-- Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
    (hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p

Proof state:
case intro.intro.intro.intro
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Tactic:
have hc₂p : 0 < c₂ ^ p := Real.rpow_pos_of_pos hc₂_mem _

Result:
case intro.intro.intro.intro
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p ≤ c₁ ^ p * f x ^ p

Tactic:
calc
(f u)^p ≤ (c₁ * f x)^p := by
exact rpow_le_rpow_of_exponent_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
_ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]

Result:
no goals

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ f u ^ p ≤ (c₂ * f x) ^ p

Tactic:
gcongr

Result:
case h₁
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg✝ : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
h✝ : 0 ≤ p
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_nonneg : 0 ≤ f x
hf_nonneg₂ : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
fu_nonneg : 0 ≤ f u
⊢ f u ≤ c₂ * f x

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
a✝ : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ c₂ ^ p * f x ^ p ≤ f u ^ p

Tactic:
calc
c₂^p * (f x)^p = (c₂ * f x)^p := by rw [mul_rpow (le_of_lt hc₂_mem) (le_of_lt hf_pos)]
_ ≤ _ := rpow_le_rpow_of_exponent_nonpos (hf_pos₂ u hu.1) (hf₁ u hu).2 (le_of_lt hp)

Result:
no goals

Proof state:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hneg : ∀ᶠ (x : ℝ) in atTop, f x < 0
this : atTop = ⊥
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Tactic:
exact False.elim <| (atTop_neBot).ne this

Result:
no goals

Proof state:
case intro.intro.intro.intro
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x => f x ^ p) u ∈ Set.Icc (c₁ * (fun x => f x ^ p) x) (c₂ * (fun x => f x ^ p) x)

Tactic:
cases le_or_lt 0 p with
| inl => refine ⟨c₁^p, hc₁p, ?_⟩
  refine ⟨c₂^p, hc₂p, ?_⟩
  filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
      (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
      with x _ hf₁ hf_nonneg hf_nonneg₂
  intro u hu
  have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁^p * (f x)^p = (c₁ * f x)^p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
      _ ≤ _ := by gcongr; exact (hf₁ u hu).1
  case ub => calc
    (f u)^p ≤ (c₂ * f x)^p := by gcongr; exact (hf₁ u hu).2
      _ = _ := by rw [← mul_rpow (le_of_lt hc₂_mem) hf_nonneg]
| inr hp => match hf.eventually_atTop_zero_or_pos_or_neg with
  | .inl hzero => refine ⟨1, by norm_num, 1, by norm_num, ?_⟩
    filter_upwards [hzero, hfnew] with x hx hx'
    intro u hu
    simp only [hx, ne_eq, zero_rpow (ne_of_lt hp), mul_zero, le_refl, not_true, lt_self_iff_false,
      Set.Icc_self, Set.mem_singleton_iff]
    simp only [hx, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx'
    rw [hx' u hu, zero_rpow (ne_of_lt hp)]
  | .inr (.inl hpos) => refine ⟨c₂^p, hc₂p, ?_⟩
    refine ⟨c₁^p, hc₁p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hpos,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hpos]
        with x _ hf₁ hf_pos hf_pos₂
    intro u hu
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₂^p * (f x)^p = (c₂ * f x)^p := by rw [mul_rpow (le_of_lt hc₂_mem) (le_of_lt hf_pos)]
        _ ≤ _ := rpow_le_rpow_of_exponent_nonpos (hf_pos₂ u hu.1) (hf₁ u hu).2 (le_of_lt hp)
    case ub => calc
      (f u)^p ≤ (c₁ * f x)^p := by
            exact rpow_le_rpow_of_exponent_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
        _ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
  | .inr (.inr hneg) => have : ∀ᶠ (_ : ℝ) in atTop, False := by
      filter_upwards [hf_nonneg, hneg] with x hx hx'; linarith
    rw [Filter.eventually_false_iff_eq_bot] at this
    exact False.elim <| (atTop_neBot).ne this

Result:
no goals

Proof state:
case h
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hneg : ∀ᶠ (x : ℝ) in atTop, f x < 0
x : ℝ
hx : 0 ≤ f x
hx' : f x < 0
⊢ False

Tactic:
linarith

Result:
no goals