cases Nat.lt_or_ge x t with
| inl hxt => have hfxt : round (t - x) = t - x := by
    apply round_eq_of_trunc_eq npos hfaithful₀
    apply hfut le_rfl hxt hfx
    rewrite [hfx']
    exact Nat.le_trans hxt.le htx'
  rw [hs₁ hxt, hfxt, add_tsub_cancel_of_le hxt.le]
| inr htx => have hftx : round (x - t) = x - t := by
    apply round_eq_of_trunc_eq npos hfaithful₀
    rewrite [ht] at htx hfty ⊢
    exact s₁ npos htx hxy (trunc_round npos hfaithful₁ _) hfy hfx hfty
  rw [hs₂ htx, hftx, tsub_tsub_cancel_of_le htx]
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfz : trunc n z = z
hfw : trunc n w = w
hzx : z ≤ x
hzy : z ≤ y
hyxulp : ulp n y ≤ 2 * x
ht : t = round (y - round (y - z))
hs₁ : x < t → s = round (x + round (t - x))
hs₂ : t ≤ x → s = round (x - round (x - t))
hxw : x ≤ w
hwy : w ≤ y
hxy : x ≤ y
ht' : t = y - round (y - z)
htx' : t ≤ y - round (y - x)
hzy' : round (y - z) ≤ y
hft : trunc n t = t
hfx' : round (y - round (y - x)) = y - round (y - x)
hfut : ∀ {u : ℕ}, x ≤ u → u < t → trunc n u = u → u ≤ round (y - round (y - x)) → trunc n (t - u) = t - u
hfty : trunc n (y - t) = y - t
⊢ s = round t
no goals