-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
protected lemma GrowsPolynomially.add {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hg : GrowsPolynomially g) (hf' : 0 ≤ᶠ[atTop] f) (hg' : 0 ≤ᶠ[atTop] g) :
    GrowsPolynomially fun x => f x + g x

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ > 0
hg : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
⊢ ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun x => f x + g x) u ∈ Set.Icc ((c₁ ⊓ c₃) * (fun x => f x + g x) x) (c₂ * (fun x => f x + g x) x)

Tactic:
refine ⟨max c₂ c₄, by show 0 < max c₂ c₄; positivity, ?_⟩

Result:
case intro.intro.intro.intro.intro.intro.intro.intro
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ > 0
hg : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun x => f x + g x) u ∈ Set.Icc ((c₁ ⊓ c₃) * (fun x => f x + g x) x) ((c₂ ⊔ c₄) * (fun x => f x + g x) x)

Proof state:
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf : ∃ c₁ > 0, ∃ c₂ > 0, ∀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:
have hg := hg b hb

Result:
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₁ * g x) (c₂ * g 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✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ > 0
hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
hf' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ f y
hg' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ g y
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
fx_nonneg : 0 ≤ f x
gx_nonneg : 0 ≤ g x
⊢ (c₁ ⊓ c₃) * f x + (c₁ ⊓ c₃) * g x ≤ c₁ * f x + c₃ * g x

Tactic:
gcongr

Result:
case h₁.h
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ > 0
hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
hf' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ f y
hg' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ g y
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
fx_nonneg : 0 ≤ f x
gx_nonneg : 0 ≤ g x
⊢ c₁ ⊓ c₃ ≤ c₁

case h₂.h
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ > 0
hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
hf' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ f y
hg' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ g y
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
fx_nonneg : 0 ≤ f x
gx_nonneg : 0 ≤ g x
⊢ c₁ ⊓ c₃ ≤ c₃

Proof state:
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₁ * g x) (c₂ * g 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:
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf

Result:
case intro.intro.intro.intro
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hg : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₁ * g x) (c₂ * g x)
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
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)

Proof state:
case h₁
f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
hf'✝ : 0 ≤ᶠ[atTop] f
hg'✝ : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
left✝ : c₄ > 0
hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
x : ℝ
hf : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₃ * g x) (c₄ * g x)
hf' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ f y
hg' : ∀ (y : ℝ), b * id x ≤ y → 0 y ≤ g y
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
fx_nonneg : 0 ≤ f x
gx_nonneg : 0 ≤ g x
⊢ f u ≤ c₂ * f x

Tactic:
exact (hf u hu).2

Result:
no goals