-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
    (∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0)

Proof state:
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hlt : c₁ < c₂
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x
hu✝ : f (3 / 4 * x) ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hu : c₁ * f x ≤ c₂ * f x
⊢ 0 ≤ f x

Tactic:
have hu' : 0 ≤ (c₂ - c₁) * f x := by linarith

Result:
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hlt : c₁ < c₂
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x
hu✝ : f (3 / 4 * x) ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hu : c₁ * f x ≤ c₂ * f x
hu' : 0 ≤ (c₂ - c₁) * f x
⊢ 0 ≤ f x

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
z_nonneg : 0 ≤ z
le_2n : n₀ ⊔ 2 ≤ 2 ^ n * (n₀ ⊔ 2)
n₀_le_z : n₀ ≤ z
fz_eq_c₂fz : f z = c₂ * f z
z_to_half_z' : f (1 / 2 * z) = c₂ * f z
z_to_half_z : f (1 / 2 * z) = f z
half_z_to_base : f (1 / 2 * z) = f (n₀ ⊔ 2)
⊢ f z = f (n₀ ⊔ 2)

Tactic:
rw [← z_to_half_z, half_z_to_base]

Result:
no goals

Proof state:
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hlt : c₁ < c₂
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x
hu✝ : f (3 / 4 * x) ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hu : (Set.Icc (c₁ * f x) (c₂ * f x)).Nonempty
⊢ 0 ≤ f x

Tactic:
rw [Set.nonempty_Icc] at hu

Result:
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hlt : c₁ < c₂
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x
hu✝ : f (3 / 4 * x) ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hu : c₁ * f x ≤ c₂ * f x
⊢ 0 ≤ f x

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hgt : c₂ < c₁
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
⊢ 1 / 2 * x ≤ 3 / 4 * x ∧ 3 / 4 * x ≤ x

Tactic:
exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩

Result:
no goals

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
z_nonneg : 0 ≤ z
le_2n : n₀ ⊔ 2 ≤ 2 ^ n * (n₀ ⊔ 2)
n₀_le_z : n₀ ≤ z
fz_eq_c₂fz : f z = c₂ * f z
z_to_half_z' : f (1 / 2 * z) = c₂ * f z
⊢ f z = f (n₀ ⊔ 2)

Tactic:
have z_to_half_z : f (1/2 * z) = f z := by rwa [← fz_eq_c₂fz] at z_to_half_z'

Result:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
z_nonneg : 0 ≤ z
le_2n : n₀ ⊔ 2 ≤ 2 ^ n * (n₀ ⊔ 2)
n₀_le_z : n₀ ≤ z
fz_eq_c₂fz : f z = c₂ * f z
z_to_half_z' : f (1 / 2 * z) = c₂ * f z
z_to_half_z : f (1 / 2 * z) = f z
⊢ f z = f (n₀ ⊔ 2)

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
heq : c₁ = c₂
⊢ ∃ c, ∀ᶠ (x : ℝ) in atTop, f x = c

Tactic:
simp only [heq, Set.Icc_self, Set.mem_singleton_iff, one_mul] at h

Result:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u = c₂ * f x
⊢ ∃ c, ∀ᶠ (x : ℝ) in atTop, f x = c

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
z_nonneg : 0 ≤ z
le_2n : n₀ ⊔ 2 ≤ 2 ^ n * (n₀ ⊔ 2)
n₀_le_z : n₀ ≤ z
fz_eq_c₂fz : f z = c₂ * f z
z_to_half_z' : f (1 / 2 * z) = c₂ * f z
z_to_half_z : f (1 / 2 * z) = f z
⊢ 1 / 2 * z < 2 ^ n * (n₀ ⊔ 2)

Tactic:
have h₁ : (2 : ℝ)^n = ((1 : ℝ)/(2 : ℝ)) * (2 : ℝ)^(n+1) := by
  rw [one_div, pow_add, pow_one]
  ring

Result:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
z_nonneg : 0 ≤ z
le_2n : n₀ ⊔ 2 ≤ 2 ^ n * (n₀ ⊔ 2)
n₀_le_z : n₀ ≤ z
fz_eq_c₂fz : f z = c₂ * f z
z_to_half_z' : f (1 / 2 * z) = c₂ * f z
z_to_half_z : f (1 / 2 * z) = f z
h₁ : 2 ^ n = 1 / 2 * 2 ^ (n + 1)
⊢ 1 / 2 * z < 2 ^ n * (n₀ ⊔ 2)

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
heq : c₁ = c₂
hmain : ∃ c, ∀ᶠ (x : ℝ) in atTop, f x = c
⊢ (∀ᶠ (x : ℝ) in atTop, 0 ≤ f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x ≤ 0

Tactic:
obtain ⟨c, hc⟩ := hmain

Result:
case intro
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
heq : c₁ = c₂
c : ℝ
hc : ∀ᶠ (x : ℝ) in atTop, f x = c
⊢ (∀ᶠ (x : ℝ) in atTop, 0 ≤ f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x ≤ 0

Proof state:
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hlt : c₁ < c₂
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x
hu : f (3 / 4 * x) ∈ Set.Icc (c₁ * f x) (c₂ * f x)
⊢ 0 ≤ f x

Tactic:
have hu := Set.nonempty_of_mem hu

Result:
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hlt : c₁ < c₂
x : ℝ
hx : ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hx_nonneg : 0 ≤ x
h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x
hu✝ : f (3 / 4 * x) ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hu : (Set.Icc (c₁ * f x) (c₂ * f x)).Nonempty
⊢ 0 ≤ f x

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
z_nonneg : 0 ≤ z
le_2n : n₀ ⊔ 2 ≤ 2 ^ n * (n₀ ⊔ 2)
⊢ n₀ ≤ z

Tactic:
calc n₀ ≤ max n₀ 2 := by simp
      _ ≤ (2 : ℝ)^n * max n₀ 2 := le_2n
      _ ≤ _ := by exact_mod_cast hz.1

Result:
no goals