Eq i Union

lemma eq_iUnion (σ τ : Ω → ℕ∞) :    stochIoc σ τ = ⋃ i : ℕ, Set.Ioc i (i + 1) ×ˢ {ω | σ ω ≤ i ∧ i < τ ω} := by  ext ⟨t, ω⟩  simp only [mem_iUnion, Set.mem_prod, Set.mem_Ioc, Set.mem_ofPred_eq]  constructor  · rintro ⟨hlt, hle⟩    obtain ⟨hsub, ht1⟩ := lt_coe_iff_le_coe_sub_one.mp hlt    refine ⟨t - 1, ⟨by lia, by lia⟩, hsub, ?_⟩    refine lt_of_lt_of_le ?_ hle    refine coe_lt_iff_coe_succ_le.mpr ?_    simp only [ht1, Nat.sub_add_cancel, ENat.some_eq_natCast]    norm_cast  · rintro ⟨i, ⟨hi_lt, hi_le⟩, hσi, hiτ⟩    have ht : t = i + 1 := by lia    subst ht    exact ⟨lt_of_le_of_lt hσi (Nat.cast_lt.mpr (by lia)),      (coe_lt_iff_coe_succ_le).mp hiτ⟩