-- 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:
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 ≥ f u - ‖g u‖

Tactic:
rw [sub_eq_add_neg, norm_eq_abs]

Result:
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 ≥ f u + -|g u|

Proof state:
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, 0 ≤ f x
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 → 0 ≤ f y
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ 1 / 2 * f x
hx_ub : f x + g x ≤ 3 / 2 * f x
hx_lb : 1 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonneg : 0 ≤ f u
hfg₃ : ‖g u‖ ≤ 1 / 2 * f u
⊢ f u - ‖g u‖ ≥ f u - 1 / 2 * f u

Tactic:
gcongr

Result:
no goals

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
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)
⊢ 0 < 1 / 2

Tactic:
norm_num

Result:
no goals

Proof state:
case inl.intro.intro.intro.intro
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
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
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)
⊢ ∃ 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:
specialize hfg (c := 1/2) (by norm_num)

Result:
case inl.intro.intro.intro.intro
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, 0 ≤ f x
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‖
⊢ ∃ 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)

Proof state:
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, 0 ≤ f x
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 → 0 ≤ f y
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ 1 / 2 * f x
hx_ub : f x + g x ≤ 3 / 2 * f x
hx_lb : 1 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonneg : 0 ≤ f u
⊢ ‖g u‖ ≤ 1 / 2 * f u

Tactic:
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]

Result:
no goals

Proof state:
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, 0 ≤ f x
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‖
⊢ c₁ / 3 > 0

Tactic:
positivity

Result:
no goals

Proof state:
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, 0 ≤ f x
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 → 0 ≤ f y
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ 1 / 2 * f x
hx_ub : f x + g x ≤ 3 / 2 * f x
hx_lb : 1 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
hfu_nonneg : 0 ≤ f u
hfg₃ : ‖g u‖ ≤ 1 / 2 * f u
⊢ 3 * c₂ * (1 / 2 * f x) ≤ 3 * c₂ * (f x + g x)

Tactic:
gcongr

Result:
no goals

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, 0 ≤ f x
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 → 0 ≤ f y
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
⊢ ∀ u ∈ Set.Icc (b * x) x, f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))

Tactic:
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)

Result:
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, 0 ≤ f x
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 → 0 ≤ f y
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ 1 / 2 * f x
⊢ ∀ u ∈ Set.Icc (b * x) x, f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))