l3lab/ntp-mathlib huggingface dataset:
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      

Proof state:
f✝ 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
⊢ 1 / 2 * f x ≤ f x + g x

Next tactic:
calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by ring
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                

Proof state:
f✝ 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
⊢ f x + g x ≥ f x - ‖g x‖

Next tactic:
rw [sub_eq_add_neg, norm_eq_abs]
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; 

Proof state:
f✝ 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
⊢ f x + g x ≥ f x + -|g x|

Next tactic:
gcongr
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; 

Proof state:
case bc
f✝ 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
⊢ -|g x| ≤ g x

Next tactic:
exact neg_abs_le_self (g x)
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by 

Proof state:
f✝ 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
⊢ f x - ‖g x‖ ≥ f x - 1 / 2 * f x

Next tactic:
gcongr
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by 

Proof state:
f✝ 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
⊢ f x - 1 / 2 * f x = 1 / 2 * f x

Next tactic:
ring
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by ring
    

Proof state:
case h
f✝ 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 ∈ Set.Icc (b * x) x, f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))

Next tactic:
intro u ⟨hu_lb, hu_ub⟩
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by ring
    intro u ⟨hu_lb, hu_ub⟩
    

Proof state:
case h
f✝ 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
⊢ f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))

Next tactic:
have hfu_nonneg : 0 ≤ f u := hf₂ _ hu_lb
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by ring
    intro u ⟨hu_lb, hu_ub⟩
    have hfu_nonneg : 0 ≤ f u := hf₂ _ hu_lb
    

Proof state:
case h
f✝ 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
⊢ f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))

Next tactic:
have hfg₃ : ‖g u‖ ≤ 1/2 * f u := by
      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]
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by ring
    intro u ⟨hu_lb, hu_ub⟩
    have hfu_nonneg : 0 ≤ f u := hf₂ _ hu_lb
    have hfg₃ : ‖g u‖ ≤ 1/2 * f u := by
      

Proof state:
f✝ 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

Next 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]
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
    (hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x := by
  intro b hb
  have hb_ub := hb.2
  rw [isLittleO_iff] at hfg
  rcases hf.eventually_atTop_nonneg_or_nonpos with hf'|hf'
  case inl =>  -- f is eventually nonneg
    have hf := hf b hb
    obtain ⟨c₁, hc₁_mem : 0 < c₁, c₂, hc₂_mem : 0 < c₂, hf⟩ := hf
    specialize hfg (c := 1/2) (by norm_num)
    refine ⟨c₁ / 3, by positivity, 3*c₂, by positivity, ?_⟩
    filter_upwards [hf,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg,
                    (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf',
                    eventually_ge_atTop 0] with x hf₁ hfg' hf₂ hx_nonneg
    have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
    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)
    have hx_ub : f x + g x ≤ 3/2 * f x := by
      calc _ ≤ f x + ‖g x‖ := by gcongr; exact le_norm_self (g x)
           _ ≤ f x + 1/2 * f x := by gcongr
           _ = 3/2 * f x := by ring
    have hx_lb : 1/2 * f x ≤ f x + g x := by
      calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le_self (g x)
           _ ≥ f x - 1/2 * f x := by gcongr
           _ = 1/2 * f x := by ring
    intro u ⟨hu_lb, hu_ub⟩
    have hfu_nonneg : 0 ≤ f u := hf₂ _ hu_lb
    have hfg₃ : ‖g u‖ ≤ 1/2 * f u := by
      calc ‖g u‖ ≤ 1/2 * ‖f u‖ := hfg' _ hu_lb
           _ = 1/2 * f u := by 

Proof state:
f✝ 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
⊢ 1 / 2 * ‖f u‖ = 1 / 2 * f u

Next tactic:
congr