Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
succLtTwoPow
[60, 1]
[71, 38]
apply Nat.one_lt_pow <;> linarith
case intro.succ u : ℕ h : Nat.succ (2 + u) < 2 ^ (2 + u) ⊢ 1 < 2 ^ (2 + u)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
simp
n : ℝ hn : 32 ≤ n ⊢ 2 * n = (2 * n) ^ (6⁻¹ * 6)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply Real.rpow_mul (by linarith)
n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ (6⁻¹ * 6) = ((2 * n) ^ 6⁻¹) ^ 6
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
linarith
n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply Real.rpow_lt_rpow _ _ (by norm_num)
n : ℝ hn : 32 ≤ n ⊢ ((2 * n) ^ 6⁻¹) ^ 6 < (↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6
n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
. apply Real.rpow_nonneg_of_nonneg; linarith
n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1
n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
. apply Nat.lt_floor_add_one
n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_num
n : ℝ hn : 32 ≤ n ⊢ 0 < 6
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply Real.rpow_nonneg_of_nonneg
n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹
case hx n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
linarith
case hx n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply Nat.lt_floor_add_one
n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_cast
n : ℝ hn : 32 ≤ n ⊢ (↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊)
n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ⌊(2 * n) ^ 6⁻¹⌋₊)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
conv => rhs; rhs; rw [mul_comm]
n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ⌊(2 * n) ^ 6⁻¹⌋₊)
n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (⌊(2 * n) ^ 6⁻¹⌋₊ * 6)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [Nat.pow_mul]
n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (⌊(2 * n) ^ 6⁻¹⌋₊ * 6)
n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < (2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊) ^ 6
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply Nat.pow_lt_pow_of_lt_left _ (by norm_num)
n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < (2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊) ^ 6
n : ℝ hn : 32 ≤ n ⊢ ⌊(2 * n) ^ 6⁻¹⌋₊ + 1 < 2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply succLtTwoPow
n : ℝ hn : 32 ≤ n ⊢ ⌊(2 * n) ^ 6⁻¹⌋₊ + 1 < 2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊
case h2 n : ℝ hn : 32 ≤ n ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
have h0 : 0 ≤ (2 * n) ^ 6⁻¹
case h2 n : ℝ hn : 32 ≤ n ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊
case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
. apply le_of_lt; apply Real.rpow_pos_of_pos; linarith
case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [Nat.le_floor_iff h0]
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑2 ≤ (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_cast
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑2 ≤ (2 * n) ^ 6⁻¹
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [← Real.rpow_le_rpow_iff (by norm_num) h0 (by norm_num : (0:ℝ) < 6)]
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ (2 * n) ^ 6⁻¹
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ ((2 * n) ^ 6⁻¹) ^ 6
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [← Real.rpow_mul (by linarith)]
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ ((2 * n) ^ 6⁻¹) ^ 6
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ (2 * n) ^ (6⁻¹ * 6)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
simp
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ (2 * n) ^ (6⁻¹ * 6)
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ 2 * n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [mul_comm]
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ 2 * n
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ n * 2
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [← div_le_iff (by norm_num)]
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ n * 2
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 / 2 ≤ n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_cast
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 / 2 ≤ n
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑(2 ^ 6) / 2 ≤ n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_num
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑(2 ^ 6) / 2 ≤ n
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 32 ≤ n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
assumption
case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 32 ≤ n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply le_of_lt
case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹
case h0.a n : ℝ hn : 32 ≤ n ⊢ 0 < (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
apply Real.rpow_pos_of_pos
case h0.a n : ℝ hn : 32 ≤ n ⊢ 0 < (2 * n) ^ 6⁻¹
case h0.a.hx n : ℝ hn : 32 ≤ n ⊢ 0 < 2 * n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
linarith
case h0.a.hx n : ℝ hn : 32 ≤ n ⊢ 0 < 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_num
n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 ≤ 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_num
n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 < 6
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
linarith
n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 ≤ 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_num
n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 < 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [Real.rpow_le_rpow_left_iff (by norm_num)]
n : ℝ hn : 32 ≤ n ⊢ 2 ^ (6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊) ≤ 2 ^ (6 * (2 * n) ^ 6⁻¹)
n : ℝ hn : 32 ≤ n ⊢ 6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ 6 * (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
rw [mul_le_mul_left (by norm_num)]
n : ℝ hn : 32 ≤ n ⊢ 6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ 6 * (2 * n) ^ 6⁻¹
n : ℝ hn : 32 ≤ n ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
have h0 : 0 ≤ (2 * n) ^ 6⁻¹
n : ℝ hn : 32 ≤ n ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹
case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
. apply le_of_lt; apply Real.rpow_pos_of_pos; linarith
case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹
n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
exact Nat.floor_le h0
n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
twoMulLtTwoPow
[73, 1]
[104, 28]
norm_num
n : ℝ hn : 32 ≤ n ⊢ 1 < 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
rw [← div_lt_iff' (by norm_num)]
n : ℝ h : 81 / 2 < n ⊢ 18 < 2 * Real.sqrt (2 * n)
n : ℝ h : 81 / 2 < n ⊢ 18 / 2 < Real.sqrt (2 * n)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
rw [Real.lt_sqrt (by norm_num)]
n : ℝ h : 81 / 2 < n ⊢ 18 / 2 < Real.sqrt (2 * n)
n : ℝ h : 81 / 2 < n ⊢ (18 / 2) ^ 2 < 2 * n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
norm_num
n : ℝ h : 81 / 2 < n ⊢ (18 / 2) ^ 2 < 2 * n
n : ℝ h : 81 / 2 < n ⊢ 81 < 2 * n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
rw [← div_lt_iff' (by norm_num)]
n : ℝ h : 81 / 2 < n ⊢ 81 < 2 * n
n : ℝ h : 81 / 2 < n ⊢ 81 / 2 < n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
linarith
n : ℝ h : 81 / 2 < n ⊢ 81 / 2 < n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
norm_num
n : ℝ h : 81 / 2 < n ⊢ 0 < 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
ltTwoMulSqrtTwoMul
[106, 1]
[111, 11]
norm_num
n : ℝ h : 81 / 2 < n ⊢ 0 ≤ 18 / 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
rw [← Real.rpow_mul (by norm_num)]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ (4 ^ (n / 3)) ^ 3
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ 4 ^ (n / 3 * 3)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
simp
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ 4 ^ (n / 3 * 3)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
norm_num
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
apply Real.rpow_le_rpow _ h (by linarith)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (4 ^ (n / 3)) ^ 3 ≤ ((2 * n) ^ (1 + Real.sqrt (2 * n))) ^ 3
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4 ^ (n / 3)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
apply Real.rpow_nonneg_of_nonneg (by norm_num)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4 ^ (n / 3)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 3
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
rw [Real.rpow_mul (by linarith)]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ ((2 * n) ^ (1 + Real.sqrt (2 * n))) ^ 3 = (2 * n) ^ ((1 + Real.sqrt (2 * n)) * 3)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
apply Real.rpow_lt_rpow (by linarith) (twoMulLtTwoPow (by linarith))
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 * n) ^ ((1 + Real.sqrt (2 * n)) * 3) < (2 ^ (6 * (2 * n) ^ 6⁻¹)) ^ ((1 + Real.sqrt (2 * n)) * 3)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (1 + Real.sqrt (2 * n)) * 3
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
simp
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (1 + Real.sqrt (2 * n)) * 3
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1 + Real.sqrt (2 * n)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
exact lt_add_of_pos_of_le (by norm_num) (Real.sqrt_nonneg _)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1 + Real.sqrt (2 * n)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 32 ≤ n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
norm_num
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
rw [← Real.rpow_mul (by linarith)]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 ^ (6 * (2 * n) ^ 6⁻¹)) ^ ((1 + Real.sqrt (2 * n)) * 3) < 2 ^ (20 * (2 * n) ^ (2 / 3))
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 2 ^ (6 * (2 * n) ^ 6⁻¹ * ((1 + Real.sqrt (2 * n)) * 3)) < 2 ^ (20 * (2 * n) ^ (2 / 3))
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
rw [Real.rpow_lt_rpow_left_iff (by linarith)]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 2 ^ (6 * (2 * n) ^ 6⁻¹ * ((1 + Real.sqrt (2 * n)) * 3)) < 2 ^ (20 * (2 * n) ^ (2 / 3))
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 6 * (2 * n) ^ 6⁻¹ * ((1 + Real.sqrt (2 * n)) * 3) < 20 * (2 * n) ^ (2 / 3)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
calc _ = (18 * (1 + Real.sqrt (2*n))) * ((2*n)^6⁻¹) := by linarith _ < (2 + 18) * Real.sqrt (2*n) * (2*n)^6⁻¹ := by rw [mul_lt_mul_right] . simp [mul_add, add_mul] apply ltTwoMulSqrtTwoMul linarith . apply Real.rpow_pos_of_pos linarith _ = 20 * (Real.sqrt (2*n) * (2*n)^6⁻¹) := by linarith _ = _ := by simp [Real.sqrt_eq_rpow] rw [← Real.rpow_add (by linarith)] norm_num
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 6 * (2 * n) ^ 6⁻¹ * ((1 + Real.sqrt (2 * n)) * 3) < 20 * (2 * n) ^ (2 / 3)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 1 < 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 6 * (2 * n) ^ 6⁻¹ * ((1 + Real.sqrt (2 * n)) * 3) = 18 * (1 + Real.sqrt (2 * n)) * (2 * n) ^ 6⁻¹
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
rw [mul_lt_mul_right]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 * (1 + Real.sqrt (2 * n)) * (2 * n) ^ 6⁻¹ < (2 + 18) * Real.sqrt (2 * n) * (2 * n) ^ 6⁻¹
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 * (1 + Real.sqrt (2 * n)) < (2 + 18) * Real.sqrt (2 * n) n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
. simp [mul_add, add_mul] apply ltTwoMulSqrtTwoMul linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 * (1 + Real.sqrt (2 * n)) < (2 + 18) * Real.sqrt (2 * n) n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
. apply Real.rpow_pos_of_pos linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
simp [mul_add, add_mul]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 * (1 + Real.sqrt (2 * n)) < (2 + 18) * Real.sqrt (2 * n)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 < 2 * Real.sqrt (2 * n)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
apply ltTwoMulSqrtTwoMul
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 < 2 * Real.sqrt (2 * n)
case h n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 81 / 2 < n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
case h n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 81 / 2 < n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
apply Real.rpow_pos_of_pos
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹
case hx n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
case hx n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 + 18) * Real.sqrt (2 * n) * (2 * n) ^ 6⁻¹ = 20 * (Real.sqrt (2 * n) * (2 * n) ^ 6⁻¹)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
simp [Real.sqrt_eq_rpow]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 20 * (Real.sqrt (2 * n) * (2 * n) ^ 6⁻¹) = 20 * (2 * n) ^ (2 / 3)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 * n) ^ 2⁻¹ * (2 * n) ^ 6⁻¹ = (2 * n) ^ (2 / 3)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
rw [← Real.rpow_add (by linarith)]
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 * n) ^ 2⁻¹ * (2 * n) ^ 6⁻¹ = (2 * n) ^ (2 / 3)
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 * n) ^ (2⁻¹ + 6⁻¹) = (2 * n) ^ (2 / 3)
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
norm_num
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ (2 * n) ^ (2⁻¹ + 6⁻¹) = (2 * n) ^ (2 / 3)
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
fourPowLtOf
[113, 1]
[141, 15]
linarith
n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [(by norm_num : (4 : ℝ) = 2 ^ 2)]
n : ℝ hn : 0 < n ⊢ 4 ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000
n : ℝ hn : 0 < n ⊢ (2 ^ 2) ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [← Real.rpow_mul (by norm_num)]
n : ℝ hn : 0 < n ⊢ (2 ^ 2) ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000
n : ℝ hn : 0 < n ⊢ 2 ^ (2 * n) < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [Real.rpow_lt_rpow_left_iff (by linarith)]
n : ℝ hn : 0 < n ⊢ 2 ^ (2 * n) < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000
n : ℝ hn : 0 < n ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
have h_nonneg : 0 < (2 * n) ^ (2 / 3) := by apply Real.rpow_pos_of_pos; linarith
n : ℝ hn : 0 < n ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [← @Real.rpow_lt_rpow_iff _ _ 3 (by linarith) (by linarith) (by norm_num)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < (20 * (2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [@Real.mul_rpow 20 _ _ (by norm_num) (by linarith)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < (20 * (2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * ((2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [← Real.rpow_mul (by linarith)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * ((2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ (2 / 3 * 3) ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
simp
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ (2 / 3 * 3) ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ 2 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [(by norm_num : (3:ℝ) = 1 + 2)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ 2 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ (1 + 2) < 20 ^ (1 + 2) * (2 * n) ^ 2 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [@Real.rpow_add (2*n) (by linarith)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ (1 + 2) < 20 ^ (1 + 2) * (2 * n) ^ 2 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 1 * (2 * n) ^ 2 < 20 ^ (1 + 2) * (2 * n) ^ 2 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [@Real.rpow_add 20 (by norm_num)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 1 * (2 * n) ^ 2 < 20 ^ (1 + 2) * (2 * n) ^ 2 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 1 * (2 * n) ^ 2 < 20 ^ 1 * 20 ^ 2 * (2 * n) ^ 2 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
norm_num
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 1 * (2 * n) ^ 2 < 20 ^ 1 * 20 ^ 2 * (2 * n) ^ 2 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n * (2 * n) ^ 2 < 8000 * (2 * n) ^ 2 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [mul_lt_mul_right (by apply sq_pos_of_pos; linarith)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n * (2 * n) ^ 2 < 8000 * (2 * n) ^ 2 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 8000 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [mul_comm]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 8000 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n * 2 < 8000 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
rw [← lt_div_iff (by norm_num)]
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n * 2 < 8000 ↔ n < 4000
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n < 8000 / 2 ↔ n < 4000
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
norm_num
n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n < 8000 / 2 ↔ n < 4000
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
norm_num
n : ℝ hn : 0 < n ⊢ 4 = 2 ^ 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
norm_num
n : ℝ hn : 0 < n ⊢ 0 ≤ 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
linarith
n : ℝ hn : 0 < n ⊢ 1 < 2
no goals
https://github.com/jvlmdr/from_the_book.git
9fb6080539a2f32bb24719600a9e7531abf2328d
FromTheBook/Ch02/Bertrand/Bertrand.lean
lt4kIff
[143, 1]
[160, 11]
apply Real.rpow_pos_of_pos
n : ℝ hn : 0 < n ⊢ 0 < (2 * n) ^ (2 / 3)
case hx n : ℝ hn : 0 < n ⊢ 0 < 2 * n