Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	constructor	case h z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (w - z) < r case h.right z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs w < Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case h z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	use 0	case pos z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊢ ∃ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z	case h z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊢ Complex.abs (0 - z) < r ∧ Complex.abs 0 < Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊢ ∃ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	simp only [zero_sub, map_neg_eq_map, rz, map_zero, AbsoluteValue.pos_iff, ne_eq, nz, not_false_eq_true, and_self]	case h z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊢ Complex.abs (0 - z) < r ∧ Complex.abs 0 < Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊢ Complex.abs (0 - z) < r ∧ Complex.abs 0 < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	rw [←ha, sub_nonneg, div_le_one azp]	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ 0 ≤ a	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ r / 2 ≤ Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ 0 ≤ a TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	exact _root_.trans (by bound) rz	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ r / 2 ≤ Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ r / 2 ≤ Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	bound	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ r / 2 ≤ r	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊢ r / 2 ≤ r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	rw [←ha, sub_lt_self_iff]	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a ⊢ a < 1	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a ⊢ 0 < r / 2 / Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a ⊢ a < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	positivity	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a ⊢ 0 < r / 2 / Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a ⊢ 0 < r / 2 / Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	rw [←hw,←ha]	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (w - z) < r	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (↑(1 - r / 2 / Complex.abs z) * z - z) < r	Please generate a tactic in lean4 to solve the state. STATE: case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (w - z) < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	simp only [Complex.ofReal_sub, Complex.ofReal_one, Complex.ofReal_div]	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (↑(1 - r / 2 / Complex.abs z) * z - z) < r	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs ((1 - ↑r / ↑2 / ↑(Complex.abs z)) * z - z) < r	Please generate a tactic in lean4 to solve the state. STATE: case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (↑(1 - r / 2 / Complex.abs z) * z - z) < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	rw [mul_sub_right_distrib]	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs ((1 - ↑r / ↑2 / ↑(Complex.abs z)) * z - z) < r	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (1 * z - ↑r / ↑2 / ↑(Complex.abs z) * z - z) < r	Please generate a tactic in lean4 to solve the state. STATE: case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs ((1 - ↑r / ↑2 / ↑(Complex.abs z)) * z - z) < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	simp only [one_mul, sub_sub_cancel_left, AbsoluteValue.map_neg, AbsoluteValue.map_mul, map_div₀, Complex.abs_ofReal, Complex.abs_two, Complex.abs_abs, abs_of_pos rp, div_mul (r/2), div_mul_cancel₀ _ azp.ne', div_one, abs_two, div_self azp.ne']	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (1 * z - ↑r / ↑2 / ↑(Complex.abs z) * z - z) < r	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ r / 2 < r	Please generate a tactic in lean4 to solve the state. STATE: case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs (1 * z - ↑r / ↑2 / ↑(Complex.abs z) * z - z) < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	bound	case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ r / 2 < r	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.left z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ r / 2 < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	simp only [←hw, AbsoluteValue.map_mul, Complex.abs_ofReal, abs_of_nonneg a0]	case h.right z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs w < Complex.abs z	case h.right z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ a * Complex.abs z < Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case h.right z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	calc a * abs z < 1 * abs z := mul_lt_mul_of_pos_right a1 azp _ = abs z := by ring	case h.right z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ a * Complex.abs z < Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.right z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ a * Complex.abs z < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	slightly_smaller	[254, 1]	[276, 27]	ring	z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ 1 * Complex.abs z = Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ nz : z ≠ 0 r : ℝ rp : 0 < r rz : r ≤ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≤ a a1 : a < 1 w : ℂ hw : ↑a * z = w ⊢ 1 * Complex.abs z = Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	frequently_smaller	[279, 1]	[282, 74]	simp only [Filter.Frequently, Metric.eventually_nhds_iff, not_exists, not_forall, not_not, Complex.dist_eq, not_and]	z : ℂ z0 : z ≠ 0 ⊢ ∃ᶠ (w : ℂ) in 𝓝 z, Complex.abs w < Complex.abs z	z : ℂ z0 : z ≠ 0 ⊢ ∀ x > 0, ∃ x_1, ∃ (_ : Complex.abs (x_1 - z) < x), Complex.abs x_1 < Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ z0 : z ≠ 0 ⊢ ∃ᶠ (w : ℂ) in 𝓝 z, Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	frequently_smaller	[279, 1]	[282, 74]	intro r rp	z : ℂ z0 : z ≠ 0 ⊢ ∀ x > 0, ∃ x_1, ∃ (_ : Complex.abs (x_1 - z) < x), Complex.abs x_1 < Complex.abs z	z : ℂ z0 : z ≠ 0 r : ℝ rp : r > 0 ⊢ ∃ x, ∃ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ z0 : z ≠ 0 ⊢ ∀ x > 0, ∃ x_1, ∃ (_ : Complex.abs (x_1 - z) < x), Complex.abs x_1 < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	frequently_smaller	[279, 1]	[282, 74]	rcases slightly_smaller z0 rp with ⟨w, b, lt⟩	z : ℂ z0 : z ≠ 0 r : ℝ rp : r > 0 ⊢ ∃ x, ∃ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z	case intro.intro z : ℂ z0 : z ≠ 0 r : ℝ rp : r > 0 w : ℂ b : Complex.abs (w - z) < r lt : Complex.abs w < Complex.abs z ⊢ ∃ x, ∃ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ z0 : z ≠ 0 r : ℝ rp : r > 0 ⊢ ∃ x, ∃ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	frequently_smaller	[279, 1]	[282, 74]	use w, b, lt	case intro.intro z : ℂ z0 : z ≠ 0 r : ℝ rp : r > 0 w : ℂ b : Complex.abs (w - z) < r lt : Complex.abs w < Complex.abs z ⊢ ∃ x, ∃ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro.intro z : ℂ z0 : z ≠ 0 r : ℝ rp : r > 0 w : ℂ b : Complex.abs (w - z) < r lt : Complex.abs w < Complex.abs z ⊢ ∃ x, ∃ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	by_cases nz : z = 0	f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z ⊢ Complex.abs (f z) ≤ c * Complex.abs z	case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs (f z) ≤ c * Complex.abs z case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 ⊢ Complex.abs (f z) ≤ c * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z ⊢ Complex.abs (f z) ≤ c * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	apply le_of_forall_small_le_add zero_lt_one	case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 ⊢ Complex.abs (f z) ≤ c * Complex.abs z	case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 ⊢ ∀ (e : ℝ), 0 < e → e < 1 → Complex.abs (f z) ≤ c * Complex.abs z + e	Please generate a tactic in lean4 to solve the state. STATE: case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 ⊢ Complex.abs (f z) ≤ c * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	intro e ep _	case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 ⊢ ∀ (e : ℝ), 0 < e → e < 1 → Complex.abs (f z) ≤ c * Complex.abs z + e	case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	Please generate a tactic in lean4 to solve the state. STATE: case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 ⊢ ∀ (e : ℝ), 0 < e → e < 1 → Complex.abs (f z) ≤ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	rcases Metric.continuous_near fc rp e ep with ⟨s,sp,_,sc⟩	case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	case neg.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	Please generate a tactic in lean4 to solve the state. STATE: case neg f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	rcases slightly_smaller nz sp with ⟨w,wz,awz⟩	case neg.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	case neg.intro.intro.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	Please generate a tactic in lean4 to solve the state. STATE: case neg.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	have wr : abs w < r := lt_of_lt_of_le awz zr	case neg.intro.intro.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	case neg.intro.intro.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	Please generate a tactic in lean4 to solve the state. STATE: case neg.intro.intro.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	calc abs (f z) = abs (f w - (f w - f z)) := by ring_nf _ ≤ abs (f w) + abs (f w - f z) := by bound _ ≤ c * abs w + e := by bound [h w wr, sc wz] _ ≤ c * abs z + e := by bound	case neg.intro.intro.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg.intro.intro.intro.intro.intro f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f z) ≤ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	refine h z ?_	case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs (f z) ≤ c * Complex.abs z	case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs z < r	Please generate a tactic in lean4 to solve the state. STATE: case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs (f z) ≤ c * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	rw [nz]	case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs z < r	case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs 0 < r	Please generate a tactic in lean4 to solve the state. STATE: case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs z < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	simpa	case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs 0 < r	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : z = 0 ⊢ Complex.abs 0 < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	ring_nf	f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f z) = Complex.abs (f w - (f w - f z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f z) = Complex.abs (f w - (f w - f z)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	bound	f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f w - (f w - f z)) ≤ Complex.abs (f w) + Complex.abs (f w - f z)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f w - (f w - f z)) ≤ Complex.abs (f w) + Complex.abs (f w - f z) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	bound [h w wr, sc wz]	f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f w) + Complex.abs (f w - f z) ≤ c * Complex.abs w + e	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ Complex.abs (f w) + Complex.abs (f w - f z) ≤ c * Complex.abs w + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_to_strong_small	[284, 1]	[296, 34]	bound	f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ c * Complex.abs w + e ≤ c * Complex.abs z + e	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ z : ℂ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≤ r fc : ContinuousAt f z h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z nz : ¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≤ r sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e w : ℂ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊢ c * Complex.abs w + e ≤ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	by_cases r0 : r ≤ 0	z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : ¬r ≤ 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	simp only [not_le] at r0	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : ¬r ≤ 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : ¬r ≤ 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	have fc : ContinuousAt (fun z ↦ log (1 + z)) z := by apply ContinuousAt.clog; apply ContinuousAt.add; exact continuousAt_const; exact continuousAt_id refine mem_slitPlane_of_near_one ?_ simp only [add_sub_cancel_left, lt_of_le_of_lt zr r1]	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	apply weak_to_strong_small r0 (by bound) zr fc	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ ∀ (z : ℂ), Complex.abs z < r → Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	intro w wr	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ ∀ (z : ℂ), Complex.abs z < r → Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ Complex.abs (1 + w).log ≤ 1 / (1 - r) * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ ∀ (z : ℂ), Complex.abs z < r → Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	exact @weak_log1p_small w r (by bound) wr	case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ Complex.abs (1 + w).log ≤ 1 / (1 - r) * Complex.abs w	no goals	Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ Complex.abs (1 + w).log ≤ 1 / (1 - r) * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	have z0 := le_antisymm (le_trans zr r0) (Complex.abs.nonneg _)	case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 z0 : Complex.abs z = 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	simp only [map_eq_zero] at z0	case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 z0 : Complex.abs z = 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 z0 : z = 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 z0 : Complex.abs z = 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	simp only [z0, add_zero, Complex.log_one, map_zero, r0, sub_zero, ne_eq, one_ne_zero, not_false_eq_true, div_self, mul_zero, le_refl]	case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 z0 : z = 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : r ≤ 0 z0 : z = 0 ⊢ Complex.abs (1 + z).log ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	apply ContinuousAt.clog	z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun z => (1 + z).log) z	case h₁ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun t => 1 + t) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun z => (1 + z).log) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	apply ContinuousAt.add	case h₁ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun t => 1 + t) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	case h₁.hf z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => 1) z case h₁.hg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => x) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	Please generate a tactic in lean4 to solve the state. STATE: case h₁ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun t => 1 + t) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	exact continuousAt_const	case h₁.hf z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => 1) z case h₁.hg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => x) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	case h₁.hg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => x) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	Please generate a tactic in lean4 to solve the state. STATE: case h₁.hf z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => 1) z case h₁.hg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => x) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	exact continuousAt_id	case h₁.hg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => x) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	Please generate a tactic in lean4 to solve the state. STATE: case h₁.hg z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ ContinuousAt (fun x => x) z case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	refine mem_slitPlane_of_near_one ?_	case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane	case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ Complex.abs (1 + z - 1) < 1	Please generate a tactic in lean4 to solve the state. STATE: case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	simp only [add_sub_cancel_left, lt_of_le_of_lt zr r1]	case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ Complex.abs (1 + z - 1) < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₂ z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r ⊢ Complex.abs (1 + z - 1) < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	bound	z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ 1 / (1 - r) > 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ 1 / (1 - r) > 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small'	[298, 1]	[312, 44]	bound	z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ r < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ r < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small	[314, 1]	[315, 68]	norm_num	z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / 2 < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log1p_small	[314, 1]	[315, 68]	norm_num	z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * Complex.abs z = 2 * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * Complex.abs z = 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small'	[318, 1]	[324, 62]	set z := (x : ℂ)	x r : ℝ r1 : r < 1 xr : \|x\| ≤ r ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\|	x r : ℝ r1 : r < 1 xr : \|x\| ≤ r z : ℂ := ↑x ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\|	Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 xr : \|x\| ≤ r ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\| TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small'	[318, 1]	[324, 62]	have zx : abs z = \|x\| := Complex.abs_ofReal _	x r : ℝ r1 : r < 1 xr : \|x\| ≤ r z : ℂ := ↑x ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\|	x r : ℝ r1 : r < 1 xr : \|x\| ≤ r z : ℂ := ↑x zx : Complex.abs z = \|x\| ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\|	Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 xr : \|x\| ≤ r z : ℂ := ↑x ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\| TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small'	[318, 1]	[324, 62]	simp only [← Complex.log_ofReal_re, ← zx] at xr ⊢	x r : ℝ r1 : r < 1 xr : \|x\| ≤ r z : ℂ := ↑x zx : Complex.abs z = \|x\| ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\|	x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = \|x\| xr : Complex.abs z ≤ r ⊢ \|(↑(1 + x)).log.re\| ≤ 1 / (1 - r) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 xr : \|x\| ≤ r z : ℂ := ↑x zx : Complex.abs z = \|x\| ⊢ \|(1 + x).log\| ≤ 1 / (1 - r) * \|x\| TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small'	[318, 1]	[324, 62]	refine _root_.trans (_root_.trans (Complex.abs_re_le_abs _) ?_) (_root_.log1p_small' r1 xr)	x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = \|x\| xr : Complex.abs z ≤ r ⊢ \|(↑(1 + x)).log.re\| ≤ 1 / (1 - r) * Complex.abs z	x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = \|x\| xr : Complex.abs z ≤ r ⊢ Complex.abs (↑(1 + x)).log ≤ Complex.abs (1 + z).log	Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = \|x\| xr : Complex.abs z ≤ r ⊢ \|(↑(1 + x)).log.re\| ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small'	[318, 1]	[324, 62]	simp only [Complex.ofReal_add, Complex.ofReal_one, le_refl]	x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = \|x\| xr : Complex.abs z ≤ r ⊢ Complex.abs (↑(1 + x)).log ≤ Complex.abs (1 + z).log	no goals	Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = \|x\| xr : Complex.abs z ≤ r ⊢ Complex.abs (↑(1 + x)).log ≤ Complex.abs (1 + z).log TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small	[327, 1]	[328, 73]	norm_num	x : ℝ xr : \|x\| ≤ 1 / 2 ⊢ 1 / 2 < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xr : \|x\| ≤ 1 / 2 ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	Real.log1p_small	[327, 1]	[328, 73]	norm_num	x : ℝ xr : \|x\| ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * \|x\| = 2 * \|x\|	no goals	Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xr : \|x\| ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * \|x\| = 2 * \|x\| TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	generalize zw : z - 1 = z1	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ⊢ Complex.abs z.log ≤ 2 * Complex.abs (z - 1)	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ⊢ Complex.abs z.log ≤ 2 * Complex.abs (z - 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	have wz : z = 1 + z1 := by rw [← zw]; ring	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	rw [wz]	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (1 + z1).log ≤ 2 * Complex.abs z1	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	refine log1p_small ?_	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (1 + z1).log ≤ 2 * Complex.abs z1	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (1 + z1).log ≤ 2 * Complex.abs z1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	rw [← zw]	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	assumption	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	rw [← zw]	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + z1	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1)	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + z1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	log_small	[331, 1]	[333, 56]	ring	z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1)	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	have hr : 0 ≤ (1 : ℝ) := by norm_num	z : ℂ h : Complex.abs z < 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	have L := Complex.locally_lipschitz_exp hr (by bound) 0 z (by simpa only [sub_zero, Complex.norm_eq_abs])	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : ‖z.exp - exp 0‖ ≤ (1 + 1) * ‖exp 0‖ * ‖z - 0‖ ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	simp only [Complex.exp_zero, Complex.norm_eq_abs, norm_one, mul_one, sub_zero] at L	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : ‖z.exp - exp 0‖ ≤ (1 + 1) * ‖exp 0‖ * ‖z - 0‖ ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : ‖z.exp - exp 0‖ ≤ (1 + 1) * ‖exp 0‖ * ‖z - 0‖ ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	have t : 1 + 1 = (2 : ℝ) := by norm_num	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	rw [t] at L	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	assumption	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	norm_num	z : ℂ h : Complex.abs z < 1 ⊢ 0 ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 ⊢ 0 ≤ 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	bound	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ 1 ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ 1 ≤ 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	simpa only [sub_zero, Complex.norm_eq_abs]	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ ‖z - 0‖ < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ ‖z - 0‖ < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	weak_exp_small	[335, 1]	[341, 26]	norm_num	z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ 1 + 1 = 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ 1 + 1 = 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	have rp : (1 : ℝ) > 0 := by norm_num	z : ℂ zs : Complex.abs z ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	have cp : (2 : ℝ) > 0 := by norm_num	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	have fc : ContinuousAt (fun z ↦ exp z - 1) z := by apply ContinuousAt.sub; apply ContinuousAt.cexp exact continuousAt_id; exact continuousAt_const	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	apply weak_to_strong_small rp cp zs fc	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ ∀ (z : ℂ), Complex.abs z < 1 → Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	intro w wr	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ ∀ (z : ℂ), Complex.abs z < 1 → Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z w : ℂ wr : Complex.abs w < 1 ⊢ Complex.abs (w.exp - 1) ≤ 2 * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ ∀ (z : ℂ), Complex.abs z < 1 → Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	exact weak_exp_small wr	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z w : ℂ wr : Complex.abs w < 1 ⊢ Complex.abs (w.exp - 1) ≤ 2 * Complex.abs w	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z w : ℂ wr : Complex.abs w < 1 ⊢ Complex.abs (w.exp - 1) ≤ 2 * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	norm_num	z : ℂ zs : Complex.abs z ≤ 1 ⊢ 1 > 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 ⊢ 1 > 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	norm_num	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ 2 > 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ 2 > 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	apply ContinuousAt.sub	z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun z => z.exp - 1) z	case hf z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => x.exp) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z	Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun z => z.exp - 1) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	apply ContinuousAt.cexp	case hf z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => x.exp) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z	case hf.h z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun y => y) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z	Please generate a tactic in lean4 to solve the state. STATE: case hf z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => x.exp) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	exact continuousAt_id	case hf.h z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun y => y) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z	case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z	Please generate a tactic in lean4 to solve the state. STATE: case hf.h z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun y => y) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	exp_small	[344, 1]	[351, 38]	exact continuousAt_const	case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	have z1 : 1 + z ≠ 0 := by rw [←Complex.abs.ne_zero_iff]; apply ne_of_gt calc abs (1 + z) ≥ abs (1 : ℂ) - abs z := by bound _ ≥ abs (1 : ℂ) - 1/2 := by bound _ > 0 := by norm_num	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	rw [Complex.cpow_def_of_ne_zero z1]	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	have ls := log1p_small zs	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	have eas : abs (log (1 + z) * w) ≤ 1 := by rw [Complex.abs.map_mul] calc abs (log (1 + z)) * abs w ≤ 2 * abs z * abs w := by bound _ ≤ 2 * (1/2) * 1 := by bound _ = 1 := by norm_num	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	have es := exp_small eas	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs ((1 + z).log * w) ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	rw [Complex.abs.map_mul, ←mul_assoc] at es	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs ((1 + z).log * w) ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs ((1 + z).log * w) ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	trans 2 * abs (log (1 + z)) * abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	exact es	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	calc 2 * abs (log (1 + z)) * abs w ≤ 2 * (2 * abs z) * abs w := by bound _ = 4 * abs z * abs w := by ring	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w	no goals	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Misc/Bounds.lean	pow1p_small	[353, 1]	[372, 37]	rw [←Complex.abs.ne_zero_iff]	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 1 + z ≠ 0	z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (1 + z) ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 1 + z ≠ 0 TACTIC:

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