝ≥0` such that `d x x = 0` and `d x y = d y x` for all `x`,
`y`. Let `dist` be the largest pseudometric distance such that `dist x y ≤ d x y`, see
`PseudoMetricSpace.ofPreNNDist`. Suppose that `d` satisfies the following triangle-like
inequality: `d x₁ x₄ ≤ 2 * max (d x₁ x₂, d x₂ x₃, d x₃ x₄)`. Then `d x y ≤ 2 * dist x y` for all
`x`, `y`. -/
theorem le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
    (dist_comm : ∀ x y, d x y = d y x)
    (hd : ∀ x₁ x₂ x₃ x₄, d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
    ↑(d x y) ≤ 2 * @dist X
      (@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x y := by
  /- We need to show that `d x y` is at most twice the sum `L` of `d xᵢ xᵢ₊₁` over a path
    `x₀=x, ..., xₙ=y`. We prove it by induction on the length `n` of the sequence. Find an edge that
    splits the path into two parts of almost equal length: both `d x₀ x₁ + ... + d xₖ₋₁ xₖ` and
    `d xₖ₊₁ xₖ₊₂ + ... + d xₙ₋₁ xₙ` are less than or equal to `L / 2`.
    Then `d x₀ xₖ ≤ L`, `d xₖ xₖ₊₁ ≤ L`, and `d xₖ₊₁ xₙ ≤ L`, thus `d x₀ xₙ ≤ 2 * L`. -/
  rw [dist_ofPreNNDist, ← NNReal.coe_two, ← NNReal.coe_mul, NNReal.mul_iInf, NNReal.coe_le_coe]
  refine' le_ciInf fun l => _
  have hd₀_trans : Transitive fun x y => d x y = 0 := by
    intro a b c hab hbc
    rw [← nonpos_iff_eq_zero]
    simpa only [nonpos_iff_eq_zero, hab, hbc, dist_self c, max_self, mul_zero] using hd a b c c
  haveI : IsTrans X fun x y => d x y = 0 := ⟨hd₀_trans⟩
  induction' hn : length l using Nat.strong_induction_on with n ihn generalizing x y l
  simp only at ihn
  subst n
  set L := zipWith d (x::l) (l ++ [y])
  have hL_len : length L = length l + 1 := by simp
  rcases eq_or_ne (d x y) 0 with hd₀ | hd₀
  · simp only [hd₀, zero_le]
  rsuffices ⟨z, z', hxz, hzz', hz'y⟩ : ∃ z z' : X, d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
  · exact (hd x z z' y).trans (mul_le_mul_left' (max_le hxz (max_le hzz' hz'y)) _)
  set s : Set ℕ := { m : ℕ | 2 * (take m L).sum ≤ L.sum }
  have hs₀ : 0 ∈ s := by simp
  have hsne : s.Nonempty := ⟨0, hs₀⟩
  obtain ⟨M, hMl, hMs⟩ : ∃ M ≤ length l, IsGreatest s M := by
    have hs_ub : length l ∈ upperBounds s := by
      intro m hm
      rw [← not_lt, Nat.lt_iff_add_one_le, ← hL_len]
      intro hLm
      rw [mem_setOf_eq, take_all_of_le hLm, two_mul, add_le_iff_nonpos_left, nonpos_iff_eq_zero,
          sum_eq_zero_iff, ← forall_iff_forall_mem, forall_zipWith,
          ← chain_append_singleton_iff_forall₂]
          at hm <;>
        [skip; simp]
      exact hd₀ (hm.rel (mem_append.2 <| Or.inr <| mem_singleton_self _))
    

Proof state:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < length l, ∀ (x y : X) (l : List X), length l = m → d x y ≤ 2 * sum (zipWith d (x :: l) (l ++ [y]))
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : length L = length l + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m | 2 * sum (take m L) ≤ sum L}
hs₀ : 0 ∈ s
hsne : Set.Nonempty s
hs_ub : length l ∈ upperBounds s
⊢ ∃ M ≤ length l, IsGreatest s M

Next tactic:
have hs_bdd : BddAbove s := ⟨length l, hs_ub⟩