Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files Community
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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
have cw : abs (c * w ^ d) ≤ (abs z)⁻¹ := by simp only [Complex.abs.map_mul, Complex.abs.map_pow] calc abs c * abs w ^ d _ ≤ abs z * (abs z)⁻¹ ^ d := by bound _ ≤ abs z * (abs z)⁻¹ ^ 2 := by bound _ = (abs z)⁻¹ := by rw [pow_two]; field_simp [z0]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
have cw2 : abs (c * w ^ d) ≤ 1 / 2 := le_trans cw (le_trans i8 (by norm_num))
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
simp only [gl_f, gl]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [Complex.inv_cpow, ← Complex.cpow_neg]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (1 + c * w ^ d).arg ≠ Real.pi
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
swap
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (1 + c * w ^ d).arg ≠ Real.pi
case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (1 + c * w ^ d).arg ≠ Real.pi c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (1 + c * w ^ d).arg ≠ Real.pi TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d ⊢ 0 < 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d ⊢ 0 < 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [one_div]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ (Complex.abs z)⁻¹ ≤ 1 / 8 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
apply inv_le_inv_of_le
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
case ha c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 0 < 8 case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
case ha c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 0 < 8 case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ Complex.abs z
case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case ha c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 0 < 8 case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
exact le_trans (by norm_num) (le_trans c16.le cz)
case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 ⊢ 8 ≤ 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ 1 / 8 ≤ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ 1 / 8 ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [map_inv₀] at wc
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ Complex.abs w ≤ (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
exact le_trans wc (mul_le_of_le_one_left (inv_nonneg.mpr (Complex.abs.nonneg _)) (by bound))
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ Complex.abs w ≤ (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ (5 / 8) ^ n ≤ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w ⊢ (5 / 8) ^ n ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
simp only [Complex.abs.map_mul, Complex.abs.map_pow]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
calc abs c * abs w ^ d _ ≤ abs z * (abs z)⁻¹ ^ d := by bound _ ≤ abs z * (abs z)⁻¹ ^ 2 := by bound _ = (abs z)⁻¹ := by rw [pow_two]; field_simp [z0]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs c * Complex.abs w ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ d
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs c * Complex.abs w ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [pow_two]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 = (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 = (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
field_simp [z0]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ ⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ ⊢ 1 / 8 ≤ 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ ⊢ 1 / 8 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
refine (lt_of_le_of_lt (le_abs_self _) (lt_of_le_of_lt ?_ (half_lt_self Real.pi_pos))).ne
case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (1 + c * w ^ d).arg ≠ Real.pi
case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2
Please generate a tactic in lean4 to solve the state. STATE: case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (1 + c * w ^ d).arg ≠ Real.pi TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [Complex.abs_arg_le_pi_div_two_iff, Complex.add_re, Complex.one_re]
case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2
case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 0 ≤ 1 + (c * w ^ d).re
Please generate a tactic in lean4 to solve the state. STATE: case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
calc 1 + (c * w ^ d).re _ ≥ 1 + -|(c * w ^ d).re| := by bound _ = 1 - |(c * w ^ d).re| := by ring _ ≥ 1 - abs (c * w ^ d) := by bound _ ≥ 1 - 1 / 2 := by linarith _ ≥ 0 := by norm_num
case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 0 ≤ 1 + (c * w ^ d).re
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hx c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 0 ≤ 1 + (c * w ^ d).re TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 + (c * w ^ d).re ≥ 1 + -|(c * w ^ d).re|
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 + (c * w ^ d).re ≥ 1 + -|(c * w ^ d).re| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
ring
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 + -|(c * w ^ d).re| = 1 - |(c * w ^ d).re|
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 + -|(c * w ^ d).re| = 1 - |(c * w ^ d).re| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 - |(c * w ^ d).re| ≥ 1 - Complex.abs (c * w ^ d)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 - |(c * w ^ d).re| ≥ 1 - Complex.abs (c * w ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
linarith
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 - Complex.abs (c * w ^ d) ≥ 1 - 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 - Complex.abs (c * w ^ d) ≥ 1 - 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 - 1 / 2 ≥ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ 1 - 1 / 2 ≥ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
have dn : abs (-(1 / ((d ^ (n + 1) : ℕ) : ℂ))) ≤ (1 / 2 : ℝ) ^ (n + 1) := by simp only [Nat.cast_pow, one_div, map_neg_eq_map, map_inv₀, map_pow, Complex.abs_natCast, inv_pow] bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
have d1 : abs (-(1 / ((d ^ (n + 1) : ℕ) : ℂ))) ≤ 1 := le_trans dn (by bound)
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
refine le_trans (pow_small ?_ d1) ?_
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case refine_1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs (1 + c * w ^ d - 1) ≤ 1 / 2 case refine_2 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (1 + c * w ^ d - 1) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
simp only [Nat.cast_pow, one_div, map_neg_eq_map, map_inv₀, map_pow, Complex.abs_natCast, inv_pow]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) ⊢ (1 / 2) ^ (n + 1) ≤ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) ⊢ (1 / 2) ^ (n + 1) ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [add_sub_cancel_left]
case refine_1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs (1 + c * w ^ d - 1) ≤ 1 / 2
case refine_1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs (c * w ^ d) ≤ 1 / 2
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs (1 + c * w ^ d - 1) ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
exact cw2
case refine_1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs (c * w ^ d) ≤ 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ Complex.abs (c * w ^ d) ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
rw [add_sub_cancel_left]
case refine_2 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (1 + c * w ^ d - 1) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case refine_2 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (1 + c * w ^ d - 1) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
calc 4 * abs (c * w ^ d) * abs (-(1 / ((d ^ (n + 1) : ℕ) : ℂ))) _ ≤ 4 * (abs z)⁻¹ * (1/2 : ℝ) ^ (n + 1) := by bound _ ≤ 2 * (1/2 : ℝ) ^ n * (abs z)⁻¹ := by simp only [pow_succ, ←mul_assoc, mul_comm _ (1/2:ℝ)]; norm_num simp only [mul_comm _ ((2:ℝ)^n)⁻¹, ←mul_assoc, le_refl]
case refine_2 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
bound
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
simp only [pow_succ, ←mul_assoc, mul_comm _ (1/2:ℝ)]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 1 / 2 * 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ n ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 1 / 2 * 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ n ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 2 * (Complex.abs z)⁻¹ * (2 ^ n)⁻¹ ≤ 2 * (2 ^ n)⁻¹ * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 1 / 2 * 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ n ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
term_approx
[89, 1]
[131, 66]
simp only [mul_comm _ ((2:ℝ)^n)⁻¹, ←mul_assoc, le_refl]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 2 * (Complex.abs z)⁻¹ * (2 ^ n)⁻¹ ≤ 2 * (2 ^ n)⁻¹ * (Complex.abs z)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z n : ℕ s : Super (f d) d ∞ := superF d z0 : Complex.abs z ≠ 0 i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 i1 : (Complex.abs z)⁻¹ ≤ 1 w : ℂ hw : (fl (f d) ∞ c)^[n] z⁻¹ = w wc : Complex.abs w ≤ (Complex.abs z)⁻¹ cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2 dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1) d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1 ⊢ 2 * (Complex.abs z)⁻¹ * (2 ^ n)⁻¹ ≤ 2 * (2 ^ n)⁻¹ * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
set s := superF d
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z ⊢ Complex.abs (⋯.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z ⊢ Complex.abs (⋯.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
have i8 : (abs z)⁻¹ ≤ 1 / 8 := by rw [one_div]; apply inv_le_inv_of_le; norm_num exact le_trans (by norm_num) (le_trans c16.le cz)
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
simp only [bottcher_eq_bottcherNear_z c16 cz, bottcherNear, Complex.abs.map_mul, ← mul_sub_one, pow_two, ← mul_assoc, map_inv₀, mul_comm (abs z)⁻¹]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) * (Complex.abs z)⁻¹ ≤ 16 * (Complex.abs z)⁻¹ * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
refine mul_le_mul_of_nonneg_right ?_ (inv_nonneg.mpr (Complex.abs.nonneg _))
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) * (Complex.abs z)⁻¹ ≤ 16 * (Complex.abs z)⁻¹ * (Complex.abs z)⁻¹
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) * (Complex.abs z)⁻¹ ≤ 16 * (Complex.abs z)⁻¹ * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
rcases term_prod_exists (superNearF d c) _ (inv_mem_t c16 cz) with ⟨p, h⟩
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
rw [h.tprod_eq]
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p ⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p ⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
simp only [HasProd] at h
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p ⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p) ⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p ⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
apply le_of_tendsto' (Filter.Tendsto.comp Complex.continuous_abs.continuousAt (h.sub_const 1))
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p) ⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p) ⊢ ∀ (c_1 : Finset ℕ), (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p) ⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
clear h
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p) ⊢ ∀ (c_1 : Finset ℕ), (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ ⊢ ∀ (c_1 : Finset ℕ), (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p) ⊢ ∀ (c_1 : Finset ℕ), (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
intro A
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ ⊢ ∀ (c_1 : Finset ℕ), (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) A ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ ⊢ ∀ (c_1 : Finset ℕ), (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
simp only [Function.comp]
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) A ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 16 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) A ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
rw [(by norm_num : (16 : ℝ) = 4 * 4), mul_assoc]
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 16 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 4 * (4 * (Complex.abs z)⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 16 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
refine dist_prod_one_le_abs_sum ?_ (by linarith)
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 4 * (4 * (Complex.abs z)⁻¹)
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun n => Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1)) ≤ 4 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 4 * (4 * (Complex.abs z)⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
refine le_trans (Finset.sum_le_sum fun n _ ↦ term_approx d (by linarith) cz n) ?_
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun n => Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1)) ≤ 4 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun i => 2 * (1 / 2) ^ i * (Complex.abs z)⁻¹) ≤ 4 * (Complex.abs z)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun n => Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1)) ≤ 4 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
simp only [mul_comm _ _⁻¹, ← mul_assoc, ← Finset.mul_sum]
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun i => 2 * (1 / 2) ^ i * (Complex.abs z)⁻¹) ≤ 4 * (Complex.abs z)⁻¹
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun i => (1 / 2) ^ i) ≤ (Complex.abs z)⁻¹ * 4
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun i => 2 * (1 / 2) ^ i * (Complex.abs z)⁻¹) ≤ 4 * (Complex.abs z)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
calc (abs z)⁻¹ * 2 * A.sum (fun n ↦ (1/2:ℝ)^n) _ ≤ (abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ := by gcongr; apply partial_geometric_bound; repeat norm_num _ = (abs z)⁻¹ * 4 := by ring
case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun i => (1 / 2) ^ i) ≤ (Complex.abs z)⁻¹ * 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun i => (1 / 2) ^ i) ≤ (Complex.abs z)⁻¹ * 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
rw [one_div]
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ (Complex.abs z)⁻¹ ≤ 1 / 8 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
apply inv_le_inv_of_le
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
case ha c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 0 < 8 case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
norm_num
case ha c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 0 < 8 case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ Complex.abs z
case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case ha c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 0 < 8 case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
exact le_trans (by norm_num) (le_trans c16.le cz)
case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d ⊢ 8 ≤ 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
norm_num
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 16 = 4 * 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 16 = 4 * 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
linarith
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 4 * (Complex.abs z)⁻¹ ≤ 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 4 * (Complex.abs z)⁻¹ ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
linarith
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ n : ℕ x✝ : n ∈ A ⊢ 16 < Complex.abs c
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ n : ℕ x✝ : n ∈ A ⊢ 16 < Complex.abs c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
gcongr
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun n => (1 / 2) ^ n) ≤ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹
case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun n => (1 / 2) ^ n) ≤ (1 - 1 / 2)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun n => (1 / 2) ^ n) ≤ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
apply partial_geometric_bound
case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun n => (1 / 2) ^ n) ≤ (1 - 1 / 2)⁻¹
case h.a0 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 0 ≤ 1 / 2 case h.a1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 1 / 2 < 1
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (A.sum fun n => (1 / 2) ^ n) ≤ (1 - 1 / 2)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
repeat norm_num
case h.a0 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 0 ≤ 1 / 2 case h.a1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 1 / 2 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.a0 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 0 ≤ 1 / 2 case h.a1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
norm_num
case h.a1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 1 / 2 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.a1 c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_approx_z
[134, 1]
[153, 33]
ring
c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ = (Complex.abs z)⁻¹ * 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ c16 : 16 < Complex.abs c cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d ∞ := superF d i8 : (Complex.abs z)⁻¹ ≤ 1 / 8 p : ℂ A : Finset ℕ ⊢ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ = (Complex.abs z)⁻¹ * 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
rw [HasDerivAt, HasDerivAtFilter, bottcher, hasFDerivAtFilter_iff_isLittleO, coe_zero, inv_zero', fill_inf]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ HasDerivAt (fun z => bottcher d (↑z)⁻¹) 1 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - 0 - (ContinuousLinearMap.smulRight 1 1) (x' - 0)) =o[𝓝 0] fun x' => x' - 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ HasDerivAt (fun z => bottcher d (↑z)⁻¹) 1 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [sub_zero, ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, smul_eq_mul, mul_one]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - 0 - (ContinuousLinearMap.smulRight 1 1) (x' - 0)) =o[𝓝 0] fun x' => x' - 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - x') =o[𝓝 0] fun x' => x'
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - 0 - (ContinuousLinearMap.smulRight 1 1) (x' - 0)) =o[𝓝 0] fun x' => x' - 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
rw [Asymptotics.isLittleO_iff]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - x') =o[𝓝 0] fun x' => x'
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ c * ‖x‖
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - x') =o[𝓝 0] fun x' => x' TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
intro k k0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ c * ‖x‖
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ k * ‖x‖
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ c * ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
rw [Metric.eventually_nhds_iff]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ k * ‖x‖
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∃ ε > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ k * ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
refine ⟨min 16⁻¹ (k / 16), by bound, ?_⟩
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∃ ε > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∀ ⦃y : ℂ⦄, dist y 0 < min 16⁻¹ (k / 16) → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∃ ε > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
intro z le
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∀ ⦃y : ℂ⦄, dist y 0 < min 16⁻¹ (k / 16) → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : dist z 0 < min 16⁻¹ (k / 16) ⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ ∀ ⦃y : ℂ⦄, dist y 0 < min 16⁻¹ (k / 16) → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [Complex.dist_eq, sub_zero, lt_min_iff] at le
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : dist z 0 < min 16⁻¹ (k / 16) ⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 ⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : dist z 0 < min 16⁻¹ (k / 16) ⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [Complex.norm_eq_abs]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 ⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 ⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
by_cases z0 : z = 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
case pos c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : z = 0 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [inv_coe z0, fill_coe]
case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
have b : abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ (16:ℝ) * (abs z⁻¹)⁻¹ ^ 2 := bottcher_approx d ?_
case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z case neg.refine_1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ 16 < Complex.abs z⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
bound
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ min 16⁻¹ (k / 16) > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k ⊢ min 16⁻¹ (k / 16) > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [z0, coe_zero, inv_zero', fill_inf, sub_zero, Complex.abs.map_zero, MulZeroClass.mul_zero, le_refl]
case pos c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : z = 0 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : z = 0 ⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [inv_inv] at b
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
apply le_trans b
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ≤ k * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
simp only [map_inv₀, inv_inv, pow_two, ← mul_assoc]
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ≤ k * Complex.abs z
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z * Complex.abs z ≤ k * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
refine mul_le_mul_of_nonneg_right ?_ (Complex.abs.nonneg _)
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z * Complex.abs z ≤ k * Complex.abs z
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z ≤ k
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z * Complex.abs z ≤ k * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
calc 16 * abs z _ ≤ 16 * (k / 16) := by linarith [le.2] _ = k := by ring
case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z ≤ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_2 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z ≤ k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
linarith [le.2]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z ≤ 16 * (k / 16)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * Complex.abs z ≤ 16 * (k / 16) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
ring
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * (k / 16) = k
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ⊢ 16 * (k / 16) = k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
rw [map_inv₀, lt_inv (by norm_num) (Complex.abs.pos_iff.mpr z0)]
case neg.refine_1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ 16 < Complex.abs z⁻¹
case neg.refine_1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs z < 16⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ 16 < Complex.abs z⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
exact le.1
case neg.refine_1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs z < 16⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg.refine_1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ Complex.abs z < 16⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_hasDerivAt_one
[161, 1]
[181, 81]
norm_num
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ 0 < 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) k : ℝ k0 : 0 < k z : ℂ le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16 z0 : ¬z = 0 ⊢ 0 < 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_mfderiv_inf_ne_zero
[184, 1]
[195, 40]
simp only [mfderiv, (bottcherHolomorphic d _ multibrotExt_inf).mdifferentiableAt, if_pos, writtenInExtChartAt, bottcher_inf, extChartAt_inf, extChartAt_eq_refl, Function.comp, PartialEquiv.refl_coe, id, PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, RiemannSphere.inv_inf, coePartialEquiv_symm_apply, toComplex_zero, PartialEquiv.coe_trans_symm, PartialEquiv.symm_symm, coePartialEquiv_apply, Equiv.toPartialEquiv_symm_apply, invEquiv_symm, ModelWithCorners.Boundaryless.range_eq_univ, fderivWithin_univ]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ mfderiv I I (bottcher d) ∞ ≠ 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fderiv ℂ (fun x => bottcher d (↑x)⁻¹) 0 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ mfderiv I I (bottcher d) ∞ ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_mfderiv_inf_ne_zero
[184, 1]
[195, 40]
rw [bottcher_hasDerivAt_one.hasFDerivAt.fderiv]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fderiv ℂ (fun x => bottcher d (↑x)⁻¹) 0 ≠ 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fderiv ℂ (fun x => bottcher d (↑x)⁻¹) 0 ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Bottcher.lean
bottcher_mfderiv_inf_ne_zero
[184, 1]
[195, 40]
rw [Ne, ContinuousLinearMap.ext_iff, not_forall]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∃ x, ¬(ContinuousLinearMap.smulRight 1 1) x = 0 x
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0 TACTIC: