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/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	intro zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ z ∈ t → Complex.abs (f z) ≤ 5 / 8 * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f z) ≤ 5 / 8 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ z ∈ t → Complex.abs (f z) ≤ 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	rw [s.fg]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f z) ≤ 5 / 8 * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (z ^ d * g f d z) ≤ 5 / 8 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f z) ≤ 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	simp	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (z ^ d * g f d z) ≤ 5 / 8 * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (z ^ d * g f d z) ≤ 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	have gs : abs (g f d z) ≤ 5 / 4 := by calc abs (g f d z) _ = abs (g f d z - 1 + 1) := by ring_nf _ ≤ abs (g f d z - 1) + abs (1 : ℂ) := by bound _ ≤ 1 / 4 + abs (1 : ℂ) := by linarith [s.gs zt] _ ≤ 5 / 4 := by norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	have az1 : abs z ≤ 1 := le_trans (s.t2 zt) (by norm_num)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	calc abs z ^ d * abs (g f d z) _ ≤ abs z ^ 2 * (5 / 4) := by bound _ = abs z * abs z * (5 / 4) := by ring_nf _ ≤ 1 / 2 * abs z * (5 / 4) := by bound [s.t2 zt] _ = 5 / 8 * abs z := by ring	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	calc abs (g f d z) _ = abs (g f d z - 1 + 1) := by ring_nf _ ≤ abs (g f d z - 1) + abs (1 : ℂ) := by bound _ ≤ 1 / 4 + abs (1 : ℂ) := by linarith [s.gs zt] _ ≤ 5 / 4 := by norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z) ≤ 5 / 4	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z) ≤ 5 / 4 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	ring_nf	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z) = Complex.abs (g f d z - 1 + 1)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z) = Complex.abs (g f d z - 1 + 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z - 1 + 1) ≤ Complex.abs (g f d z - 1) + Complex.abs 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z - 1 + 1) ≤ Complex.abs (g f d z - 1) + Complex.abs 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	linarith [s.gs zt]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z - 1) + Complex.abs 1 ≤ 1 / 4 + Complex.abs 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z - 1) + Complex.abs 1 ≤ 1 / 4 + Complex.abs 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ 1 / 4 + Complex.abs 1 ≤ 5 / 4	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ 1 / 4 + Complex.abs 1 ≤ 5 / 4 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ 1 / 2 ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ 1 / 2 ≤ 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ Complex.abs z ^ 2 * (5 / 4)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ Complex.abs z ^ 2 * (5 / 4) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	ring_nf	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ 2 * (5 / 4) = Complex.abs z * Complex.abs z * (5 / 4)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ 2 * (5 / 4) = Complex.abs z * Complex.abs z * (5 / 4) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	bound [s.t2 zt]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z * Complex.abs z * (5 / 4) ≤ 1 / 2 * Complex.abs z * (5 / 4)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z * Complex.abs z * (5 / 4) ≤ 1 / 2 * Complex.abs z * (5 / 4) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	f_converges	[263, 1]	[277, 33]	ring	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ 1 / 2 * Complex.abs z * (5 / 4) = 5 / 8 * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ 1 / 2 * Complex.abs z * (5 / 4) = 5 / 8 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_le	[279, 1]	[280, 78]	intro rp	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ ⊢ r > 0 → (5 / 8) ^ n * r ≤ r	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ r	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ ⊢ r > 0 → (5 / 8) ^ n * r ≤ r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_le	[279, 1]	[280, 78]	trans (1:ℝ) ^ n * r	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ r	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ 1 ^ n * r f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_le	[279, 1]	[280, 78]	bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ 1 ^ n * r f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ 1 ^ n * r f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_le	[279, 1]	[280, 78]	simp only [one_pow, one_mul, le_refl]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_lt	[282, 1]	[285, 88]	intro rp np	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ ⊢ r > 0 → n ≠ 0 → (5 / 8) ^ n * r < r	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ (5 / 8) ^ n * r < r	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ ⊢ r > 0 → n ≠ 0 → (5 / 8) ^ n * r < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_lt	[282, 1]	[285, 88]	have h : (5 / 8 : ℝ) ^ n < 1 := pow_lt_one (by norm_num) (by norm_num) np	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ (5 / 8) ^ n * r < r	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ (5 / 8) ^ n * r < r	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ (5 / 8) ^ n * r < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_lt	[282, 1]	[285, 88]	exact lt_of_lt_of_le (mul_lt_mul_of_pos_right h rp) (by simp only [one_mul, le_refl])	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ (5 / 8) ^ n * r < r	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ (5 / 8) ^ n * r < r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_lt	[282, 1]	[285, 88]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ 0 ≤ 5 / 8	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ 0 ≤ 5 / 8 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_lt	[282, 1]	[285, 88]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ 5 / 8 < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ 5 / 8 < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	five_eights_pow_lt	[282, 1]	[285, 88]	simp only [one_mul, le_refl]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ 1 * r ≤ r	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ 1 * r ≤ r TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	SuperNear.mapsTo	[288, 1]	[290, 49]	induction' n with n h	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ MapsTo f^[n] t t	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ MapsTo f^[0] t t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ MapsTo f^[n] t t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	SuperNear.mapsTo	[288, 1]	[290, 49]	simp only [Set.mapsTo_id, Function.iterate_zero]	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ MapsTo f^[0] t t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t	Please generate a tactic in lean4 to solve the state. STATE: case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ MapsTo f^[0] t t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	SuperNear.mapsTo	[288, 1]	[290, 49]	rw [Function.iterate_succ']	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo (f ∘ f^[n]) t t	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	SuperNear.mapsTo	[288, 1]	[290, 49]	exact s.ft.comp h	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo (f ∘ f^[n]) t t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo (f ∘ f^[n]) t t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	intro n zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	induction' n with n nh	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f^[0] z) ≤ (5 / 8) ^ 0 * Complex.abs z case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs (f^[n + 1] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	simp only [Function.iterate_zero, id, pow_zero, one_mul, Nat.cast_one, le_refl]	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f^[0] z) ≤ (5 / 8) ^ 0 * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f^[0] z) ≤ (5 / 8) ^ 0 * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	rw [Function.iterate_succ']	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs (f^[n + 1] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs (f^[n + 1] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	trans (5/8 : ℝ) * abs (f^[n] z)	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ 5 / 8 * Complex.abs (f^[n] z) f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	exact f_converges s (s.mapsTo n zt)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ 5 / 8 * Complex.abs (f^[n] z)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ 5 / 8 * Complex.abs (f^[n] z) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	calc (5/8 : ℝ) * abs (f^[n] z) _ ≤ (5/8 : ℝ) * ((5/8 : ℝ) ^ n * abs z) := by bound _ = 5/8 * (5/8 : ℝ) ^ n * abs z := by ring _ = (5/8 : ℝ) ^ (n + 1) * abs z := by rw [← pow_succ'] _ = (5/8 : ℝ) ^ n.succ * abs z := rfl	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ 5 / 8 * ((5 / 8) ^ n * Complex.abs z)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ 5 / 8 * ((5 / 8) ^ n * Complex.abs z) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	ring	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * ((5 / 8) ^ n * Complex.abs z) = 5 / 8 * (5 / 8) ^ n * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * ((5 / 8) ^ n * Complex.abs z) = 5 / 8 * (5 / 8) ^ n * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_converge	[293, 1]	[305, 46]	rw [← pow_succ']	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * (5 / 8) ^ n * Complex.abs z = (5 / 8) ^ (n + 1) * Complex.abs z	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * (5 / 8) ^ n * Complex.abs z = (5 / 8) ^ (n + 1) * Complex.abs z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	intro n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), AnalyticOn ℂ f^[n] t	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ AnalyticOn ℂ f^[n] t	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), AnalyticOn ℂ f^[n] t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	induction' n with n h	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ AnalyticOn ℂ f^[n] t	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ f^[0] t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ f^[n + 1] t	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ AnalyticOn ℂ f^[n] t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	simp only [Function.iterate_zero]	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ f^[0] t	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ id t	Please generate a tactic in lean4 to solve the state. STATE: case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ f^[0] t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	exact analyticOn_id _	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ id t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ id t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	rw [Function.iterate_succ']	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ f^[n + 1] t	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ (f ∘ f^[n]) t	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ f^[n + 1] t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	intro z zt	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ (f ∘ f^[n]) t	case succ f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (f ∘ f^[n]) z	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ (f ∘ f^[n]) t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_analytic	[308, 1]	[310, 90]	exact (s.fa _ (s.mapsTo n zt)).comp (h z zt)	case succ f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (f ∘ f^[n]) z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (f ∘ f^[n]) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_analytic	[313, 1]	[317, 90]	intro n z zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), AnalyticOn ℂ (term f d n) t	f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (term f d n) z	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), AnalyticOn ℂ (term f d n) t TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_analytic	[313, 1]	[317, 90]	refine AnalyticAt.cpow ?_ analyticAt_const ?_	f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (term f d n) z	case refine_1 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (fun z => g f d (f^[n] z)) z case refine_2 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ g f d (f^[n] z) ∈ Complex.slitPlane	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (term f d n) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_analytic	[313, 1]	[317, 90]	exact (s.ga _ (s.mapsTo n zt)).comp (iterates_analytic s n z zt)	case refine_1 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (fun z => g f d (f^[n] z)) z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (fun z => g f d (f^[n] z)) z TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_analytic	[313, 1]	[317, 90]	exact mem_slitPlane_of_near_one (lt_of_le_of_lt (s.gs (s.mapsTo n zt)) (by norm_num))	case refine_2 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ g f d (f^[n] z) ∈ Complex.slitPlane	no goals	Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ g f d (f^[n] z) ∈ Complex.slitPlane TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_analytic	[313, 1]	[317, 90]	norm_num	f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ 1 / 4 < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ 1 / 4 < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	intro n zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	rw [term]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 1 / 2 * (1 / 2) ^ n	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	trans 4 * abs (g f d (f^[n] z) - 1) * abs (1 / (d ^ (n + 1) : ℕ) : ℂ)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	apply pow_small	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1)))	case zs f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 2 case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	exact le_trans (s.gs (s.mapsTo n zt)) (by norm_num)	case zs f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: case zs f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 / 4 ≤ 1 / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 / 4 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	simp only [one_div, map_inv₀, Complex.abs_pow, Complex.abs_natCast, Nat.cast_pow]	case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1	case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ (↑d ^ (n + 1))⁻¹ ≤ 1	Please generate a tactic in lean4 to solve the state. STATE: case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	apply inv_le_one	case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ (↑d ^ (n + 1))⁻¹ ≤ 1	case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ ↑d ^ (n + 1)	Please generate a tactic in lean4 to solve the state. STATE: case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ (↑d ^ (n + 1))⁻¹ ≤ 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	have hd : 1 ≤ (d : ℝ) := le_trans (by norm_num) s.dr2	case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ ↑d ^ (n + 1)	case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t hd : 1 ≤ ↑d ⊢ 1 ≤ ↑d ^ (n + 1)	Please generate a tactic in lean4 to solve the state. STATE: case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ ↑d ^ (n + 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	exact one_le_pow_of_one_le hd _	case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t hd : 1 ≤ ↑d ⊢ 1 ≤ ↑d ^ (n + 1)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t hd : 1 ≤ ↑d ⊢ 1 ≤ ↑d ^ (n + 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	have gs : abs (g f d (f^[n] z) - 1) ≤ 1 / 4 := s.gs (s.mapsTo n zt)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	have ps : abs (1 / (d:ℂ) ^ (n + 1) : ℂ) ≤ 1/2 * (1/2 : ℝ) ^ n := by have nn : (1/2:ℝ) * (1/2 : ℝ) ^ n = (1/2 : ℝ) ^ (n + 1) := (pow_succ' _ _).symm rw [nn] simp only [one_div, map_inv₀, map_pow, Complex.abs_natCast, inv_pow, ge_iff_le] bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	calc (4:ℝ) * abs (g f d (f^[n] z) - 1) * abs ((1:ℂ) / (d ^ (n + 1) : ℕ) : ℂ) _ = (4:ℝ) * abs (g f d (f^[n] z) - 1) * abs ((1:ℂ) / (d:ℂ) ^ (n + 1) : ℂ) := by rw [Nat.cast_pow] _ ≤ 4 * (1 / 4) * (1 / 2 * (1 / 2 : ℝ) ^ n) := by bound _ = 1 / 2 * (1 / 2 : ℝ) ^ n := by ring	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	have nn : (1/2:ℝ) * (1/2 : ℝ) ^ n = (1/2 : ℝ) ^ (n + 1) := (pow_succ' _ _).symm	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	rw [nn]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ (1 / 2) ^ (n + 1)	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	simp only [one_div, map_inv₀, map_pow, Complex.abs_natCast, inv_pow, ge_iff_le]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ (1 / 2) ^ (n + 1)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ (1 / 2) ^ (n + 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹ TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	rw [Nat.cast_pow]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) = 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑d ^ (n + 1))	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) = 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑d ^ (n + 1)) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	bound	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑d ^ (n + 1)) ≤ 4 * (1 / 4) * (1 / 2 * (1 / 2) ^ n)	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑d ^ (n + 1)) ≤ 4 * (1 / 4) * (1 / 2 * (1 / 2) ^ n) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_converges	[322, 1]	[341, 45]	ring	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * (1 / 4) * (1 / 2 * (1 / 2) ^ n) = 1 / 2 * (1 / 2) ^ n	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * (1 / 4) * (1 / 2 * (1 / 2) ^ n) = 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	intro n zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → term f d n z ≠ 0	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ term f d n z ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → term f d n z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	have h := term_converges s n zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ term f d n z ≠ 0	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ term f d n z ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ term f d n z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	have o : 1 / 2 * (1 / 2 : ℝ) ^ n < 1 := by have p : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one n (by norm_num) (by linarith) calc 1 / 2 * (1 / 2 : ℝ) ^ n ≤ 1 / 2 * 1 := by linarith _ < 1 := by norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ term f d n z ≠ 0	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n o : 1 / 2 * (1 / 2) ^ n < 1 ⊢ term f d n z ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ term f d n z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	exact near_one_avoids_zero (lt_of_le_of_lt h o)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n o : 1 / 2 * (1 / 2) ^ n < 1 ⊢ term f d n z ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n o : 1 / 2 * (1 / 2) ^ n < 1 ⊢ term f d n z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	have p : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one n (by norm_num) (by linarith)	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 1 / 2 * (1 / 2) ^ n < 1	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n < 1	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 1 / 2 * (1 / 2) ^ n < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	calc 1 / 2 * (1 / 2 : ℝ) ^ n ≤ 1 / 2 * 1 := by linarith _ < 1 := by norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 0 ≤ 1 / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 0 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	linarith	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 1 / 2 ≤ 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 1 / 2 ≤ 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	linarith	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n ≤ 1 / 2 * 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n ≤ 1 / 2 * 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_nonzero	[344, 1]	[352, 50]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * 1 < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * 1 < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_prod	[355, 1]	[361, 33]	have c12 : (1 / 2 : ℝ) ≤ 1 / 2 := by norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_prod	[355, 1]	[361, 33]	have a0 : 0 ≤ (1 / 2 : ℝ) := by norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_prod	[355, 1]	[361, 33]	exact fast_products_converge' s.o c12 a0 (by linarith) (term_analytic s) fun n z ↦ term_converges s n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_prod	[355, 1]	[361, 33]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ 1 / 2 ≤ 1 / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ 1 / 2 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_prod	[355, 1]	[361, 33]	norm_num	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ 0 ≤ 1 / 2	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ 0 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	term_prod	[355, 1]	[361, 33]	linarith	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ 1 / 2 < 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	simp_rw [bottcherNear]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f z) = bottcherNear f d z ^ d	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f z) = bottcherNear f d z ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	have pe := (term_prod_exists s) z zt	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	simp only [mul_pow, product_pow' pe]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	have pe : ProdExists fun n ↦ term f d n z ^ d := by rcases pe with ⟨g, hg⟩; exact ⟨_, product_pow d hg⟩	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe✝ : ProdExists fun n => term f d n z pe : ProdExists fun n => term f d n z ^ d ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	simp only [product_split pe, ← term_eqn s, ← mul_assoc, ← mul_pow, ← term_base s]	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe✝ : ProdExists fun n => term f d n z pe : ProdExists fun n => term f d n z ^ d ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d	no goals	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe✝ : ProdExists fun n => term f d n z pe : ProdExists fun n => term f d n z ^ d ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	rcases pe with ⟨g, hg⟩	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ ProdExists fun n => term f d n z ^ d	case intro f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t g : ℂ hg : HasProd (fun n => term f d n z) g ⊢ ProdExists fun n => term f d n z ^ d	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ ProdExists fun n => term f d n z ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn	[376, 1]	[383, 84]	exact ⟨_, product_pow d hg⟩	case intro f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t g : ℂ hg : HasProd (fun n => term f d n z) g ⊢ ProdExists fun n => term f d n z ^ d	no goals	Please generate a tactic in lean4 to solve the state. STATE: case intro f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t g : ℂ hg : HasProd (fun n => term f d n z) g ⊢ ProdExists fun n => term f d n z ^ d TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn_iter	[386, 1]	[390, 43]	induction' n with n h	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ ⊢ bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f^[0] z) = bottcherNear f d z ^ d ^ 0 case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1)	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ ⊢ bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn_iter	[386, 1]	[390, 43]	simp only [Function.iterate_zero, id, pow_zero, pow_one]	case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f^[0] z) = bottcherNear f d z ^ d ^ 0 case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1)	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1)	Please generate a tactic in lean4 to solve the state. STATE: case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f^[0] z) = bottcherNear f d z ^ d ^ 0 case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	bottcherNear_eqn_iter	[386, 1]	[390, 43]	simp only [Function.comp, Function.iterate_succ', pow_succ, pow_mul, bottcherNear_eqn s (s.mapsTo n zt), h]	case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1) TACTIC:
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/BottcherNear.lean	iterates_at_zero	[393, 1]	[395, 67]	intro n	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), f^[n] 0 = 0	f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ f^[n] 0 = 0	Please generate a tactic in lean4 to solve the state. STATE: f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), f^[n] 0 = 0 TACTIC:

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