mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: case ub₁
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
ih : f (2 ^ (-↑k - 1) * x₀) = 0
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Tactic:
case ub₁ =>
        have := hz.2
        simp only [neg_add, ← sub_eq_add_neg] at this
        exact this

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c : ℝ := b (max_bi b) + (1 - b (max_bi b)) / 2
h_max_bi_pos : 0 < b (max_bi b)
h_max_bi_lt_one : 0 < 1 - b (max_bi b)
hc_pos : 0 < c
h₁ : 0 < (1 - b (max_bi b)) / 2
⊢ 1 * (1 / 2) + 1 / 2 = 1

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ 1 < 2

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ 0 < 2

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ 2 ≠ 1

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ ↑⌈-logb 2 (x / x₀)⌉₊ ≤ 1 + ↑⌊-logb 2 (x / x₀)⌋₊

Tactic:
rw [add_comm]

Goal state after:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ ↑⌈-logb 2 (x / x₀)⌉₊ ≤ ↑⌊-logb 2 (x / x₀)⌋₊ + 1
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf : GrowsPolynomially f
⊢ 1 / 2 ∈ Set.Ioo 0 1

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: ⊢ 2 ≠ 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: case hbc
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
⊢ 1 / 2 ≤ 3 / 4

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: case hbc
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 ≤ 3 / 4

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
n : ℕ
hn : 1 < n
hn' : ∀ (i : α), 0 < log (b i * ↑n)
⊢ 1 / log (b i * ↑n) - 1 / log ↑n = (log ↑n - log (b i * ↑n)) / (log (b i * ↑n) * log ↑n)

Tactic:
have h_log_pos : 0 < log n := Real.log_pos <| by simp_all

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
n : ℕ
hn : 1 < n
hn' : ∀ (i : α), 0 < log (b i * ↑n)
h_log_pos : 0 < log ↑n
⊢ 1 / log (b i * ↑n) - 1 / log ↑n = (log ↑n - log (b i * ↑n)) / (log (b i * ↑n) * log ↑n)
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ (b : ℝ), n₀ ≤ b → ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
⊢ 1 < 2

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: c x✝¹ : ℝ
x✝ : x✝¹ ∈ Set.Ioo 0 1
⊢ 1 > 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ 1 > 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
⊢ 1 / 2 < 1

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ c * ‖f x‖
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)
⊢ 0 < 1 / 2

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: case e_a.e_f.e_a.e_a.e_a
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound1 : ∀ (i : α), ↑(r i n) ^ p a b * (1 - ε ↑(r i n)) ≤ b i ^ p a b * ↑n ^ p a b * (1 - ε ↑n)
h_smoothing_pos : 0 < 1 - ε ↑n
h_sumTransform : ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁⁻¹
h_ind : ∀ m ∈ Ico n₀ n, T m ≤ C * ((1 - ε ↑m) * asympBound g a b m)
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
i : α
⊢ ∑ u ∈ range (r i n), g ↑u / ↑u ^ (p a b + 1) + ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1) =
    ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)

Tactic:
rw [add_comm]

Goal state after:
case e_a.e_f.e_a.e_a.e_a
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound1 : ∀ (i : α), ↑(r i n) ^ p a b * (1 - ε ↑(r i n)) ≤ b i ^ p a b * ↑n ^ p a b * (1 - ε ↑n)
h_smoothing_pos : 0 < 1 - ε ↑n
h_sumTransform : ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁⁻¹
h_ind : ∀ m ∈ Ico n₀ n, T m ≤ C * ((1 - ε ↑m) * asympBound g a b m)
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
i : α
⊢ ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1) + ∑ u ∈ range (r i n), g ↑u / ↑u ^ (p a b + 1) =
    ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound1 : ∀ (i : α), ↑(r i n) ^ p a b * (1 - ε ↑(r i n)) ≤ b i ^ p a b * ↑n ^ p a b * (1 - ε ↑n)
h_smoothing_pos : 0 < 1 - ε ↑n
h_sumTransform : ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁⁻¹
h_ind : ∀ m ∈ Ico n₀ n, T m ≤ C * ((1 - ε ↑m) * asympBound g a b m)
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
⊢ ∑ i,
        C * a i *
          (b i ^ p a b * (1 - ε ↑n) *
            (↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)) -
              ↑n ^ p a b * ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1))) +
      g ↑n =
    ∑ i, C * a i * (b i ^ p a b * (1 - ε ↑n) * (asympBound g a b n - sumTransform (p a b) g (r i n) n)) + g ↑n

Tactic:
asympBound_def',

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound1 : ∀ (i : α), ↑(r i n) ^ p a b * (1 - ε ↑(r i n)) ≤ b i ^ p a b * ↑n ^ p a b * (1 - ε ↑n)
h_smoothing_pos : 0 < 1 - ε ↑n
h_sumTransform : ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁⁻¹
h_ind : ∀ m ∈ Ico n₀ n, T m ≤ C * ((1 - ε ↑m) * asympBound g a b m)
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
⊢ ∑ i,
        C * a i *
          (b i ^ p a b * (1 - ε ↑n) *
            (↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)) -
              ↑n ^ p a b * ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1))) +
      g ↑n =
    ∑ x,
        C * a x *
          (b x ^ p a b * (1 - ε ↑n) *
            (↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)) - sumTransform (p a b) g (r x n) n)) +
      g ↑n
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound1 : ∀ (i : α), ↑(r i n) ^ p a b * (1 - ε ↑(r i n)) ≤ b i ^ p a b * ↑n ^ p a b * (1 - ε ↑n)
h_smoothing_pos : 0 < 1 - ε ↑n
h_sumTransform : ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁⁻¹
h_ind : ∀ m ∈ Ico n₀ n, T m ≤ C * ((1 - ε ↑m) * asympBound g a b m)
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
⊢ 1 ≤ 2 * 1 * (1 / 2)

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
⊢ 1 / 2 < 1

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ c * ‖f x‖
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
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)
⊢ 0 < 1 / 2

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: case e_a.e_f.e_a.e_a.e_a
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound2 : ∀ (i : α), b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i n))
h_smoothing_pos : 0 < 1 + ε ↑n
h_sumTransform : ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
h_exp : ⌈rexp 1⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁
h_ind : ∀ m ∈ Ico n₀ n, (1 + ε ↑m) * asympBound g a b m ≤ C * T m
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
i : α
⊢ ∑ u ∈ range (r i n), g ↑u / ↑u ^ (p a b + 1) + ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1) =
    ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)

Tactic:
rw [add_comm]

Goal state after:
case e_a.e_f.e_a.e_a.e_a
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound2 : ∀ (i : α), b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i n))
h_smoothing_pos : 0 < 1 + ε ↑n
h_sumTransform : ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
h_exp : ⌈rexp 1⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁
h_ind : ∀ m ∈ Ico n₀ n, (1 + ε ↑m) * asympBound g a b m ≤ C * T m
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
i : α
⊢ ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1) + ∑ u ∈ range (r i n), g ↑u / ↑u ^ (p a b + 1) =
    ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)
Module: Mathlib.Computability.Language

Goal state before:
pp: case a.h.inl
α : Type u_1
l m n : Language α
hm : [] ∉ m
h : l = m * l + n
a : List α
ha : a ∈ m
b : List α
hb : b ∈ l
ih : ∀ m_1 < a.length + b.length, ∀ ⦃x : List α⦄, x ∈ l → x.length = m_1 → x ∈ m∗ * n
hx : a ++ b ∈ m * l
⊢ a ++ b ∈ m∗ * n

Tactic:
have hal : 0 < a.length := length_pos_iff.mpr <| ne_of_mem_of_not_mem ha hm

Goal state after:
case a.h.inl
α : Type u_1
l m n : Language α
hm : [] ∉ m
h : l = m * l + n
a : List α
ha : a ∈ m
b : List α
hb : b ∈ l
ih : ∀ m_1 < a.length + b.length, ∀ ⦃x : List α⦄, x ∈ l → x.length = m_1 → x ∈ m∗ * n
hx : a ++ b ∈ m * l
hal : 0 < a.length
⊢ a ++ b ∈ m∗ * n
Module: Mathlib.Computability.AkraBazzi.AkraBazzi

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound2 : ∀ (i : α), b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i n))
h_smoothing_pos : 0 < 1 + ε ↑n
h_sumTransform : ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
h_exp : ⌈rexp 1⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁
h_ind : ∀ m ∈ Ico n₀ n, (1 + ε ↑m) * asympBound g a b m ≤ C * T m
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
⊢ ∑ i,
        a i *
          (b i ^ p a b * (1 + ε ↑n) *
            (↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)) -
              ↑n ^ p a b * ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1))) +
      C * g ↑n =
    ∑ i, a i * (b i ^ p a b * (1 + ε ↑n) * (asympBound g a b n - sumTransform (p a b) g (r i n) n)) + C * g ↑n

Tactic:
asympBound_def',

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
n : ℕ
hn : n₀ ≤ n
bound2 : ∀ (i : α), b i ^ p a b * ↑n ^ p a b * (1 + ε ↑n) ≤ ↑(r i n) ^ p a b * (1 + ε ↑(r i n))
h_smoothing_pos : 0 < 1 + ε ↑n
h_sumTransform : ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
h_smoothing_gt_half : 1 / 2 < 1 - ε ↑n
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
n₀_div_le_n : ⌈↑n₀ / b'⌉₊ ≤ n
h_exp : ⌈rexp 1⌉₊ ≤ n
n₀_le_r : ∀ (i : α), n₀ ≤ r i n
r_le_n : ∀ (i : α), r i n < n
C : ℝ
hC : C ≥ 2 * c₁
h_ind : ∀ m ∈ Ico n₀ n, (1 + ε ↑m) * asympBound g a b m ≤ C * T m
C_pos : 0 ≤ C
g_pos : 0 ≤ g ↑n
⊢ ∑ i,
        a i *
          (b i ^ p a b * (1 + ε ↑n) *
            (↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)) -
              ↑n ^ p a b * ∑ u ∈ Ico (r i n) n, g ↑u / ↑u ^ (p a b + 1))) +
      C * g ↑n =
    ∑ x,
        a x *
          (b x ^ p a b * (1 + ε ↑n) *
            (↑n ^ p a b * (1 + ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)) - sumTransform (p a b) g (r x n) n)) +
      C * g ↑n
Module: Mathlib.Computability.ContextFreeGrammar

Goal state before:
pp: T : Type u_1
L : Language T
h : L.IsContextFree
⊢ L.reverse.IsContextFree

Tactic:
rcases h with ⟨g, rfl⟩

Goal state after:
T : Type u_1
g : ContextFreeGrammar T
⊢ g.language.reverse.IsContextFree
Module: Mathlib.Computability.DFA

Goal state before:
pp: α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
hx : x ∈ M.accepts
hlen : Fintype.card σ ≤ x.length
⊢ ∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts

Tactic:
by
  obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl
  use a, b, c, hx, hlen, hnil
  intro y hy
  rw [Language.mem_mul] at hy
  rcases hy with ⟨ab, hab, c', hc', rfl⟩
  rw [Language.mem_mul] at hab
  rcases hab with ⟨a', ha', b', hb', rfl⟩
  rw [Set.mem_singleton_iff] at ha' hc'
  subst ha' hc'
  have h := M.evalFrom_of_pow hb hb'
  rwa [mem_accepts, eval, evalFrom_of_append, evalFrom_of_append, h, hc]

Goal state after:
Module: Mathlib.Data.Nat.Size

Goal state before:
pp: case bit
b : Bool
m : ℕ
e : bit b m ≠ 0
IH : ∀ {n : ℕ}, m < 2 ^ n → m.size ≤ n
n : ℕ
h : bit b m < 2 ^ n
⊢ (bit b m).size ≤ n

Tactic:
rw [size_bit e]

Goal state after:
case bit
b : Bool
m : ℕ
e : bit b m ≠ 0
IH : ∀ {n : ℕ}, m < 2 ^ n → m.size ≤ n
n : ℕ
h : bit b m < 2 ^ n
⊢ m.size.succ ≤ n
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: 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 _

Goal state after:
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)
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: 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
hzero : ∀ᶠ (x : ℝ) in atTop, f x = 0
⊢ 1 > 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Data.Nat.Bitwise

Goal state before:
pp: case pos
b : Bool
hn : 0 ≠ 0 → ∃ i, testBit 0 i = true ∧ ∀ (j : ℕ), i < j → testBit 0 j = false
h : bit b 0 ≠ 0
⊢ ∃ i, (bit b 0).testBit i = true ∧ ∀ (j : ℕ), i < j → (bit b 0).testBit j = false

Tactic:
rw [show b = true by
        revert h
        cases b <;> simp]

Goal state after:
case pos
b : Bool
hn : 0 ≠ 0 → ∃ i, testBit 0 i = true ∧ ∀ (j : ℕ), i < j → testBit 0 j = false
h : bit b 0 ≠ 0
⊢ ∃ i, (bit true 0).testBit i = true ∧ ∀ (j : ℕ), i < j → (bit true 0).testBit j = false
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: 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 * f x) ^ p

Tactic:
by
              exact rpow_le_rpow_of_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
⊢ 1 / 2 > 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
⊢ 1 > 0

Tactic:
norm_num

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
⊢ 0 < 1 / 2

Tactic:
norm_num

Goal state after: