-- Module: Mathlib.Topology.Metrizable.Uniformity
Goal:
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))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
h :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
h₁ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l) (take (M + 1) (l ++ [y]))
h₂ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M).toList)
h₃ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M).toList)
h₄ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (some (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl)).toList)
h₅ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++
        Option.toList ((fun (a : X) ↦ some a) (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl)))
h₆ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl])
h₇ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l) (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl])
h₈ h₉ :
  zipWith d (x :: take M l) (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M ⋯]) =
    zipWith d (x :: take M l) (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl])
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L

Tactic:
rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]

Premises used: List.take_zipWith, List.take_append_of_le_length, List.getElem?_append_left, LT.lt.le, Option.coe_def, List.getElem?_eq_getElem, Option.toList_some, List.take_add_one, List.take, List.getElem_cons_succ