awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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‖
h :
  ∀ (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), ∀ u ∈ Set.Icc (b * x) x, f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ f x + g x) u ∈
        Set.Icc (c₁ / 3 * (fun (x : ℝ) ↦ f x + g x) x) (3 * c₂ * (fun (x : ℝ) ↦ f x + g x) x)

Tactic:
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

Premises used: Filter.Tendsto.const_mul_atTop, Filter.tendsto_id, And.left, Filter.eventually_ge_atTop, OfNat.ofNat, Filter.Tendsto.eventually_forall_ge_atTop
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
nth_rewrite : b * x ≤ 1 * x
⊢ b * x ≤ x

Tactic:
nth_rewrite 2 [← one_mul x]

Premises used: one_mul
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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 ha : 0 ≤ x
⊢ b * x ≤ 1 * x

Tactic:
gcongr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ b * x ≤ x

Tactic:
have hbx : b * x ≤ x := by nth_rewrite 2 [← one_mul x]; gcongr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ ‖g x‖ ≤ 1 / 2 * ‖f x‖

Tactic:
hfg' x hbx
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
e_a : ‖f x‖ = f x
⊢ 1 / 2 * ‖f x‖ = 1 / 2 * f x

Tactic:
congr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ ‖f x‖ = f x

Tactic:
norm_of_nonneg (hf₂ _ hbx)

Premises used: Real.norm_of_nonneg
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ ‖f x‖ = f x

Tactic:
exact norm_of_nonneg (hf₂ _ hbx)

Premises used: Real.norm_of_nonneg
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
calc : 1 / 2 * ‖f x‖ = 1 / 2 * f x
⊢ ‖g x‖ ≤ 1 / 2 * f x

Tactic:
calc ‖g x‖ ≤ 1 / 2 * ‖f x‖ := hfg' x hbx
           _ = 1 / 2 * f x := by congr; exact norm_of_nonneg (hf₂ _ hbx)
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ ‖g x‖ ≤ 1 / 2 * f x

Tactic:
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)
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
bc : g x ≤ ‖g x‖
⊢ f x + g x ≤ f x + ‖g x‖

Tactic:
gcongr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ g x ≤ ‖g x‖

Tactic:
le_norm_self (g x)

Premises used: Real.le_norm_self
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ g x ≤ ‖g x‖

Tactic:
exact le_norm_self (g x)

Premises used: Real.le_norm_self
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ f x + ‖g x‖ ≤ f x + 1 / 2 * f x

Tactic:
gcongr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ f x + 1 / 2 * f x = 3 / 2 * f x

Tactic:
ring
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
calc : f x + g x ≤ f x + ‖g x‖
calc₁ : f x + ‖g x‖ ≤ f x + 1 / 2 * f x
calc₂ : f x + 1 / 2 * f x = 3 / 2 * f x
⊢ f x + g x ≤ 3 / 2 * f x

Tactic:
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
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ f x + g x ≤ 3 / 2 * f x

Tactic:
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
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
bc : -|g x| ≤ g x
⊢ f x + g x ≥ f x + -|g x|

Tactic:
gcongr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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

Tactic:
neg_abs_le (g x)

Premises used: neg_abs_le
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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

Tactic:
exact neg_abs_le (g x)

Premises used: neg_abs_le
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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

Tactic:
gcongr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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

Tactic:
ring
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
calc : f x + g x ≥ f x - ‖g x‖
calc₁ : f x - ‖g x‖ ≥ f x - 1 / 2 * f x
calc₂ : f x - 1 / 2 * f x = 1 / 2 * f x
⊢ 1 / 2 * f x ≤ f x + g x

Tactic:
calc f x + g x ≥ f x - ‖g x‖ := by
                rw [sub_eq_add_neg, norm_eq_abs]; gcongr; exact neg_abs_le (g x)
           _ ≥ f x - 1 / 2 * f x := by gcongr
           _ = 1 / 2 * f x := by ring
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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

Tactic:
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 (g x)
           _ ≥ f x - 1 / 2 * f x := by gcongr
           _ = 1 / 2 * f x := by ring
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
h : ∀ (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))
⊢ u ∈ Set.Icc (b * x) x → f u + g u ∈ Set.Icc (c₁ / 3 * (f x + g x)) (3 * c₂ * (f x + g x))

Tactic:
⟨hu_lb, hu_ub⟩
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
⊢ 0 ≤ f u

Tactic:
have hfu_nonneg : 0 ≤ f u := hf₂ _ hu_lb
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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‖

Tactic:
hfg' _ hu_lb
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
e_a : ‖f u‖ = f u
⊢ 1 / 2 * ‖f u‖ = 1 / 2 * f u

Tactic:
congr
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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‖ = f u

Tactic:
simp only [norm_eq_abs, abs_eq_self, hfu_nonneg]

Premises used: abs_eq_self, Real.norm_eq_abs
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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
calc : 1 / 2 * ‖f u‖ = 1 / 2 * f u
⊢ ‖g u‖ ≤ 1 / 2 * f u

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]
-- Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially
Goal:
f g : ℝ → ℝ
hf_1 : GrowsPolynomially f
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
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

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]