url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | exact _root_.trans (exp_small (by linarith)) (by linarith) | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ Complex.abs ((s.sum g).exp - 1) ≤ 4 * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ Complex.abs ((s.sum g).exp - 1) ≤ 4 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | intro n m | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
⊢ ∀ n ∈ s, Complex.abs (f n - 1) ≤ c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
⊢ ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | refine _root_.trans ?_ le | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ s.sum fun n => Complex.abs (f n - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | exact Finset.single_le_sum (f := fun n ↦ abs (f n - 1)) (fun _ _ ↦ Complex.abs.nonneg _) m | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ s.sum fun n => Complex.abs (f n - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ s.sum fun n => Complex.abs (f n - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | intro n m | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
⊢ ∀ n ∈ s, f n ≠ 0 | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
n : ℕ
m : n ∈ s
⊢ f n ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
⊢ ∀ n ∈ s, f n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | specialize b n m | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
n : ℕ
m : n ∈ s
⊢ f n ≠ 0 | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : Complex.abs (f n - 1) ≤ c
⊢ f n ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
n : ℕ
m : n ∈ s
⊢ f n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | contrapose b | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : Complex.abs (f n - 1) ≤ c
⊢ f n ≠ 0 | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : ¬f n ≠ 0
⊢ ¬Complex.abs (f n - 1) ≤ c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : Complex.abs (f n - 1) ≤ c
⊢ f n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | simp only [not_not] at b | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : ¬f n ≠ 0
⊢ ¬Complex.abs (f n - 1) ≤ c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ ¬Complex.abs (f n - 1) ≤ c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : ¬f n ≠ 0
⊢ ¬Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | simp only [b, not_le] | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ ¬Complex.abs (f n - 1) ≤ c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < Complex.abs (0 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ ¬Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | norm_num | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < Complex.abs (0 - 1) | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < Complex.abs (0 - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | linarith | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | refine _root_.trans (Complex.abs.sum_le _ _) ?_ | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ Complex.abs (s.sum g) ≤ 2 * c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (s.sum fun i => Complex.abs (f i).log) ≤ 2 * c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ Complex.abs (s.sum g) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | refine _root_.trans (Finset.sum_le_sum (fun n m ↦ log_small (_root_.trans (b n m) c1))) ?_ | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (s.sum fun i => Complex.abs (f i).log) ≤ 2 * c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (s.sum fun i => 2 * Complex.abs (f i - 1)) ≤ 2 * c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (s.sum fun i => Complex.abs (f i).log) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | rw [← Finset.mul_sum] | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (s.sum fun i => 2 * Complex.abs (f i - 1)) ≤ 2 * c | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (2 * s.sum fun i => Complex.abs (f i - 1)) ≤ 2 * c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (s.sum fun i => 2 * Complex.abs (f i - 1)) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | bound | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (2 * s.sum fun i => Complex.abs (f i - 1)) ≤ 2 * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
⊢ (2 * s.sum fun i => Complex.abs (f i - 1)) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | rw [Complex.exp_sum] | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ s.prod f = (s.sum g).exp | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ s.prod f = s.prod fun x => (f x).log.exp | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ s.prod f = (s.sum g).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | apply Finset.prod_congr rfl | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ s.prod f = s.prod fun x => (f x).log.exp | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ ∀ x ∈ s, f x = (f x).log.exp | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ s.prod f = s.prod fun x => (f x).log.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | intro n m | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ ∀ x ∈ s, f x = (f x).log.exp | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
n : ℕ
m : n ∈ s
⊢ f n = (f n).log.exp | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
⊢ ∀ x ∈ s, f x = (f x).log.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | rw [Complex.exp_log (f0 n m)] | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
n : ℕ
m : n ∈ s
⊢ f n = (f n).log.exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
n : ℕ
m : n ∈ s
⊢ f n = (f n).log.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | linarith | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ Complex.abs (s.sum g) ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ Complex.abs (s.sum g) ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | linarith | f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ 2 * Complex.abs (s.sum g) ≤ 4 * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
s : Finset ℕ
c : ℝ
le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c
c1 : c ≤ 1 / 2
g : ℕ → ℂ := fun n => (f n).log
b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ 2 * Complex.abs (s.sum g) ≤ 4 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | apply ConvexOn.sup | ⊢ ConvexOn ℝ univ fun p => max p.1 p.2 | case hf
⊢ ConvexOn ℝ univ fun p => p.1
case hg
⊢ ConvexOn ℝ univ fun p => p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ConvexOn ℝ univ fun p => max p.1 p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | use convex_univ | case hf
⊢ ConvexOn ℝ univ fun p => p.1 | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).1 ≤ a • x.1 + b • y.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
⊢ ConvexOn ℝ univ fun p => p.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | intros | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).1 ≤ a • x.1 + b • y.1 | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).1 ≤ a✝³ • x✝.1 + b✝ • y✝.1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).1 ≤ a • x.1 + b • y.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | simp | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).1 ≤ a✝³ • x✝.1 + b✝ • y✝.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).1 ≤ a✝³ • x✝.1 + b✝ • y✝.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | use convex_univ | case hg
⊢ ConvexOn ℝ univ fun p => p.2 | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).2 ≤ a • x.2 + b • y.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
⊢ ConvexOn ℝ univ fun p => p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | intros | case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).2 ≤ a • x.2 + b • y.2 | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).2 ≤ a✝³ • x✝.2 + b✝ • y✝.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
⊢ ∀ ⦃x : ℝ × ℝ⦄,
x ∈ univ → ∀ ⦃y : ℝ × ℝ⦄, y ∈ univ → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y).2 ≤ a • x.2 + b • y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | convexOn_max | [20, 1] | [22, 34] | simp | case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).2 ≤ a✝³ • x✝.2 + b✝ • y✝.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x✝ : ℝ × ℝ
a✝⁵ : x✝ ∈ univ
y✝ : ℝ × ℝ
a✝⁴ : y✝ ∈ univ
a✝³ b✝ : ℝ
a✝² : 0 ≤ a✝³
a✝¹ : 0 ≤ b✝
a✝ : a✝³ + b✝ = 1
⊢ (a✝³ • x✝ + b✝ • y✝).2 ≤ a✝³ • x✝.2 + b✝ • y✝.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | induction' n with n h | A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s | case zero
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) 0) s
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | simp [fc 0] | case zero
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) 0) s
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) 0) s
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | simp | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n ⊔ f (n + 1) x) s | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) (n + 1)) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Max.lean | ContinuousOn.partialSups | [25, 1] | [28, 76] | exact ContinuousOn.max h (fc _) | case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n ⊔ f (n + 1) x) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
A : Type
inst✝ : TopologicalSpace A
f : ℕ → A → ℝ
s : Set A
fc : ∀ (n : ℕ), ContinuousOn (f n) s
n : ℕ
h : ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n) s
⊢ ContinuousOn (fun x => (_root_.partialSups fun k => f k x) n ⊔ f (n + 1) x) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_mono | [21, 1] | [23, 73] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 1 < x
xy : x ≤ y
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 1 < x
xy : x ≤ y
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | have lx := neg_pos.mpr (Real.log_neg x0 x1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
⊢ (-y.log).log < (-x.log).log ↔ x < y | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
⊢ (-y.log).log < (-x.log).log ↔ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
⊢ (-y.log).log < (-x.log).log ↔ x < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | have ly := neg_pos.mpr (Real.log_neg y0 y1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
⊢ (-y.log).log < (-x.log).log ↔ x < y | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
⊢ (-y.log).log < (-x.log).log ↔ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
⊢ (-y.log).log < (-x.log).log ↔ x < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | rw [Real.log_lt_log_iff ly lx, neg_lt_neg_iff, Real.log_lt_log_iff x0 y0] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
⊢ (-y.log).log < (-x.log).log ↔ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
⊢ (-y.log).log < (-x.log).log ↔ x < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | rw [Real.le_log_iff_exp_le (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ log 2 * x ≤ (1 + x).log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ (log 2 * x).exp ≤ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ log 2 * x ≤ (1 + x).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | have x0' : 0 ≤ 1 - x := by linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ (log 2 * x).exp ≤ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | have h := convexOn_exp.2 (mem_univ 0) (mem_univ (log 2)) x0' x0 (by abel) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : ((1 - x) • 0 + x • log 2).exp ≤ (1 - x) • exp 0 + x • (log 2).exp
⊢ (log 2 * x).exp ≤ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | simp only [smul_eq_mul, mul_zero, zero_add, Real.exp_zero, mul_one, Real.exp_log zero_lt_two] at h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : ((1 - x) • 0 + x • log 2).exp ≤ (1 - x) • exp 0 + x • (log 2).exp
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 - x + x * 2
⊢ (log 2 * x).exp ≤ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : ((1 - x) • 0 + x • log 2).exp ≤ (1 - x) • exp 0 + x • (log 2).exp
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | ring_nf at h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 - x + x * 2
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 + x
⊢ (log 2 * x).exp ≤ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 - x + x * 2
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | rwa [mul_comm] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 + x
⊢ (log 2 * x).exp ≤ 1 + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 + x
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 < 1 + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 < 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 ≤ 1 - x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 ≤ 1 - x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | abel | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ 1 - x + x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ 1 - x + x = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have hd : ∀ x, 2 ≤ x → HasDerivAt (fun x ↦ log (x-1) / log x)
((1 / (x-1) * log x - log (x-1) * x⁻¹) / (log x)^2) x := by
intro x x2
have l0 : 0 < log x := Real.log_pos (by linarith)
refine HasDerivAt.div ?_ (Real.hasDerivAt_log (by positivity)) l0.ne'
exact HasDerivAt.log ((hasDerivAt_id _).sub_const _) (by linarith) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have d : DifferentiableOn ℝ (fun x ↦ log (x-1) / log x) (Ici 2) :=
fun x m ↦ (hd x m).differentiableAt.differentiableWithinAt | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | apply monotoneOn_of_deriv_nonneg (convex_Ici _) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | case hf
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ContinuousOn (fun x => (x - 1).log / x.log) (Ici 2)
case hf'
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (interior (Ici 2))
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ∀ x ∈ interior (Ici 2), 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | intro x x2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have l0 : 0 < log x := Real.log_pos (by linarith) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | refine HasDerivAt.div ?_ (Real.hasDerivAt_log (by positivity)) l0.ne' | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log) (1 / (x - 1)) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact HasDerivAt.log ((hasDerivAt_id _).sub_const _) (by linarith) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log) (1 / (x - 1)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log) (1 / (x - 1)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x - 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x - 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact d.continuousOn | case hf
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ContinuousOn (fun x => (x - 1).log / x.log) (Ici 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ContinuousOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact d.mono interior_subset | case hf'
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (interior (Ici 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (interior (Ici 2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | intro x m | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ∀ x ∈ interior (Ici 2), 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : x ∈ interior (Ici 2)
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ∀ x ∈ interior (Ici 2), 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | simp only [nonempty_Iio, interior_Ici', mem_Ioi] at m | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : x ∈ interior (Ici 2)
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : x ∈ interior (Ici 2)
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have l0 : 0 < log x := Real.log_pos (by linarith) | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | simp only [(hd x m.le).deriv, one_div] | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ ((x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹) / x.log ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | refine div_nonneg ?_ (by positivity) | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ ((x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹) / x.log ^ 2 | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ ((x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹) / x.log ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | simp only [sub_nonneg, mul_comm] | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹ | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ * (x - 1).log ≤ (x - 1)⁻¹ * x.log | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | apply mul_le_mul | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ * (x - 1).log ≤ (x - 1)⁻¹ * x.log | case hf'_nonneg.h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ ≤ (x - 1)⁻¹
case hf'_nonneg.h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ (x - 1).log ≤ x.log
case hf'_nonneg.c0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1).log
case hf'_nonneg.b0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ * (x - 1).log ≤ (x - 1)⁻¹ * x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | positivity | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x.log ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x.log ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact inv_le_inv_of_le (by linarith) (by linarith) | case hf'_nonneg.h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ ≤ (x - 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ ≤ (x - 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 < x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 < x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x - 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x - 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact Real.log_le_log (by linarith) (by linarith) | case hf'_nonneg.h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ (x - 1).log ≤ x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ (x - 1).log ≤ x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact Real.log_nonneg (by linarith) | case hf'_nonneg.c0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.c0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 1 ≤ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 1 ≤ x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact inv_nonneg.mpr (by linarith) | case hf'_nonneg.b0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.b0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | generalize abs (f' d c z) = w at zw | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
w : ℝ
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ w
⊢ (Complex.abs z).log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | generalize abs z = x at zw cz z4 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
w : ℝ
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ w
⊢ (Complex.abs z).log.log + 0.548 ≤ w.log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
cz : Complex.abs c ≤ x
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
w : ℝ
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ w
⊢ (Complex.abs z).log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | clear z c cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
cz : Complex.abs c ≤ x
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
cz : Complex.abs c ≤ x
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have x0 : 0 < x := by linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have x1 : 1 < x := by linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | simp only [pow_one] at zw | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have lx1 : 1 < log (x-1) :=
lt_trans (by norm_num) (lt_of_lt_of_le lt_log_3 (Real.log_le_log (by norm_num) (by linarith))) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have lx : 1 < log x := lt_trans lx1 (Real.log_lt_log (by linarith) (by linarith)) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have ll0 : 0 ≤ log (x-1) / log x := by positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have ll1 : log (x-1) / log x ≤ 1 :=
div_le_one_of_le (Real.log_le_log (by linarith) (by linarith)) (by positivity) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | calc log (log w)
_ ≥ log (log ((x-1)*x)) := log_log_mono (by nlinarith) zw
_ = log (log x + log (x-1)) := by rw [mul_comm, Real.log_mul (by positivity) (by linarith)]
_ = log (log x) + log (1 + log (x-1) / log x) := by rw [log_add _ _ (by linarith) (by linarith)]
_ ≥ log (log x) + log 2 * (log (x-1) / log x) := by bound [le_log_one_add ll0 ll1]
_ ≥ log (log x) + 0.693 * 0.791 := by bound [lt_log_2]
_ ≥ log (log x) + 0.548 := by bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.548 ≤ w.log.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | refine iter_large (d := d) (n := 1) ?_ ?_ ?_ cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z) | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 2 ≤ Complex.abs z
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 2 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 2 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | exact le_refl _ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 1 < 1.098 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 1 < 1.098
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 3 ≤ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 3 ≤ x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ 0 < x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ 0 < x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x - 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x - 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | apply log_ratio_mono ?_ ?_ z4 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (x - 1).log / x.log ≥ (4 - 1).log / log 4 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 4 ∈ Ici 2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x ∈ Ici 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (x - 1).log / x.log ≥ (4 - 1).log / log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | simp only [mem_Ici] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 4 ∈ Ici 2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 4 ∈ Ici 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | simp only [mem_Ici] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x ∈ Ici 2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x ∈ Ici 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ x
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.