-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x

Proof state:
case h
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonpos : f u ≤ 0
hfg₃ : ‖g u‖ ≤ -1 / 2 * f u
⊢ f u + g u ∈ Set.Icc (3 * c₁ * (f x + g x)) (c₂ / 3 * (f x + g x))

Tactic:
refine ⟨?lb, ?ub⟩

Result:
case lb
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonpos : f u ≤ 0
hfg₃ : ‖g u‖ ≤ -1 / 2 * f u
⊢ 3 * c₁ * (f x + g x) ≤ f u + g u

case ub
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonpos : f u ≤ 0
hfg₃ : ‖g u‖ ≤ -1 / 2 * f u
⊢ f u + g u ≤ c₂ / 3 * (f x + g x)

Proof state:
f g : ℝ → ℝ
hf : GrowsPolynomially f
hfg : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ c * ‖f x‖
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x => f x + g x) u ∈ Set.Icc (c₁ * (fun x => f x + g x) x) (c₂ * (fun x => f x + g x) x)

Tactic:
cases hf.eventually_atTop_nonneg_or_nonpos with
| inl hf' => have hf := hf b hb
  obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
  specialize hfg (c := 1/2) (by norm_num)
  refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
  filter_upwards [hf,
                  (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                  (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                  eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
  have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
  have hfg₂ : ‖g x‖ ≤ 1/2 * f x := by
    calc ‖g x‖ ≤ 1/2 * ‖f x‖ := hfg' x hbx
         _ = 1/2 * f x := by congr; exact norm_of_nonneg (hf₂ _ hbx)
  have hx_ub : f x + g x ≤ 3/2 * f x := by
    calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
         _ ≤ f x + 1/2 * f x := by gcongr
         _ = 3/2 * f x := by ring
  have hx_lb : 1/2 * f x ≤ f x + g x := by
    calc f x + g x ≥ f x - ‖g x‖ := by
              rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le (g x)
         _ ≥ f x - 1/2 * f x := by gcongr
         _ = 1/2 * f x := by ring
  intro u ⟨hu_lb, hu_ub⟩
  have hfu_nonneg : 0 ≤ f u := hf₂ _ hu_lb
  have hfg₃ : ‖g u‖ ≤ 1/2 * f u := by
    calc ‖g u‖ ≤ 1/2 * ‖f u‖ := hfg' _ hu_lb
         _ = 1/2 * f u := by congr; simp only [norm_eq_abs, abs_eq_self, hfu_nonneg]
  refine ⟨?lb, ?ub⟩
  case lb =>
    calc f u + g u ≥ f u - ‖g u‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le _
         _ ≥ f u - 1/2 * f u := by gcongr
         _ = 1/2 * f u := by ring
         _ ≥ 1/2 * (c₁ * f x) := by gcongr; exact (hf₁ u ⟨hu_lb, hu_ub⟩).1
         _ = c₁/3 * (3/2 * f x) := by ring
         _ ≥ c₁/3 * (f x + g x) := by gcongr
  case ub =>
    calc _ ≤ f u + ‖g u‖ := by gcongr; exact le_norm_self (g u)
         _ ≤ f u + 1/2 * f u := by gcongr
         _ = 3/2 * f u := by ring
         _ ≤ 3/2 * (c₂ * f x) := by gcongr; exact (hf₁ u ⟨hu_lb, hu_ub⟩).2
         _ = 3*c₂ * (1/2 * f x) := by ring
         _ ≤ 3*c₂ * (f x + g x) := by gcongr
| inr hf' => have hf := hf b hb
  obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
  specialize hfg (c := 1/2) (by norm_num)
  refine ⟨3*c₁, by positivity, c₂/3, by positivity, ?_⟩
  filter_upwards [hf,
                  (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                  (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                  eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
  have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
  have hfg₂ : ‖g x‖ ≤ -1/2 * f x := by
    calc ‖g x‖ ≤ 1/2 * ‖f x‖ := hfg' x hbx
         _ = 1/2 * (-f x) := by congr; exact norm_of_nonpos (hf₂ x hbx)
         _ = _ := by ring
  have hx_ub : f x + g x ≤ 1/2 * f x := by
    calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
         _ ≤ f x + (-1/2 * f x) := by gcongr
         _ = 1/2 * f x := by ring
  have hx_lb : 3/2 * f x ≤ f x + g x := by
    calc f x + g x ≥ f x - ‖g x‖ := by
              rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le (g x)
         _ ≥ f x + 1/2 * f x := by
                rw [sub_eq_add_neg]
                gcongr
                refine le_of_neg_le_neg ?bc.a
                rwa [neg_neg, ← neg_mul, ← neg_div]
         _ = 3/2 * f x := by ring
  intro u ⟨hu_lb, hu_ub⟩
  have hfu_nonpos : f u ≤ 0 := hf₂ _ hu_lb
  have hfg₃ : ‖g u‖ ≤ -1/2 * f u := by
    calc ‖g u‖ ≤ 1/2 * ‖f u‖ := hfg' _ hu_lb
         _ = 1/2 * (-f u) := by congr; exact norm_of_nonpos hfu_nonpos
         _ = -1/2 * f u := by ring
  refine ⟨?lb, ?ub⟩
  case lb =>
    calc f u + g u ≥ f u - ‖g u‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le _
         _ ≥ f u + 1/2 * f u := by
                rw [sub_eq_add_neg]
                gcongr
                refine le_of_neg_le_neg ?_
                rwa [neg_neg, ← neg_mul, ← neg_div]
         _ = 3/2 * f u := by ring
         _ ≥ 3/2 * (c₁ * f x) := by gcongr; exact (hf₁ u ⟨hu_lb, hu_ub⟩).1
         _ = 3*c₁ * (1/2 * f x) := by ring
         _ ≥ 3*c₁ * (f x + g x) := by gcongr
  case ub =>
    calc _ ≤ f u + ‖g u‖ := by gcongr; exact le_norm_self (g u)
         _ ≤ f u - 1/2 * f u := by
              rw [sub_eq_add_neg]
              gcongr
              rwa [← neg_mul, ← neg_div]
         _ = 1/2 * f u := by ring
         _ ≤ 1/2 * (c₂ * f x) := by gcongr; exact (hf₁ u ⟨hu_lb, hu_ub⟩).2
         _ = c₂/3 * (3/2 * f x) := by ring
         _ ≤ c₂/3 * (f x + g x) := by gcongr

Result:
no goals

Proof state:
f g : ℝ → ℝ
hf : GrowsPolynomially f
hfg : g =o[atTop] f
⊢ GrowsPolynomially fun x => f x + g x

Tactic:
intro b hb

Result:
f g : ℝ → ℝ
hf : GrowsPolynomially f
hfg : g =o[atTop] f
b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x => f x + g x) u ∈ Set.Icc (c₁ * (fun x => f x + g x) x) (c₂ * (fun x => f x + g x) x)