/-- An auxiliary lemma that combines two double exponential estimates into a similar estimate
on the difference of the functions. -/
theorem isBigO_sub_exp_exp {a : ℝ} {f g : ℂ → E} {l : Filter ℂ} {u : ℂ → ℝ}
    (hBf : ∃ c < a, ∃ B, f =O[l] fun z => expR (B * expR (c * |u z|)))
    (hBg : ∃ c < a, ∃ B, g =O[l] fun z => expR (B * expR (c * |u z|))) :
    ∃ c < a, ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|)) := by
  have : ∀ {c₁ c₂ B₁ B₂}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ z,
      ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖ := fun hc hB₀ hB z ↦ by
    simp only [Real.norm_eq_abs, Real.abs_exp]; gcongr
  rcases hBf with ⟨cf, hcf, Bf, hOf⟩; rcases hBg with ⟨cg, hcg, Bg, hOg⟩
  refine' ⟨max cf cg, max_lt hcf hcg, max 0 (max Bf Bg), _⟩
  refine' (hOf.trans_le <| this _ _ _).sub (hOg.trans_le <| this _ _ _)
  

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))
⊢ cf ≤ max cf cg
case intro.intro.intro.intro.intro.intro.refine'_2
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))
⊢ 0 ≤ max 0 (max Bf Bg)
case intro.intro.intro.intro.intro.intro.refine'_3
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))
⊢ Bf ≤ max 0 (max Bf Bg)
case intro.intro.intro.intro.intro.intro.refine'_4
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))
⊢ cg ≤ max cf cg
case intro.intro.intro.intro.intro.intro.refine'_5
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))
⊢ 0 ≤ max 0 (max Bf Bg)
case intro.intro.intro.intro.intro.intro.refine'_6
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))
⊢ Bg ≤ max 0 (max Bf Bg)

Next tactic:
exacts [le_max_left _ _, le_max_left _ _, (le_max_left _ _).trans (le_max_right _ _),
    le_max_right _ _, le_max_left _ _, (le_max_right _ _).trans (le_max_right _ _)]