pkuAI4M/lean_github · Datasets at Hugging Face

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/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	Please generate a tactic in lean4 to solve the state. STATE: case intro.succ u : ℕ h : Nat.succ (2 + u) < 2 ^ (2 + u) ⊢ 1 < 2 ^ (2 + u) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 2 * n = (2 * n) ^ (6⁻¹ * 6) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ (6⁻¹ * 6) = ((2 * n) ^ 6⁻¹) ^ 6 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ ((2 * n) ^ 6⁻¹) ^ 6 < (↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 < 6 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case hx n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1 TACTIC:
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⌋₊)	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊) TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ⌊(2 * n) ^ 6⁻¹⌋₊) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (⌊(2 * n) ^ 6⁻¹⌋₊ * 6) TACTIC:
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⌋₊	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < (2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊) ^ 6 TACTIC:
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⌋₊	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ ⌊(2 * n) ^ 6⁻¹⌋₊ + 1 < 2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊ TACTIC:
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⌋₊	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊ TACTIC:
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⌋₊	Please generate a tactic in lean4 to solve the state. STATE: 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⌋₊ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑2 ≤ (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ (2 * n) ^ 6⁻¹ TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ ((2 * n) ^ 6⁻¹) ^ 6 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ (2 * n) ^ (6⁻¹ * 6) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ n * 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 / 2 ≤ n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑(2 ^ 6) / 2 ≤ n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 32 ≤ n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h0.a n : ℝ hn : 32 ≤ n ⊢ 0 < (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h0.a.hx n : ℝ hn : 32 ≤ n ⊢ 0 < 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 ≤ 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 < 6 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 ≤ 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 < 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 2 ^ (6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊) ≤ 2 ^ (6 * (2 * n) ^ 6⁻¹) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ 6 * (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 1 < 2 TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 18 < 2 * Real.sqrt (2 * n) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 18 / 2 < Real.sqrt (2 * n) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ (18 / 2) ^ 2 < 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 81 < 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 81 / 2 < n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 0 < 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 0 ≤ 18 / 2 TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ (4 ^ (n / 3)) ^ 3 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ 4 ^ (n / 3 * 3) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4 TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4 ^ (n / 3) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 3 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (1 + Real.sqrt (2 * n)) * 3 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1 + Real.sqrt (2 * n) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 32 ≤ n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1 TACTIC:
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))	Please generate a tactic in lean4 to solve the state. STATE: 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)) TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: 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)) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/Bertrand.lean	fourPowLtOf	[113, 1]	[141, 15]	calc _ = (18 * (1 + Real.sqrt (2n))) ((2n)^6⁻¹) := by linarith _ < (2 + 18) Real.sqrt (2n) (2n)^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 (2n) (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	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 1 < 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹ TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 < 2 * Real.sqrt (2 * n) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case h n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 81 / 2 < n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹ TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: case hx n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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)	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 4 ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ (2 ^ 2) ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 2 ^ (2 * n) < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < (20 * (2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * ((2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ (2 / 3 * 3) ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ 2 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ (1 + 2) < 20 ^ (1 + 2) * (2 * n) ^ 2 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n * (2 * n) ^ 2 < 8000 * (2 * n) ^ 2 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 8000 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n * 2 < 8000 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n < 8000 / 2 ↔ n < 4000 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 4 = 2 ^ 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 0 ≤ 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 1 < 2 TACTIC:
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	Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 0 < (2 * n) ^ (2 / 3) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case intro.succ u : ℕ h : Nat.succ (2 + u) < 2 ^ (2 + u) ⊢ 1 < 2 ^ (2 + u) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 2 * n = (2 * n) ^ (6⁻¹ * 6) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ (6⁻¹ * 6) = ((2 * n) ^ 6⁻¹) ^ 6 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ ((2 * n) ^ 6⁻¹) ^ 6 < (↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 < 6 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case hx n : ℝ hn : 32 ≤ n ⊢ 0 ≤ 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (2 * n) ^ 6⁻¹ < ↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1 TACTIC:

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⌋₊)

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (↑⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊) TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (6 * ⌊(2 * n) ^ 6⁻¹⌋₊) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < 2 ^ (⌊(2 * n) ^ 6⁻¹⌋₊ * 6) TACTIC:

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⌋₊

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ (⌊(2 * n) ^ 6⁻¹⌋₊ + 1) ^ 6 < (2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊) ^ 6 TACTIC:

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⌋₊

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ ⌊(2 * n) ^ 6⁻¹⌋₊ + 1 < 2 ^ ⌊(2 * n) ^ 6⁻¹⌋₊ TACTIC:

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⌋₊

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊ TACTIC:

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⌋₊

Please generate a tactic in lean4 to solve the state. STATE: 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⌋₊ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ ⌊(2 * n) ^ 6⁻¹⌋₊ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑2 ≤ (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ≤ (2 * n) ^ 6⁻¹ TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ ((2 * n) ^ 6⁻¹) ^ 6 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ (2 * n) ^ (6⁻¹ * 6) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 ≤ n * 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 2 ^ 6 / 2 ≤ n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑(2 ^ 6) / 2 ≤ n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h2 n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 32 ≤ n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h0 n : ℝ hn : 32 ≤ n ⊢ 0 ≤ (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h0.a n : ℝ hn : 32 ≤ n ⊢ 0 < (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h0.a.hx n : ℝ hn : 32 ≤ n ⊢ 0 < 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 ≤ 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 < 6 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 ≤ 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ 0 < 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 2 ^ (6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊) ≤ 2 ^ (6 * (2 * n) ^ 6⁻¹) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 6 * ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ 6 * (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n h0 : 0 ≤ (2 * n) ^ 6⁻¹ ⊢ ↑⌊(2 * n) ^ 6⁻¹⌋₊ ≤ (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 32 ≤ n ⊢ 1 < 2 TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 18 < 2 * Real.sqrt (2 * n) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 18 / 2 < Real.sqrt (2 * n) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ (18 / 2) ^ 2 < 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 81 < 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 81 / 2 < n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 0 < 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ h : 81 / 2 < n ⊢ 0 ≤ 18 / 2 TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ (4 ^ (n / 3)) ^ 3 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 4 ^ n ≤ 4 ^ (n / 3 * 3) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4 TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 4 ^ (n / 3) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 3 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (1 + Real.sqrt (2 * n)) * 3 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1 + Real.sqrt (2 * n) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 32 ≤ n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 1 TACTIC:

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))

Please generate a tactic in lean4 to solve the state. STATE: 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)) TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: 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)) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 ≤ 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 1 < 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹ TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 18 < 2 * Real.sqrt (2 * n) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case h n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 81 / 2 < n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < (2 * n) ^ 6⁻¹ TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: case hx n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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)

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 81 / 2 < n h : 4 ^ (n / 3) ≤ (2 * n) ^ (1 + Real.sqrt (2 * n)) ⊢ 0 < 2 * n TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 4 ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ (2 ^ 2) ^ n < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 2 ^ (2 * n) < 2 ^ (20 * (2 * n) ^ (2 / 3)) ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 20 * (2 * n) ^ (2 / 3) ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < (20 * (2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * ((2 * n) ^ (2 / 3)) ^ 3 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ (2 / 3 * 3) ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ 3 < 20 ^ 3 * (2 * n) ^ 2 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ (2 * n) ^ (1 + 2) < 20 ^ (1 + 2) * (2 * n) ^ 2 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n * (2 * n) ^ 2 < 8000 * (2 * n) ^ 2 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ 2 * n < 8000 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n * 2 < 8000 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n h_nonneg : 0 < (2 * n) ^ (2 / 3) ⊢ n < 8000 / 2 ↔ n < 4000 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 4 = 2 ^ 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 0 ≤ 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 1 < 2 TACTIC:

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

Please generate a tactic in lean4 to solve the state. STATE: n : ℝ hn : 0 < n ⊢ 0 < (2 * n) ^ (2 / 3) TACTIC: