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

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖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)
⊢ 3 * c₂ > 0

Tactic:
positivity

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

Goal state before:
pp: case hd
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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 ≤ p a b + 1
u : ℕ
hu : r i n ≤ u ∧ u < n
this : 0 < u
⊢ 0 < ↑u ^ (p a b + 1)

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 ≤ p a b + 1
⊢ ↑n ≠ 0

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 ≤ p a b + 1
⊢ ↑n ^ (p a b + 1) ≠ 0

Tactic:
positivity

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
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
⊢ c₁ ≠ 0

Tactic:
positivity

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
n : ℕ
hn : 0 < asympBound g a b n
hn₂ : 0 < 1 + ε ↑n
⊢ 0 ≤ (1 + ε ↑n) * asympBound g a b n

Tactic:
positivity

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

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖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)
⊢ 3 * c₁ > 0

Tactic:
positivity

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

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖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)
⊢ c₂ / 3 > 0

Tactic:
positivity

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

Goal state before:
pp: case hd
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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
u : ℕ
hu : r i n ≤ u ∧ u < n
this : 0 < u
⊢ 0 < ↑u ^ (p a b + 1)

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
u : ℕ
hu : r i n ≤ u ∧ u < n
⊢ 0 < ↑(r i n)

Tactic:
positivity

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
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
⊢ ↑n ≥ 0

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ 0 < c₁ * ↑n

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ 0 ≤ c₁

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ 0 ≤ ↑n

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ ↑n ≠ 0

Tactic:
positivity

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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ ↑n ^ (p a b + 1) ≠ 0

Tactic:
positivity

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
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
a✝ : i ∈ univ
j : ℕ
x✝ : j ∈ range (r i n)
⊢ ↑j ≥ 0

Tactic:
positivity

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
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
a✝ : i ∈ univ
j : ℕ
x✝ : j ∈ range (r i n)
⊢ 0 ≤ ↑j ^ (p a b + 1)

Tactic:
positivity

Goal state after:
Module: Mathlib.Computability.DFA

Goal state before:
pp: case refine_1
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ x.length
hx : M.evalFrom s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : M.evalFrom s (List.take (↑n) x) = M.evalFrom s (List.take (↑m) x)
hle : ↑n ≤ ↑m
⊢ x = List.take (↑n) (List.take (↑m) x) ++ List.drop (↑n) (List.take (↑m) x) ++ List.drop (↑m) x

Tactic:
List.take_append_drop,

Goal state after:
case refine_1
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ x.length
hx : M.evalFrom s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : M.evalFrom s (List.take (↑n) x) = M.evalFrom s (List.take (↑m) x)
hle : ↑n ≤ ↑m
⊢ x = List.take (↑m) x ++ List.drop (↑m) x
Module: Mathlib.Computability.DFA

Goal state before:
pp: case right
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ x.length
hx : M.evalFrom s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : M.evalFrom s (List.take (↑n) x) = M.evalFrom s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hq :
  M.evalFrom (M.evalFrom s (List.take (↑n) (List.take (↑m) x))) (List.drop (↑n) (List.take (↑m) x)) =
    M.evalFrom s (List.take (↑n) (List.take (↑m) x))
⊢ M.evalFrom s (List.take (↑n) (List.take (↑m) x) ++ List.drop (↑n) (List.take (↑m) x) ++ List.drop (↑m) x) = t

Tactic:
List.take_append_drop,

Goal state after:
case right
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ x.length
hx : M.evalFrom s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : M.evalFrom s (List.take (↑n) x) = M.evalFrom s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hq :
  M.evalFrom (M.evalFrom s (List.take (↑n) (List.take (↑m) x))) (List.drop (↑n) (List.take (↑m) x)) =
    M.evalFrom s (List.take (↑n) (List.take (↑m) x))
⊢ M.evalFrom s (List.take (↑m) x ++ List.drop (↑m) x) = t
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: case inl
f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ ∀ᶠ (x : ℝ) in atTop, f x ≠ 0

Tactic:
filter_upwards [H] with _ hx

Goal state after:
f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, 0 < f x
a✝ : ℝ
hx : 0 < f a✝
⊢ f a✝ ≠ 0
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: case inr
f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ ∀ᶠ (x : ℝ) in atTop, f x ≠ 0

Tactic:
filter_upwards [H] with _ hx

Goal state after:
f : ℝ → ℝ
hf : GrowsPolynomially f
H : ∀ᶠ (x : ℝ) in atTop, f x < 0
a✝ : ℝ
hx : f a✝ < 0
⊢ f a✝ ≠ 0
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x ↦ |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)
⊢ 0 < c₂⁻¹

Tactic:
positivity

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

Goal state before:
pp: f : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0
hf✝ : GrowsPolynomially fun x ↦ |f x|
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_pos : 0 < b
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)
⊢ 0 < c₁⁻¹

Tactic:
positivity

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

Goal state before:
pp: f : Bool → Bool → Bool
h : f false false = false
a : Bool
m : ℕ
b : Bool
n : ℕ
⊢ (if (if a = true then 2 * m + 1 else 2 * m) = 0 then
      if f false true = true then if b = true then 2 * n + 1 else 2 * n else 0
    else
      if (if b = true then 2 * n + 1 else 2 * n) = 0 then
        if f true false = true then if a = true then 2 * m + 1 else 2 * m else 0
      else
        if
            f (decide ((if a = true then 2 * m + 1 else 2 * m) % 2 = 1))
                (decide ((if b = true then 2 * n + 1 else 2 * n) % 2 = 1)) =
              true then
          bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) +
              bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) +
            1
        else
          bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) +
            bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2)) =
    if f a b = true then 2 * bitwise f m n + 1 else 2 * bitwise f m n

Tactic:
have h4 x : (x + x + 1) / 2 = x := by rw [← two_mul, add_comm]; simp [add_mul_div_left]

Goal state after:
f : Bool → Bool → Bool
h : f false false = false
a : Bool
m : ℕ
b : Bool
n : ℕ
h4 : ∀ (x : ℕ), (x + x + 1) / 2 = x
⊢ (if (if a = true then 2 * m + 1 else 2 * m) = 0 then
      if f false true = true then if b = true then 2 * n + 1 else 2 * n else 0
    else
      if (if b = true then 2 * n + 1 else 2 * n) = 0 then
        if f true false = true then if a = true then 2 * m + 1 else 2 * m else 0
      else
        if
            f (decide ((if a = true then 2 * m + 1 else 2 * m) % 2 = 1))
                (decide ((if b = true then 2 * n + 1 else 2 * n) % 2 = 1)) =
              true then
          bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) +
              bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) +
            1
        else
          bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2) +
            bitwise f ((if a = true then 2 * m + 1 else 2 * m) / 2) ((if b = true then 2 * n + 1 else 2 * n) / 2)) =
    if f a b = true then 2 * bitwise f m n + 1 else 2 * bitwise f m n
Module: Mathlib.Data.Nat.Size

Goal state before:
pp: b : Bool
m n : ℕ
IH : shiftLeft' b m n ≠ 0 → (shiftLeft' b m n).size = m.size + n
h : ¬bit b (shiftLeft' b m n) = 0
s0 : ¬shiftLeft' b m n = 0
⊢ (shiftLeft' b m n).size.succ = (m.size + n).succ

Tactic:
rw [IH s0]

Goal state after:
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
⊢ 0 < c₁ * f x

Tactic:
positivity

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

Goal state before:
pp: case inl.refine_1
a b c : ℕ
v : ℕ := a ^^^ b ^^^ c
h✝ : v ≠ 0
hv : v = a ^^^ b ^^^ c
hab : a ^^^ b = c ^^^ v
hbc : b ^^^ c = a ^^^ v
hca : c ^^^ a = b ^^^ v
i : ℕ
hi : v.testBit i = true
hi' : ∀ (j : ℕ), i < j → v.testBit j = false
h : a.testBit i = true
⊢ (a ^^^ v).testBit i = false

Tactic:
rw [testBit_xor, h, hi]

Goal state after:
case inl.refine_1
a b c : ℕ
v : ℕ := a ^^^ b ^^^ c
h✝ : v ≠ 0
hv : v = a ^^^ b ^^^ c
hab : a ^^^ b = c ^^^ v
hbc : b ^^^ c = a ^^^ v
hca : c ^^^ a = b ^^^ v
i : ℕ
hi : v.testBit i = true
hi' : ∀ (j : ℕ), i < j → v.testBit j = false
h : a.testBit i = true
⊢ (true ^^ true) = false
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
h_tendsto : Tendsto (fun x ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ x ≠ 0

Tactic:
positivity

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

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
h_tendsto : Tendsto (fun x ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ b ≠ 0

Tactic:
positivity

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

Goal state before:
pp: case inr.inl.refine_1
a b c : ℕ
v : ℕ := a ^^^ b ^^^ c
h✝ : v ≠ 0
hv : v = a ^^^ b ^^^ c
hab : a ^^^ b = c ^^^ v
hbc : b ^^^ c = a ^^^ v
hca : c ^^^ a = b ^^^ v
i : ℕ
hi : v.testBit i = true
hi' : ∀ (j : ℕ), i < j → v.testBit j = false
h : b.testBit i = true
⊢ (b ^^^ v).testBit i = false

Tactic:
rw [testBit_xor, h, hi]

Goal state after:
case inr.inl.refine_1
a b c : ℕ
v : ℕ := a ^^^ b ^^^ c
h✝ : v ≠ 0
hv : v = a ^^^ b ^^^ c
hab : a ^^^ b = c ^^^ v
hbc : b ^^^ c = a ^^^ v
hca : c ^^^ a = b ^^^ v
i : ℕ
hi : v.testBit i = true
hi' : ∀ (j : ℕ), i < j → v.testBit j = false
h : b.testBit i = true
⊢ (true ^^ true) = false
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
h_tendsto : Tendsto (fun x ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ 0 < b * x

Tactic:
positivity

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

Goal state before:
pp: 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)
⊢ 0 < c₁ * c₂⁻¹ * c₃

Tactic:
positivity

Goal state after: