l3lab/ntp-mathlib huggingface dataset:
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by 

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₂⁻¹ * c₂ * c₁ * (c₃ * ‖g x‖) = c₁ * (c₃ * ‖g x‖)

Next tactic:
simp [c₂_cancel]
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by 

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₁ * (c₃ * ‖g x‖) ≤ c₁ * ‖g u‖

Next tactic:
gcongr
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by gcongr; 

Proof state:
case h
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₃ * ‖g x‖ ≤ ‖g u‖

Next tactic:
exact (hg_bound u hu).1
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by gcongr; exact (hg_bound u hu).1
      _ ≤ f u := by
          

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₁ * ‖g u‖ ≤ f u

Next tactic:
rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by gcongr; exact (hg_bound u hu).1
      _ ≤ f u := by
          rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
          

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₁ * ‖g u‖ ≤ ‖f u‖

Next tactic:
exact h_lb u hu.1
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by gcongr; exact (hg_bound u hu).1
      _ ≤ f u := by
          rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
          exact h_lb u hu.1
  case ub => calc
    f u ≤ c₂ * ‖g u‖ := by 

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ f u ≤ c₂ * ‖g u‖

Next tactic:
rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by gcongr; exact (hg_bound u hu).1
      _ ≤ f u := by
          rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
          exact h_lb u hu.1
  case ub => calc
    f u ≤ c₂ * ‖g u‖ := by rw [← Real.norm_of_nonneg (hf_pos u hu.1)]; 

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ ‖f u‖ ≤ c₂ * ‖g u‖

Next tactic:
exact h_ub u hu.1
lemma GrowsPolynomially.of_isTheta {f g : ℝ → ℝ} (hg : GrowsPolynomially g) (hf : f =Θ[atTop] g)
    (hf' : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially f := by
  intro b hb
  have hb_pos := hb.1
  have hf_lb := isBigO_iff''.mp hf.isBigO_symm
  have hf_ub := isBigO_iff'.mp hf.isBigO
  obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
  obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
  have hg := hg.norm b hb
  obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
  obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
  have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
  have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
  refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
  refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
  have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel (by positivity)
  have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel (by positivity)
  filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
  intro u hu
  have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x    := by gcongr; exact le_of_lt hb.2
             _ = x        := by rw [one_mul]
  have hg_bound := hg_bound x hbx
  refine ⟨?lb, ?ub⟩
  case lb => calc
    c₁ * c₂⁻¹ * c₃ * f x ≤ c₁ * c₂⁻¹ * c₃ * (c₂ * ‖g x‖) := by
          rw [← Real.norm_of_nonneg (hf_pos x hbx)]; gcongr; exact h_ub x hbx
      _ = (c₂⁻¹ * c₂) * c₁ * (c₃ * ‖g x‖) := by ring
      _ = c₁ * (c₃ * ‖g x‖) := by simp [c₂_cancel]
      _ ≤ c₁ * ‖g u‖ := by gcongr; exact (hg_bound u hu).1
      _ ≤ f u := by
          rw [← Real.norm_of_nonneg (hf_pos u hu.1)]
          exact h_lb u hu.1
  case ub => calc
    f u ≤ c₂ * ‖g u‖ := by rw [← Real.norm_of_nonneg (hf_pos u hu.1)]; exact h_ub u hu.1
      _ ≤ c₂ * (c₄ * ‖g x‖) := by 

Proof state:
f✝ f g : ℝ → ℝ
hg✝ : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x => ‖g x‖) u ∈ Set.Icc (c₃ * (fun x => ‖g x‖) x) (c₄ * (fun x => ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound✝ : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hbx : b * x ≤ x
hg_bound : ∀ u ∈ Set.Icc (b * x) x, ‖g u‖ ∈ Set.Icc (c₃ * ‖g x‖) (c₄ * ‖g x‖)
⊢ c₂ * ‖g u‖ ≤ c₂ * (c₄ * ‖g x‖)

Next tactic:
gcongr